Curvature of Space and Time, with an Introduction to Geometric Analysis 1470456281, 9781470456283

Citation preview

STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES VOLUME 93

Curvature of Space and Time, with an Introduction to Geometric Analysis Iva Stavrov

American Mathematical Society Institute for Advanced Study

Curvature of Space and Time, with an Introduction to Geometric Analysis

S T U D E N T M AT H E M AT I C A L L I B R A R Y IAS /PAR K CITY MATHEMATICAL SUBSERIES Volume 93

Curvature of Space and Time, with an Introduction to Geometric Analysis Iva Stavrov

American Mathematical Society Institute for Advanced Study

Editorial Board of the Student Mathematical Library John McCleary Rosa C. Orellana

Kavita Ramanan John Stillwell (Chair)

Series Editor for the Park City Mathematics Institute Rafe Mazzeo 2020 Mathematics Subject Classification. Primary 83-01, 53-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-93 Library of Congress Cataloging-in-Publication Data Names: Stavrov, Iva, 1976- author. Title: Curvature of space and time, with an introduction to geometric analysis / Iva Stavrov. Description: Providence, Rhode Island : American Mathematical Society ; Princeton, New Jersey : Institute for Advanced Study, [2020] | Series: Student mathematical library, 1520-9121 ; volume 93 | Includes bibliographical references and index. Identifiers: LCCN 2020035487 | ISBN 9781470456283 (paperback) | ISBN 9781470463137 (ebook) Subjects: LCSH: General relativity (Physics)–Mathematics. | Curvature. | Space and time. | Geometry, Riemannian. — Geometry, Differential. | Geometric analysis. | AMS: Relativity and gravitational theory – Instructional exposition (textbooks, tutorial papers, etc.). | Differential geometry For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx – Instructional exposition (textbooks, tutorial papers, etc.). Classification: LCC QC173.6 .S735 2020 — DDC 530.1101/516373–dc23 LC record available at https://lccn.loc.gov/2020035487 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission @ams.org.

c 2020 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

25 24 23 22 21 20

Contents

IAS/Park City Mathematics Institute

vii

Preface

ix

Chapter 1. Introduction to Riemannian geometry §1.1. Riemann’s Habilitation lecture in examples

1 1

§1.2. The framework of Riemannian geometry

16

§1.3. Geodesics

29

Chapter 2. Differential calculus with tensors

45

§2.1. Introduction to differential calculus

45

§2.2. Tensors

60

§2.3. Differentiation of tensors

76

Chapter 3. Curvature §3.1. Intuiting curvature via Jacobi equation §3.2. Ricci and scalar curvature Chapter 4. General relativity

89 89 104 123

§4.1. The framework of special relativity

123

§4.2. Gravity and general relativity

144

§4.3. Geometry of Schwarzschild space-time

162 v

vi

Contents §4.4. Kruskal-Szekeres extension of Schwarzschild space-time 178

Chapter 5. Introduction to geometric analysis

199

§5.1. The (relativistic) Poisson problem

199

§5.2. On the concept of mass

218

Bibliography

239

Index

241

IAS/Park City Mathematics Institute

The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National Science Foundation. In mid-1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The annual PCMI summer programs take place in Park City, Utah. Each year’s PCMI summer program is an intensive three-week session where several different activities take place in parallel. These include individual programs for researchers, graduate students, undergraduate faculty, undergraduate students, and K-12 teachers, as well as a workshop devoted to issues surrounding equity in the mathematics classroom. Over 300 people are in attendance each year. A main goal of PCMI is to make all participants aware of the broad spectrum of mathematical activities and to promote interactions between these groups, often leading to new collaboration and new mentoring arrangements. Each summer a different research topic is chosen as the focus of the Research Program and Graduate Summer School. The Undergraduate Program typically focuses on closely related material. Lecture notes from the Graduate Summer School are published each year

vii

viii

IAS/Park City Mathematics Institute

in the IAS/Park City Mathematics Series. Course material for the Undergraduate Program at PCMI is published intermittently under the IAS/Park City Mathematical Subseries in the Student Mathematical Library. We are very pleased to make available to a wide audience the expanded versions of these undergraduate lectures, which we believe should be of great value to students everywhere. Rafe Mazzeo, Series Editor July 2020

Preface

In 2013 Park City Math Institute (PCMI) ran a summer program on geometric analysis, and I was invited to give a course at an undergraduate level. I agreed to teach a course introducing students to mathematical general relativity. While the mathematical foundation of general relativity is in differential and (semi-)Riemannian geometry, an active researcher in the field regularly uses techniques from geometric analysis. It is for this reason that the class started with a treatment of Riemannian metrics, tensors and curvature, and ended with a brief introduction into geometric analysis. The overall strategy of the course was to explain the concept of curvature via the Jacobi equation which, through discussion of tidal forces, further helped motivate the Einstein field equations. The book you have in your hands is a modified and expanded version of my PCMI lecture notes. Since 2013 these materials were used extensively by students at Lewis & Clark College. In the Fall of 2015, on the occasion of general relativity’s 100th birthday, I taught a semester-long class at Lewis & Clark College based on the PCMI lecture notes. The class included extra material on general relativity, and omitted the geometric analysis portion of the lecture notes. The prerequisites for the class were sophomorelevel Multivariable and Vector Calculus and sophomore-level Linear Algebra. It was recommended, but not required, that students had

ix

x

Preface

also taken an upper-level (preferably proof-based) class. The class attracted both physics and mathematics majors. Although a small amount of physics was discussed there were no physics prerequisites. The course was primarily mathematical and did not go into any up-todate astrophysical information. It ended in addressing the geometry of the Schwarzschild body (black hole). The expansion of the general relativity chapter in this book is based on the material developed for the 2015 course. The geometric analysis chapter of this book is also changed from the original set of PCMI lecture notes. The chapter is now based on analysis techniques I taught those of the Lewis & Clark College students who pursued independent research as undergraduates. Two of their theorems are also included in the chapter. Courses in differential and/or Riemannian geometry are usually associated with a high upfront cost of prerequisites. A typical undergraduate route departs very little from Gauss’s original “Disquisitiones generales circa superficies curvas” (“General Investigations of Curved Surfaces”) as it rests on the theory of curves and surfaces in R3 . In contrast, these lecture notes begin with an analysis of one of ¨ the central paragraphs of Riemann’s Habilitation lecture “Uber die Hypothesen, welche der Geometrie zu Grunde liegen” (“On Hypotheses Which Lie at the Bases of Geometry”). As a matter of principle, geometry and curvature in these lecture notes are treated exclusively from the intrinsic standpoint. Most of us have been taught Riemannian geometry only after we have learned the basics of differential topology. This is a demanding approach because concepts such as tangent and tensor bundles are quite mathematically sophisticated. I have made a deliberate choice to bluntly avoid differential topology altogether. As a result, one will find neither a definition of a manifold nor a definition of tangent vector in these lecture notes. Instead, the narrative relies exclusively on a reader’s familiarity with curvilinear coordinatization from Multivariable Calculus. Connections and Christoffel symbols can be explained to anyone fluent in using polar or spherical coordinates in Euclidean space. Results such as asymptotic formulae for volumes of small balls in arbitrary dimension are local in nature and convey much about

Preface

xi

concepts of curvature without relying on the exact definition of a tangent vector. I am certain that my choice to avoid differential topology will make some readers see my notes as “lacking mathematical rigor”. I now take a moment to write to such readers. Over the many years of teaching at a small liberal arts school I came to realize just how non-linear the process of learning mathematics is. The practice of organizing information in a deductive manner, which as mathematicians we are committed to, is not always conducive to learning mathematics and developing intuition. Throughout history people discovered new mathematics without having a complete understanding of subtleties involved in the mathematical foundations underneath. An approach to teaching and learning mathematics which relies on several passes through a subject, each at a higher level of mathematical rigor, is in a sense historically proven to be pedagogically optimal. My personal experience is that of working with students who very rarely start out as “being a part of the math choir”. I dedicate my lecture notes to all those oddball students because it was they who retaught me mathematics.

Chapter 1

Introduction to Riemannian geometry

1.1. Riemann’s Habilitation lecture in examples 1.1.1. Introduction. The German academic system to this day recognizes a post-PhD qualification called Habilitation. Bernhard Riemann did his Habilitation work at the University of G¨ottingen in the early 1850’s where he was supervised by Carl Friedrich Gauss. The practice at the time was to have the Habilitation candidate prepare several public lectures (not necessarily related to the Habilitation dissertation), and it was Gauss who chose the one to be given. ¨ Gauss’s choice was “Uber die Hypothesen, welche der Geometrie zu Grunde liegen” (“On Hypotheses Which Lie at the Bases of Geometry”), which Riemann gave on June 10th, 1854. The following quote1 from Riemann’s lecture summarizes the key ideas behind what is now called Riemannian geometry. “Position-fixing being reduced to quantity-fixings, and the position of a point in the n-dimensional manifoldness being consequently expressed by

1 The translation used here is due to W. K. Clifford; see Nature, Vol. VIII. Nos. 183, 184, pp. 14–17, 36, 37.

1

2

1. Introduction means of n variables x1 , x2 , x3 , ..., xn , the determination of a line comes to the giving of these quantities as functions of one variable. The problem consists then in establishing a mathematical expression for the length of a line, and to this end we must consider the quantities x as expressible in terms of certain units. I shall treat this problem only under certain restrictions, and I shall confine myself in the first place to lines in which the ratios of the increments dx of the respective variables vary continuously. We may then conceive these lines broken up into elements within which the ratios of quantities dx may be regarded as constant; and the problem is then reduced to establishing for each point a general expression for the linear element ds starting from that point, an expression which will thus contain the quantities x and quantities dx.”

We begin these lectures with a detailed analysis of the quoted paragraph. Our narrative is based on two familiar examples: Euclidean space Rn and spheres in Euclidean spaces. 1.1.2. Euclidean Rn . The position of a point in Rn can be shown in terms of n Cartesian, real-valued coordinates2 : (x1 , x2 , ..., xn ). A path/curve γ in Rn is best described parametrically   γ(t) = x1 (t), x2 (t), ....., xn (t) , a ≤ t ≤ b, although it should be noted that each geometric path γ permits different parametrizations in t (which depend on the “traversing speed”). For each 1 ≤ j ≤ n the paths  t, if i = j, i x (t) = const, if i = j, define coordinate grid lines. The tangent vector field to the above indicates the direction in which the xj -coordinates increase; such vector 2 We intentionally depart from the notation used in Riemann’s original paper and adopt modern notational conventions according to which the indices on coordinate variables are superscripts.

1.1. Riemann’s Habilitation lecture

3

fields will be denoted by ∂xj , or ∂j for short3 . At each point of Rn the set of vectors {∂j 1 ≤ j ≤ n} provides a basis for the space of vectors based at that point. In particular,  i dx γ˙ = dt ∂i i

  represents the tangent vector field to γ(t) = x1 (t), x2 (t), ....., xn (t) . The length of this curve can be conceptualized in two equivalent ways. As suggested by Riemann, one can compute the length of the curve by breaking it up into infinitesimal pieces (line elements) which are effectively straight (so that the ratios of infinitesimal coordinate increments dxi “may be regarded as constant”), by computing the length ds of the resulting line element using the Pythagorean Theorem     i 2 i 2 dx 2 ds = (dx ) = ( dt ) dt2 i

i

and by adding ds’s up using definite integrals. Of course, this is the i same as integrating the magnitude of the velocity vector i ( dx dt ) ∂i with respect to t. y

θ

3

(9/6)π



2

(5/6)π 1

x = r cos θ y = r sin θ

(5/6)π (1/6)π 0

1

2

3

(1/6)π

x

r (9/6)π

Figure 1.1. Polar coordinates in R2

It should be noted that there are many ways of coordinatizing Rn , or its subsets4 . For instance, in R2 one may want to use polar 3 In multivariable calculus in R3 these vector fields are commonly denoted by i, j and k. 4 In order to avoid situations when a point in R2 is described by multiple pairs of coordinates one may wish to restrict the domain of polar coordinates to open rectangles

4

1. Introduction

coordinates (r, θ). Parametrization of R2 using polar coordinates is represented visually in Figures 1.1 and 1.2. Note that ∂r and ∂θ are tangent vector fields to polar coordinate grid lines. The vector field ∂r points in directions in which r increases and θ remains constant while ∂θ points in directions in which θ increases and r remains constant. y

∂r

∂r ∂r

∂θ x ∂θ

∂θ

Figure 1.2. Vector fields ∂r and ∂θ

It is important to have a means of relating expressions coming from different coordinate systems. In our situation one can use the formulae x = r cos θ, y = r sin θ to relate Cartesian expressions to those given in terms of polar coordinates. For example, the expression for the Euclidean line element ds2 = dx2 + dy 2 becomes ds2 = dr 2 + r 2 dθ 2 . This brings us to the following quote from Riemann’s lecture: “.... the problem is then reduced to establishing for each point a general expression for the linear element ds starting from that point, an expression which will thus contain the quantities x and quantities dx.” As the illustration of the coordinate grids from Figure 1.1 shows, the curvilinear mapping (r, θ) → (r cos θ, r sin θ) distorts distances: segments r = 1 and r = 3 appear to be equally long on the coordinate such as (0, ∞) × (−π, π). To cover the entire R2 one may want to use several sets of polar coordinates.

1.1. Riemann’s Habilitation lecture

5

grid on the left but the corresponding lengths in Euclidean R2 are very different. The reason for this can be tracked back to the fact that the expression for ds2 in polar coordinates varies from point to point and, more specifically, dθ 2 contributes differently to ds2 when r = 1 and when r = 3. Generally speaking, when using curvilinear (e.g. polar, cylindrical, spherical) coordinates (y 1 , ..., y n ) for the Euclidean Rn we should expect the line elements to take the form of ds2 =



gij dy i dy j ,

ij

for some functions gij = gij (y 1 , y 2 , ..., y n ). (In fact, the functions gij are in this case determined by the Euclidean inner-products of coordinate vector fields ∂yi and ∂yj ; see exercise (10) following this lecture.) The differences in the sizes of different coordinate cells are accounted for in the expression for the line element.

1.1.3. Spheres in Euclidean spaces. The map-making process results in coordinatization of the surface of the Earth. In general, coordinates on a surface give rise to a parametrization, i.e., a mapping which transforms a map/rectangle in an atlas to the surface that is being coordinatized. An extremely common example of a parametrization comes from spherical coordinates. More specifically, the curvilinear mapping (θ, φ) → ( sin φ cos θ,  sin φ sin θ,  cos φ) transforms a rectangle in our atlas (the portion of the θφ-plane5 where 0 < θ < 2π and 0 < φ < π) to the surface of the sphere of radius  in R3 . The nature of spherical coordinates is presented in Figures 1.3 and 1.4. Figure 1.4 also illustrates the coordinate vector fields ∂θ , ∂φ .

5 Although in some sense the “North Pole” corresponds to φ = 0, it is not described by a unique pair of coordinates (θ, φ); similar comments apply to the “South Pole” φ = π and the “Greenwich meridian” θ ∈ {0, 2π}. For this reason it is advantageous to restrict our attention to the open rectangle (0, 2π) × (0, π).

6

1. Introduction z φ = π/6 φ = π/3 φ = π/2 x

θ = π/3

θ=0

y

Figure 1.3. Parametrization of a sphere using spherical coordinates

z

x

z

y x

y

Figure 1.4. Vector field ∂θ on the left and ∂φ on the right

It is important to notice that since we need only two parameters (de-facto latitude and longitude) to describe the position of an object on its surface, the sphere is two-dimensional, despite the fact that we tend to visualize the sphere as being a part of a larger threedimensional space. This is the reason why we use S 2 to denote the standard unit sphere in R3 . In the case when a sphere is representing our planet Earth we are also interested in more detailed representations of certain regions of the Earth, and as a result we have a full atlas worth of maps (pun!!) from a rectangle (the sheet of paper on which the map is drawn) to the surface of the Earth. This atlas coordinatizes the Earth: every point is represented in some rectangle in a coordinate plane.

1.1. Riemann’s Habilitation lecture

7

Figure 1.5. Mercator world map

Once again one should observe the extent to which the distances on the sphere differ from those in a map. One can easily relate to the issue by looking at, say, the Mercator world map in which Antarctica and Greenland look huge, especially in relation to regions in the tropics. Figure 1.6 shows what a map based on spherical coordinates would look like. Note how stretched the head of the stick-human appears in the map. φ π π 2

−π Figure 1.6. “World map” in spherical coordinates

π θ

8

1. Introduction

Let us now examine explicit expressions which capture this distortion phenomenon. Assuming the sphere is embedded in Euclidean geometry coordinatized by (x, y, z), the length of any path – so including those on the surface of the sphere – are to be computed using ds2 = dx2 + dy 2 + dz 2 . Inserting x =  cos θ sin φ, y =  sin θ sin φ, z =  cos φ into the above we get

 ds2 = 2 dφ2 + sin2 (φ) dθ 2 . This is the formula for the line element on spheres of radius  in Euclidean R3 . Note that the expression for the line element is not the same everywhere on the sphere – one of the coefficients depends on φ. In other words, the expression for the line element depends on the North/South location. This type of dependance on the North/South location is exactly what we want in order to be able to account for the distorted distances on, say, the Mercator world map. The function sin2 (φ) scrunches the contributions6 of dθ 2 as one approaches the North and the South Poles in such a way that the φ = const circles close up to form a dome. There are other ways of coordinatizing spheres. One such coordinatization is based on something called the stereographic projection. This transformation bijectively maps the surface of a sphere, excluding a point, onto R2 . For instance, we may consider the excluded point to be “the North Pole” (0, 0, 1) of the unit sphere and the target plane to be the equatorial plane (x1 , x2 , 0); we require the North Pole, the point P ∈ S 2 and its projection (x1 , x2 , 0) to be collinear. (See Figure 1.7 for an illustration.) It can be computed (e.g Exercise 5 following this lecture) that for such choices the stereographic projection of a point P = (x, y, z) on S 2 has coordinates x1 =

x y , x2 = . 1−z 1−z

6 There is a lot to be learned by comparing the situation here to that of polar coordinates in Euclidean R2 where the contributions of dθ were increasing proportionally to r.

1.1. Riemann’s Habilitation lecture

9

P ∈ S2

(x1 , x2 , 0) P ∈ S2

(x1 , x2 , 0)

Figure 1.7. Stereographic projection

The inverse of the stereographic projection is a parametrization of the unit sphere (without a point). To understand the visual aspects of this parametrization observe the extent to which the “coordinate grid lines” on the surface of the sphere become denser the closer we get to the North Pole.

x1

x2

Figure 1.8. Stereographic parametrization

To cover the entire sphere we need two coordinate patches based on stereographic projection: one projecting from the North Pole and the other from the South Pole. Most points on S 2 can thus be given

10

1. Introduction

two sets of coordinates: the coordinates (x1 , x2 ) relative to the North Pole and the coordinates (y 1 , y 2 ) relative to the South Pole. It can be shown (see Exercise 5 below) that the two sets of coordinates are related as follows:

x1 x2 1 2 y 1 2 , , (y , y ) = Φx (x , x ) = (x1 )2 + (x2 )2 (x1 )2 + (x2 )2

y1 y2 , (x1 , x2 ) = Φxy (y 1 , y 2 ) = . (y 1 )2 + (y 2 )2 (y 1 )2 + (y 2 )2 The transition mappings Φyx and Φxy express how y-coordinates and x-coordinates depend on each other. In our situation Φyx (and its inverse Φxy ) is a differentiable bijection of R2 \{(0, 0)}. This particular mapping is called the inversion with respect to the unit circle. Its name is motivated by the fact that the mapping exchanges the interior of the unit circle for its exterior while keeping the unit circle fixed. More generally, equations of the form (x1 )2 + ... + (xn )2 = 2 represent spheres of radius  in Euclidean Rn . One can coordinatize such spheres using n − 1 coordinates; this can be accomplished by means of stereographic projection. As a result, spheres in Rn are (n− 1)-dimensional. The unit sphere in Euclidean Rn+1 is denoted by S n . The line element it inherits from Rn+1 , in stereographic coordinates, is computed to be  1 2  4 (dx ) + ... + (dxn )2 . ds2 = 1 2 n 2 2 (1 + (x ) + ... + (x ) ) 4 Once more, note how the multiplicative factor (1+(x1 )2 +...+(x n )2 )2 n scrunches far away regions of R . This, for example, explains why the coordinate grid lines in Figure 1.8 are effectively denser near the North Pole.

1.1.4. Real projective plane. A much more abstract example is that of the real projective plane, denoted RP 2 . As a set RP 2 consists of lines passing through the origin in R3 . The study of such an object is motivated the fact that the set of lines through a painter’s eye to some extent corresponds to the set of points on this painter’s canvas. Each element of RP 2 is a line u spanned by some non-zero vector u = (u1 , u2 , u3 )T ∈ R3 : u = {λu | λ ∈ R}.

1.1. Riemann’s Habilitation lecture

11

Note that u = v if and only if u and v are collinear non-zero vectors. By rescaling if necessary we may always assume that at least one coordinate of u = 0 is equal to 1. It now follows that the entire RP 2 can be covered by 3 maps, the domain of which is R2 : (x1 , x2 ) → (1,x1 ,x2 )T , (x1 , x2 ) → (x1 ,1,x2 )T , (x1 , x2 ) → (x1 ,x2 ,1)T . Certain regions of RP 2 are represented in several different maps. For instance, the subset {(u1 ,u2 ,u3 )T |(u1 , u2 ) = (0, 0)} ⊆ RP 2 is covered by both (x1 , x2 ) → (1,x1 ,x2 )T and (y 1 , y 2 ) → (y1 ,1,y2 )T . The two sets of coordinates for (1,x1 ,x2 )T = (y1 ,1,y2 )T are related as follows:

1 x2 1 2 y 1 2 , (y , y ) = Φx (x , x ) = , x1 x1

1 y2 , (x1 , x2 ) = Φxy (y 1 , y 2 ) = . y1 y1 In this situation transition mappings Φyx and Φxy are differentiable bijections between {(x1 , x2 )|x1 = 0} and {(y 1 , y 2 )|y 1 = 0}. What would be a line element on RP 2 ? According to Riemann, “.... the problem is then reduced to establishing for each point a general expression for the linear element ds starting from that point, an expression which will thus contain the quantities x and quantities dx.” Put bluntly, ds2 in coordinates can be almost anything of the form  gij (dxi )(dxj ), though there are some limitations. For example, we have to have ds2 > 0. Perhaps more importantly, the coordinate expressions for ds2 have to line up whenever two coordinate patches overlap. That is, the transition map Φyx has to reduce the coordinate expression for ds2 in y-coordinates to that in x-coordinates and – vice versa – Φxy has to reduce the coordinate expression for ds2 in x-coordinates to that in y-coordinates. An example of such a line element is ds2 =

(1 + (x2 )2 )(dx1 )2 − 2x1 x2 dx1 dx2 + (1 + (x1 )2 )(dx2 )2 . (1 + (x1 )2 + (x2 )2 )2

12

1. Introduction

More precisely, we claim that ds2 can be taken to have this form in any of the three coordinate patches introduced above7 . To see this first observe that ds2 > 0 because its numerator can be expressed as 2 a sum of squares. Secondly, inserting x1 = y11 , x2 = yy1 into the expression for ds2 in x-coordinates and simplifying yields the claimed expression for ds2 in y-coordinates: ds2 =

(1 + (y 2 )2 )(dy 1 )2 − 2y 1 y 2 dy 1 dy 2 + (1 + (y 1 )2 )(dy 2 )2 . (1 + (y 1 )2 + (y 2 )2 )2

The example of RP 2 can be generalized further. This could entail an increase in dimension (e.g. a study of the collection RP n of all lines in Rn+1 ) or a replacement of R by some other normed division algebra such as C; consult exercises following this lecture for further details. 1.1.5. Exercises for Lecture 1.1. Polar and/or spherical coordinates. (1) (a) Consider the coordinatization of R2 {(0, 0)} using polar coordinates (r, θ). Express the vector fields ∂r and ∂θ in terms of Cartesian ∂x and ∂y . (b) Consider the coordinatization of R3  {(0, 0, 0)} using spherical coordinates (r, θ, φ). Express the vector fields ∂r , ∂θ and ∂φ in terms of Cartesian ∂x , ∂y and ∂z . (2) Provide a computation which shows that the (Euclidean) line element for R2  {(0, 0)} in polar coordinates takes the form of ds2 = dr 2 + r 2 dθ 2 . (3) Consider the polar coordinates (r, θ) in the Euclidean plane, and consider the line element ds2 = dr 2 + f (r)2 dθ 2 for some positive function f (r). (a) For a fixed value of θ∗ compute the length of r(t) = t, θ(t) = θ∗ , 0 < t < . 7 The fact that in this example ds2 has the same form in each of the three coordinate patches is an artifact of the fact that this example is particularly natural. In general one should not expect this feature.

1.1. Riemann’s Habilitation lecture

13

(b) For a fixed value of  compute the length of r(t) = , θ(t) = t, −π < t < π in terms of f . (c) Which geometry corresponds to the following choices of f ? • f (r) = r; • f (r) = sin(r), under the restriction 0 < r < π; • f (r) = 1; (d) Based on the above provide a visual description of the geometries corresponding to the following choices of f . • f (r) = 12 r; • f (r) = 2r; • f (r) = 1 + 2r; • f (r) = 2 + sin(r) • f (r) = er ; • f (r) = sinh(r) = 12 (er − e−r ). (4) (a) Provide a computation which shows that the Euclidean line element for R3  {(0, 0, 0)} in spherical coordinates (r, θ, φ) takes the form of ds2 = dr 2 + r 2 ds2S 2 , where ds2S 2 = dφ2 + sin2 φ dθ 2 denotes the standard spherical line element. (b) In analogy to Exercise 3, provide a visual description of the geometries on R3  {(0, 0, 0)} corresponding to the following line elements. • ds2 = dr 2 + sin2 r ds2S 2 ; • ds2 = dr 2 + sinh2 r ds2S 2 .

14

1. Introduction

Stereographic projection. (5) (a) Establish the explicit formulae relating a point on S 2 whose Cartesian coordinates are (x, y, z) to its stereographic projection (x1 , x2 , 0), and vice versa. Do this both for the projection from the North Pole and for the projection from the South Pole. (b) Coordinatize the entire S 2 with two coordinate patches, one arising from the stereographic projection from the North Pole and the other arising from the stereographic projection from the South Pole. Sketch a figure on the surface of S 2 and then sketch the corresponding figures in the two maps. (c) Find the explicit formulae for the transition maps corresponding to the coordinate patches used in Exercise 5b above. Illustrate the effect of the transition maps on the polar coordinate grids in R2 {(0, 0)}, and convince yourself that the transition maps are smooth bijections of R2  {(0, 0)}. (6) (a) Develop formulas for a stereographic coordinatization of S n . (b) Provide a computation which shows that the standard line element on S n takes the form of  1 2  4 d(x ) + · · · + d(xn )2 ds2 = 1 2 n 2 2 (1 + (x ) + · · · + (x ) ) in the coordinates arising from the stereographic projection. Projective spaces. (7) (a) Based on the example of RP 2 worked out in the lecture, roughly describe the coordinatization of the ndimensional real projective space: RP n = {v | v ∈ Rn+1  {0}} for v = {λv | λ ∈ R}.

1.1. Riemann’s Habilitation lecture

15

(b) Let CP n denote the set of all complex lines in Cn+1 passing through the origin; in other words, let CP n denote the set of all v where v = {λv | λ ∈ C} for some non-zero v = (v 1 , v 2 , ..., v n+1 ) ∈ Cn+1 . Show that CP n , called the complex projective space, can be coordinatized by using 2n variables. (8) (a) It can easily be seen that ds2 =

(1 + (x2 )2 )(dx1 )2 − 2x1 x2 dx1 dx2 + (1 + (x1 )2 )(dx2 )2 (1 + (x1 )2 + (x2 )2 )2 satisfies ds2 > 0 and thus gives rise to a Riemannian metric on R2 . Show that the schematic form of this line element is invariant under the change of coordinates 2 y 1 = x11 , y 2 = xx1 .

(b) Explain why the above defines a Riemannian metric on the entire RP 2 . (9) (a) Let u, v ∈ C. Show that ds2 =

v )(dvd¯ u)] + (1 + |u|2 )|dv|2 (1 + |v|2 )|du|2 − 2Re[(u¯ 2 (1 + |u| + |v|2 )2

satisfies ds2 > 0 and thus gives rise to a 4-dimensional line element. (b) Show that the above defines a Riemannian metric on the entire CP 2 . Other. (10) Assume that (x1 , . . . , xn ) are Cartesian and assume that (y 1 , y 2 , . . . , y n ) are some (curvilinear) coordinates on Rn . (a) Show that (i) dxi = (ii) ∂yi =



∂xi k k ∂y k dy ;



∂xk k ∂y i ∂xk ;

16

1. Introduction (b) Compute ∂yi , ∂yj where ., . denotes the Euclidean inner-product in Rn ; (c) Compute gij , the coefficients of the Euclidean line element expression in y-coordinates,  gij dy i dy j . ds2 = ij

(d) Compare the answers to the last two questions.

1.2. The framework of Riemannian geometry 1.2.1. Introduction. In 1828 Gauss published “Disquisitiones generales circa superficies curvas” (“General Investigations of Curved Surfaces”), an investigation of surfaces embedded in the Euclidean space R3 , their associated line elements ds2 and the concept of curvature. This work includes Theorema Egregium which identifies a formula (with more than a dozen terms) for the curvature based solely on the (derivatives of) line element coefficients. Thus, the concept of curvature does not depend as much on the ambient space, as it does on the expression for the line element. In other words, Gauss’s work shifted the emphasis from the ambient space to the line element on the surface itself. In his Habilitation lecture Riemann discussed how the notions from Gauss’s work on the concept of curvature can be applied to the space itself. The idea is that, prima facie, space can be almost anything that is coordinatizable and line elements can be any expressions of the form  gij (x1 , x2 , ..., xn )dxi dxj , ds2 = as long as we have ds2 > 0. (Actually, we can always in addition assume that gij = gji due to dxi dxj = dxj dxi .) This line element, as we shall see, determines the curvature of the space. We begin this lecture by formulating this basic framework of Riemannian geometry somewhat more precisely, and end the lecture by investigating the concepts of distance and volume within such a framework. 1.2.2. The concept of an n-dimensional manifold. The most basic concept in Riemannian geometry is that of a manifold .

1.2. The framework

17

x2 x1

Figure 1.9. One map in an atlas

In a nutshell, the term n-dimensional manifold signifies a set M which can be coordinatized (possibly with a big atlas of maps/charts) so that locations of points represented in the same map depend on n parameters x1 , x2 , ..., xn . It is time to make a major disclaimer. To comply with the modern standards of rigor in mathematics, one first has to start by addressing the topological underpinnings of the manipulations done above. For example, how does one know that each map in an atlas requires the same number of parameters? These lectures are meant to provide an introduction to the subject and are not designed to serve as a replacement for rigorous courses in differential and Riemannian geometry which address the very foundations of the subject. In particular, these lectures do not even attempt to answer questions such as the one raised above. Those interested in a more careful treatment of differential and Riemannian geometry are encouraged to consult resources such as the trilogy of textbooks [6], [7] and [8] by John M. Lee. As illustrated in Figure 1.10, certain domains within M may be covered with a number of different maps. In such situations we always assume that the transition maps Φyx : (x1 , ..., xn ) → (y 1 , ..., y n ) and Φxy : (y 1 , ..., y n ) → (x1 , ..., xn ) are differentiable bijections between the corresponding domains in Rn .

18

1. Introduction

y2 x2

Φyx y1 x1 Φxy

Figure 1.10. Transition mappings

1.2.3. The concept of tangent vector (fields). Using coordinates, a parametrized curve γ on a manifold can be expressed as γ(t) = x1 (t), x2 (t), ..., xn (t) . The curves  t, if i = j, i x (t) = const, if i = j, define coordinate grid lines. The rough idea here is that at each point P within the scope of x-coordinates the coordinate vector fields ∂i = ∂xi provide a basis for the vector space of directions based at P , also known as the tangent space at P .

∂2

∂1

Figure 1.11. Coordinate vector fields

1.2. The framework

19

It is time for another major disclaimer. The ideas we just described are fairly intuitive in the cases when the manifold is manifestly embedded as a surface within some Rn . However, when that is not the case the concepts of tangent space and vector (fields) are considerably more abstract and, as a result, somewhat difficult to define. One of the many approaches has us establish an equivalence relation between tangent spaces in different coordinates (e.g. by using dΦyx —see below). Within such an approach the concept of the tangent space at a point of a manifold arises as the set of the corresponding equivalence classes. In the interests of brevity we shy away from the rigorous treatment of the subject. Those interested in details should consult textbooks such as those of John M. Lee referred to earlier. Ultimately, the idea is that each tangent vector field V on M has coordinate representations of the form i V i ∂i ; the functions V i are often called the components of V . In particular, tangent vector field i γ˙ to a curve γ can be expressed as i ( dx dt )∂i .

Figure 1.12. Vector field γ˙ =

dx1 ∂ dt 1

+

dx2 ∂ dt 2

Of course, different coordinate patches yield different coordinate vector fields. Considering the parametric equations for the x-grid lines in y-coordinates yields

(1.1)

∂xi =

 ∂y α α

∂xi

∂yα .

20

1. Introduction

More precisely, if Φyx : (x1 , ..., xn ) → (y 1 , ..., y n ) is a transition map then its linearization/Jacobi matrix dΦyx viewed as a linear transformation  ∂y α ∂α , dΦyx : ∂i → ∂xi α that is, dΦyx :

 i

    ∂y α V i ∂i → V i ∂α , i ∂x α i

provides a method of identifying tangent spaces in different coordi nates. Ultimately, the idea is that the coordinate expressions i V i ∂i α and α V ∂α given in terms of x- and y-coordinates respectively patch/transition “correctly” in order to define an actual vector field on M provided the components V i and V α relate as follows:  ∂y α  ∂xi i i (1.2) Vα = V i.e. V = V α. i α ∂x ∂y i i The reader is strongly encouraged to view (1.1) and (1.2) as formulae for changing bases and coordinates from linear algebra. 1.2.4. Riemannian metric. Readers who have done Exercise 10 from Lecture 1.1 will recall that the functions gij featured in the ex pression ds2 = gij dy i dy j for the Euclidean line element in curvilinear coordinates (y 1 , ..., y n ) are equal to the (Euclidean) inner-products of coordinate vector fields ∂yi and ∂yj . Due to the bilinearity of the inner-product we then have V, W = ij gij V i W j i i for all vector fields V = V ∂yi and W = W ∂yi on Rn . A contemporary standpoint on line elements is that they are inner-product operations on the set of tangent vectors based at a particular point or, more globally, on the set of vector fields. This change of perspective is mirrored in the change of terminology: in place of the phrase line element we are more likely to use the phrase Riemannian metric. A Riemannian metric g is an operation (V, W ) → g(V, W ), the outcome of which is: a number (scalar) when performed on tangent vectors based at the same point, and a function (scalar field)

1.2. The framework

21

when performed on vector fields. It is required that g be symmetric in V and W and that it be bilinear: g(αV1 + βV2 , W ) = αg(V1 , W ) + βg(V2 , W ). The bilinearity here applies not only with respect to constants, but also with respect to functions/scalar fields α and β. It is further required that g(V, V ) ≥ 0 for all V with g(V, V ) = 0 at a point when and only when V = 0 there. In relation to this last requirement, it is worth pointing out a perspective according to which Riemannian metric is simply a positive definite inner-product on each tangent space, changing smoothly from basepoint to basepoint. 1.2.5. Notation. Notationally, expect to see expressions such as gij = g(∂i , ∂j ), g(V, W ) = ij gij V i W j . Traditional notation for an inner-product, ., . , is often used in place of and/or in parallel with g. (Occasionally, one even sees ., . g .) We  use |V |g to denote the magnitude g(V, V ) of V with respect to g. The quantities gij = g(∂i , ∂j ) are called the metric components. The reader is encouraged to examine the behavior of the metric components under a change of coordinates. Specifically, in Exercise 8 following this lecture the reader is asked to prove that the components gij and gαβ of the metric g in xand y-coordinates respectively relate as  ∂xi ∂xj gαβ = gij . ∂y α ∂y β ij One often thinks of the metric components as entries of a symmetric n × n matrix field. The condition that g(V, V ) > 0 for non-zero V implies that the eigenvalues of the matrix field [gij ] are all positive at each and every base point. In particular, [gij ] is invertible at each base point. The entries of its inverse are denoted by g ij : [gij ]−1 = [g ij ]. Note that by the definition of the inverse we have:   1, if i = j, kj gik g = 0, otherwise. k

22

1. Introduction

We leave it as an exercise for the reader to prove that  ∂y α ∂y β g ij . g αβ = i ∂xj ∂x ij 1.2.6. Length and distance measurements. Riemannian metrics g and line elements ds2 are related as follows: The (square of the) magnitude of a tangent vector to γ(t) = (x1 (t), ..., xn (t)) is given by dxj i dxi dxj ds 2 |γ| ˙ 2g = g( i ( dx j ( dt ) ∂j ) = ij gij ( dt )( dt ) = ( dt ) . dt ) ∂i , To measure the length of a curve γ one needs to integrate the mag˙ nitude |γ| ˙ g of the tangent vector field γ:  b L(γ) = |γ| ˙ g dt. a

One can show that the length of a curve does not depend on its parametrization. (See Exercise 2.) Equipped with a way of measuring lengths of paths, we are now able to determine distance between points. Naturally, distance between two points is the length of the shortest path between the two points, should such a path exist. (See below for situations when such a path does not exist.) In situations when a path of shortest length exists, determining the distance between two points is an optimization problem. Like most optimization problems it can be solved by methods of calculus. We dedicate the entire next lecture to deriving and analyzing the ODE’s which solve such an optimization problem. For now let us at least announce the solution of the shortest path problem on a two-dimensional sphere.

Figure 1.13. Great circles

1.2. The framework

23

A pair of points on a two-dimensional sphere is always contained in a great circle, that is, a circle which is the intersection of the sphere with a plane containing the center of the sphere (see Figure 1.13). As we show in the next lecture, great circles serve as length minimizers on the sphere. In more practical terms, this is the reason why many intercontinental flights (e.g. between North America and Europe) take a route which goes over the Arctic. It is not at all clear that for a given pair of points there exists a path of shortest length joining them. We generally want our manifolds to be path connected, i.e., we generally want them to be such that a given pair of points can be joined by some path. However, there are easy examples which show a possibility of non-existence of shortest paths even on path-connected manifolds. Consider the Euclidean plane with a point O removed and consider two points P and Q which are collinear with, but located on opposite sides of, the excluded point O. Clearly, the shortest path between P and Q with respect to the Euclidean metric cannot be found within the plane when O is removed. Thus, in general, we cannot expect that there will always be a path of shortest length between two given points. For this reason, we define the distance d(A, B) between two points A and B as the infimum, i.e., greatest lower bound, of the set of lengths of paths between A and B. In our example the distance between P and Q would still have the same value as in the standard Euclidean plane.

P

??

O

Q

Figure 1.14. (In)completeness

A somewhat obvious remedy to the problem described above is to only consider (path-connected) manifolds without punctures, holes, or otherwise missing points. Readers familiar with elementary analysis

24

1. Introduction

probably recognize that the desired property here is that of completeness . The technical meaning of this phrase will be addressed in more detail in the following lecture. One should also notice that a path of shortest length between two points need not be unique. This is blatantly obvious in the example of a sphere where there are infinitely many paths of shortest length joining the North and South Poles: each and every meridian serves as a length minimizer. To make things even more complicated there may be many local length minimizers, in the sense that for a given pair of points there may exist paths which only have the shortest length amongst nearby paths. A good example of this comes from the circular cylinder in R3 . Since it can be obtained by a distance-preserving rolling up of the Euclidean plane, the shortest paths between points come from straight line segments in the Euclidean plane. However, due to the folding effects there are many such line segments. As a result, each of the curves in Figure 1.15 minimizes length amongst nearby curves joining the same points.

Figure 1.15. Locally Euclidean geometry on a cylinder

1.2.7. Volume measurements. A Riemannian metric also determines the volume element, the integral of which measures the volume of the domain of integration. To understand this let us revisit how we compute volumes of parallelotopes in Euclidean Rn . A simple induction argument on n shows that the square of the volume of the parallelotope spanned by the vectors a1 , . . . , an is equal to the determinant of the matrix of Euclidean inner-products: vol2 = det([ ai , aj ]).

1.2. The framework

25

Indeed, by performing column operations we can make the first column of the matrix on the right become ( h, a1 , 0, . . . , 0)T where h denotes the orthogonal component of a1 with respect to the subspace spanned by a2 , . . . , an . Since h, a1 = |h|2 , the inductive hypothesis gives us that det([ ai , aj ] is equal to the product of |h|2 and the square of the volume of the (lower-dimensional) parallelotope spanned by a2 , . . . , an . This product is simply the square of the volume of the parallelotope spanned by a1 , . . . , an . The above identity is used to define the volume of a parallelotope in general, with respect any inner-product ., . in Rn . It can be checked that the volume defined in this manner has the expected features: it is positive unless a1 , . . . , an are linearly dependent, it is additive, and it is preserved by isometries of ., . . According to this definition, the square of the volume of the parallelotope spanned by ∂1 , . . . , ∂n in the tangent space equipped with the Riemannian metric g, is simply |g| := det[gij ]. Scaling to the infinitesimal level gives us the following formula for the volume element:  dvol = |g| dx1 . . . dxn . Finally, one checks (e.g., Exercise 10 following this lecture) that (1.3)

det[gij ] = (det dΦyx )2 det[gαβ ]

under the change of coordinates Φyx and the stated expression for the volume element dvol does not depend on the choice of coordinates. Example. As an illustration, we compute the volume8 of the standard unit three-dimensional sphere S 3 . In stereographic coordinates we have

 ds2 = (1+(x1 )2 +(x42 )2 +(x3 )2 )2 (dx1 )2 + (dx2 )2 + (dx3 )2 , which can also be expressed as ds2S 3 =

4 (1+r 2 )2

2  (dr ) + r 2 ds2S 2 ,

8 Note: this does not refer to the volume enclosed by the sphere within R4 but to the surface “area”.

26

1. Introduction

where 0 < r < ∞ and where ds2S 2 denotes the standard line element on S 2 in spherical coordinates. From here we get dvolS 3 = 

8r 2 2 (1+r 2 )3 dvolS

8r 2 Vol(S ) = dr Vol(S 2 ). (1 + r 2 )3 0 The integral on the right can be easily computed using the trigonometric substitution r = tan s; its value comes out to be π2 . Thus, the volume of the standard unit three-dimensional sphere S 3 is:

so that

3

∞

Vol(S 3 ) =

2 π 2 Vol(S )

= 2π 2 .

In general, compact manifolds have finite volume. We often think of compact manifolds as being closed and bounded (e.g. closed and bounded surfaces in R3 ), although this is not a part of their definition. Technically, we say that a Riemannian manifold (M, g) is compact if every bounded sequence9 of points {Pm } has a convergent subsequence {Pmi }. One common feature that all compact manifolds share is that they can always be covered with finitely many coordinate patches. Another property that all compact manifolds have in common is that all continuous real-functions defined on compact manifolds reach their maximum and minimum. Proofs of these claims belong to elementary analysis and will not be done here. 1.2.8. Riemannian geometry and geometric analysis. By Riemannian geometry we mean a study of Riemannian manifolds, i.e., manifolds equipped with a Riemannian metric, in general; the phrase “Riemannian geometry” does not refer to any specific geometry. One of the central notions in Riemannian geometry is that of curvature; many a theorem in Riemannian geometry addresses the kind of consequences that restrictions on the curvature (e.g. curvature bounds) have on geometric quantities such as distances and volumes or on topological qualities such as homeomorphism type or the fundamental group. Imposing conditions on curvature often amounts to (a system of) differential equations on metric (components). Analysis of such differential equations is the subject of geometric analysis. 9 The phrase “bounded sequence” stands for a sequence for which there is some C > 0 and some P ∈ M such that d(Pm , P ) ≤ C for all m.

1.2. The framework

27

1.2.9. Exercises for Lecture 1.2. Lengths of paths. 2

2

(1) Consider the line element ds2 = dx y+dy on the upper half2 plane y > 0. Use this line element to compute the lengths of the following paths joining (−1, 1) and (1, 1). Which of the two paths is shorter? (a) Let c1 be the “straight” line segment: x(t) = t, y(t) = 1, −1 ≤ t ≤ 1. (b) Let c2 be the “circular arc”: √ √ x(t) = 2 cos(t), y(t) = 2 sin(t),

π 4

≤t≤

3π 4 .

(2) Show that the length L(γ) of a path γ is the same for all parametrizations of γ. (Assume continuous differentiability where needed, and that the path is completely contained in a single coordinate patch. For changes in coordinate patches see Exercise 9 below.) (3) (a) For a compact Riemannian manifold (M, g) we define the diameter to be the maximum value of d(P, Q) for P, Q ∈ M . What is the diameter of the standard unit sphere S 2 ? What about S n ? (b) We define the circle of radius r centered at the point P of a 2-dimensional Riemannian manifold (M, g) as the set of all points Q with d(P, Q) = r. Describe and compute the circumference of the circle of radius r centered at the North Pole of the standard unit S 2 . Volume measurements. (4) Compute: (a) The volume of S 4 . (b) The volume of S n .

28

1. Introduction (5) Let a(., .) be some (not necessarily Euclidean) inner-product operation on Rn . Define aij := a(∂i , ∂j ) with respect to the standard Cartesian basis {∂i }. Let A denote the corresponding matrix [aij ]. Note that A is symmetric. (a) Show that a(v, w) = Av, w , where ., . denotes the standard Euclidean inner-product. (b) By a theorem from linear algebra, the symmetric matrix A must have real eigenvalues. Show that the eigenvalues of A and the determinant of A are necessarily positive. (c) By a theorem from linear algebra, there is an orthonormal basis for the Euclidean Rn consisting of eigenvectors of A. Show that the same basis is orthogonal with respect to a(., .). (d) Consider the orthonormal basis of eigenvectors of A from the above. What do you think the lengths of these eigenvectors are with respect to a(., .)? (e) Consider the rectangular box spanned by the orthonormal basis of eigenvectors of A in the Euclidean Rn . What do you think the volume of this box is with respect to a(., .)? How does your formula relate to our discussion of the formula for the volume element? (6) In multivariable calculus classes one often sees a formula for the Euclidean volume of a parallelotope in terms of the determinant of the matrix of coordinates (with respect to some orthonormal basis) of the vectors forming said parallelotope. For example, the Euclidean volume of the parallelotope spanned by the vectors ai = (a1i , a2i , a3i )T , where 1 ≤ i ≤ 3, in R3 is     vol = det[aji ] . Show that this expression satisfies vol2 = det[ ai , aj ].

1.3. Geodesics

29

Behavior under change of coordinates. (7) To avoid pathological situations one always assumes that curves γ are parametrized so that the tangent vector field γ˙ is nowhere vanishing. Show that under this assumption a curve on a Riemannian manifold can be smoothly reparametrized so that its tangent vector is of constant length. (8) This problem investigates how a change of coordinates Φyx : (x1 , x2 , ..., xn ) → (y 1 , y 2 , ..., y n ) effects the coordinate expressions for vector fields and the components of a Riemannian metric. (a) Let gij and gαβ be the metric components with respect to the x- and y-coordinates, respectively. Prove that  α β ∂y ∂y [gij ] = (dΦyx )T [gαβ ]dΦyx i.e., gij = ∂xi ∂xj gαβ . αβ

(b) Likewise, prove that  α β ∂y ∂y ij g αβ = ∂xi ∂xj g . ij

(9) Let a path γ be contained within the scope of both the xand the y-coordinates. Show that the value of the length of γ, L(γ), does not depend on the choice of the coordinate system.  (10) Show that the expression |g| dx1 ...dxn is invariant under the change of coordinates.

1.3. Geodesics 1.3.1. Introduction. In Lecture 1.2 we saw that the distance between two points of a Riemannian manifold (M, g) is the infimum of the lengths of paths joining the two points. In many circumstances this infimum is reached by a length minimizing path. The goal of this lecture is to use methods of calculus to find such length minimizers. We end the lecture with an investigation of certain examples, one being of great historical significance.

30

1. Introduction

1.3.2. The length functional. Suppose there is a path γ0 of minimum length between points P and Q, and for simplicity assume that γ0 lies within a single coordinate patch. Let γε be a family of nearby paths, indexed by a small parameter ε, between P and Q. One may want to think of this family as perturbations or slight modifications of the path γ0 ; in other words, γε → γ0 as ε → 0.

γε

γ0 Q

P Figure 1.16. Perturbations γε of γ0

In coordinates the family γε looks like   γ(ε, t) = x1 (ε, t), . . . , xn (ε, t) . Note that for a given geometric family of curves there are infinitely many representations of such a form: intuitively speaking they correspond to different speeds of traverse, i.e., different |γ(ε, ˙ t)|g . This kind of ambiguity can be resolved by requiring that the duration of traverse be the same for all ε, and that the speeds of traverse be constant. That is, from now on we assume10 that a ≤ t ≤ b for all ε and that L(γε ) . |γ(ε, ˙ t)|g = b−a As γε all join P to Q we know that γ(ε, a) and γ(ε, b) are each constant in ε. Therefore, we have ∂xi  ∂xi  = =0   ∂ε t=a ∂ε t=b for all i and at each ε. 10

Such restrictions do not lose generality – see Exercise 7 from Lecture 1.2.

1.3. Geodesics

31

The length L(γε ) of each γε is given in coordinates by ⎛ ⎞1/2  b  i j  ∂x ∂x   ⎝ ⎠ dt L(γε ) = gij γ(ε,t) γ(ε,t) ∂t γ(ε,t) ∂t a ij If γ0 indeed is the path of shortest length between points P and Q, then the function ε → L(γε ) reaches its global minimum at ε = 0. In particular, we must have dL(γε )  = 0.  dε ε=0 In fact, if γ0 is the shortest of all paths between P and Q then the above has to hold for all possible perturbations γε of γ0 . The following question arises: what is the best way of computing this derivative? We can try differentiating under the integral sign, but the presence of the square root promises computational complications. Instead, we will consider a related functional which amongst its critical “points” includes the global minimum of the length functional γ → L(γ). 1.3.3. The energy functional. The energy of a curve γ(t), where a ≤ t ≤ b, is defined by  1 b 2 |γ| ˙ g dt. E(γ) = 2 a A big warning is in order: the value E(γ) is not a geometric feature of γ as it depends on its parametrization! (Compare with Exercise 2 following this lecture.) However, when |γ| ˙ g is constant (and it is in our underlying assumptions) then we clearly have E(γ) =

L(γ)2 2(b−a) .

In particular, minimizing length amongst paths of constant speed is equivalent to minimizing the corresponding energies. Going back to our family of curves γε and its coordinate representations γ(ε, t), we see that we should try to minimize  b  ∂xi  ∂xj  E(γε ) = gij γ(ε,t) dt. ∂t γ(ε,t) ∂t γ(ε,t)) a ij

32

1. Introduction

This can be done by imposing that the equation dE(γε )  =0  dε ε=0 hold for all possible perturbations γε of γ0 . We proceed by investigating what this requirement has to say about γ0 . The computation we are about to do is highly related to the derivation of the Euler-Lagrange equations in the field of mathematics called the calculus of variations. To simplify the notation we adopt the notational convention11 according to which repeated indices are being summed over. ε) by means of differentiation under the inteWe compute dE(γ dε gral sign. Before embarking on this path we clarify the nature of dependency of gij γ(ε,t) on ε:

   gij γ(ε,t) = gij x1 (ε, t), ..., xn (ε, t) .  Thus, the derivative of gij γ(ε,t) with respect to ε is ∂gij ∂gij ∂xl = . ∂ε ∂xl ∂ε Overall, differentiating E(γε ) under the integral sign produces:  b  b  b ∂gij ∂xl ∂xi ∂xj ∂ 2 xi ∂xj ∂xi ∂ 2 xj dt + dt + dt = 0, g gij ij l ∂t∂ε ∂t ∂t ∂t∂ε a ∂x ∂ε ∂t ∂t a a where all the integrands are evaluated along γ0 . Next, we apply integration by parts to the last two terms of this equation. For example, the middle term is equal to

 b  b ∂ 2 xi ∂xj ∂xi ∂xj t=b ∂ ∂xj ∂xi dt =gij dt gij − gij  ∂t ∂ε ∂t ∂ε ∂t t=a ∂t ∂ε a a ∂t

 b ∂ ∂xj ∂xi =− dt, gij ∂t ∂ε a ∂t 11 This convention is in fact standard across the field of general relativity, where it goes under the name of the Einstein summation convention.

1.3. Geodesics

33 i

due to the fact that ∂x ∂ε vanishes at t = a and t = b. In view of ∂gij ∂gij ∂xl = we further obtain: ∂t ∂xl ∂t  b  b  b ∂ 2 xi ∂xj ∂gij ∂xl ∂xj ∂xi ∂ 2 xj ∂xi dt = − dt − dt. gij gij 2 l ∂t ∂ε ∂t ∂t ∂ε a a ∂x ∂t ∂t ∂ε a Recall that we are summing over all i, j and (when present) l. For this reason we are able to relabel indices as follows:

 b  b ∂ 2 xi ∂xj ∂ 2 xj ∂glj ∂xi ∂xj ∂xl dt = − dt. gij glj 2 + ∂t ∂ε ∂t ∂t ∂xi ∂t ∂t ∂ε a a The same kind of manipulation can be done to the last term, namely b i 2 j ε)  g ∂x ∂ x dt, in the expression for dE(γ . Gathering like dε a ij ∂t ∂t∂ε ε=0 terms and taking care of minus signs and factors of 2 ultimately produces (1.4)



 b ∂ 2 xj 1 ∂gjl ∂gil ∂gij ∂xi ∂xj ∂xl dt = 0. + − glj 2 + ∂t 2 ∂xi ∂xj ∂xl ∂t ∂t ∂ε a Here is a crucial observation. The last equality has to hold for all perturbations γε of γ0 . Thus, we have a great deal of freedom in l how we choose our ∂x ∂ε – the only restriction is that these derivatives vanish at each (ε, a) and each (ε, b)! Exercising this freedom (details on this are in Exercise 3 below) brings us to

 d2 xj  1 ∂gjl ∂gil ∂gij dxi dxj =0 glj 2 + + − (1.5) dt 2 ∂xi ∂xj ∂xl dt dt j ij i

i

dx for each individual l. The notational departure from ∂x ∂t to dt etc, is not a typo—we have in fact arrived at a system of ODEs which the coordinate expressions (x1 (t), . . . , xn (t)) for constant speed parametrizations of a length minimizer γ0 have to satisfy! An equivalent and far more convenient form of the system is obtained by solving 2 j (1.5) for individual ddtx2 in terms of the lower-order derivatives. To 2 j do so observe that ddtx2 satisfies a linear system of equations with the coefficient matrix [glj ]. Applying the inverse matrix [g kl ] leads to

d2 xk  1 kl ∂glj ∂gil ∂gij dxi dxj g = 0, 1 ≤ k ≤ n. + + − dt2 2 ∂xi ∂xj ∂xl dt dt ij

34

1. Introduction

1.3.4. Geodesics and geodesic equations. In hindsight, coordinate expressions for any of the critical “points” (critical parametrized paths, really) of the energy functional coincide with the solutions of the above system of ODEs. Critical parametrized paths of the energy functional are called geodesics. It is important to note that since the energy functional is parametrization-dependent the concept of a geodesic is as well. In fact, readers who complete Exercises 5 and 6 below are about to learn that geodesics are necessarily parametrized so that their tangent vector fields have constant length. The system of ODEs for geodesics, also known as the geodesic equations, are most commonly expressed in an abbreviated form as d2 xk  k dxi dxj = 0, 1 ≤ k ≤ n, + Γij dt2 dt dt ij where



1 kl ∂glj ∂gil ∂gij := g + − . 2 ∂xi ∂xj ∂xl are called the Christoffel symbols12 .

Γkij The symbols Γkij

As a trivial example, consider the geodesic equations in the Euclidean plane with respect to Cartesian coordinates. The metric components are constant, and as a result the Christoffel symbols vanish. The geodesic equations, therefore, boil down to saying that the second derivatives of all xk (t) vanish. In other words, the functions xk (t) must all be linear and the geometric curve described by them must be a (subset of) a line. This is to be expected as line segments are length minimizers in the Euclidean plane. Before we move on to more interesting examples we need to address existence and other properties (e.g., uniqueness, time of existence, etc.) of solutions of geodesic equations. This can be done through basic Existence, Uniqueness and Stability Theorems for initial value problems. One such theorem is stated below. Theorem 1.1. Let F : Rn × Rn → Rn be a smooth function. Then for all (y∗ , v∗ ) ∈ Rn × Rn there exist: • a neighborhood Y × V of (y∗ , v∗ ) in Rn × Rn , 12 These symbols are named in honor of the German mathematician Elwin Bruno Christoffel.

1.3. Geodesics

35

• a number ε > 0 and • a smooth map Y : Y × V × (−ε, ε) → Rn such that for each (y0 , v0 ) ∈ Y × V the function y : t → Y(y0 , v0 , t) is the unique solution of the initial value problem ⎧ 2 d y y ⎪ ⎪ ⎨ dt2 = F (y, dt ), (1.6) y(0) = y0 , ⎪ ⎪ ⎩ dy (0) = v . dt

0

Thus, roughly speaking, solutions of initial value problems such as (1.6) exist, are unique, and vary smoothly with the initial data. It is important to notice that no claim is being made about the long-term existence of solutions; the only claim that is being made is that the solutions are defined on some small interval (−ε, ε). When applied to geodesic equations the above theorem gives us the following result. Theorem 1.2. Let (M, g) be a Riemannian manifold. For each point P ∈ M and each tangent vector V there is an ε > 0 and a unique geodesic γ(t) with −ε < t < ε which passes through P with the velocity vector V : γ(0) = P,

γ(0) ˙ = V.

It is important to note that geodesics γ(t) need not be defined for all t ∈ R. We say that a manifold is geodesically complete if every geodesic (can be extended to be) defined on all R.

P

Figure 1.17. Assigning geodesics to tangent vectors

36

1. Introduction

Geodesically complete manifolds are the subject of the HopfRinow Theorem13 . We only offer the statement of the theorem here; the proof is too difficult for us at this point. Theorem 1.3. If a path-connected Riemannian manifold (M, g) is geodesically complete, then any two points of M can be joined by a length-minimizing geodesic. For the benefit of the readers familiar with elementary analysis we point to a consequence of the Hopf-Rinow theorem, which states that a Riemannian manifold (M, g) is geodesically complete if and only if it is complete as a metric space. For this reason it is appropriate to use the phrase complete manifold whenever the manifold is geodesically complete. Unless specifically stated otherwise, we always assume our manifolds are complete and path connected. It is useful to know that every compact manifold without boundary is automatically complete. 1.3.5. Examples. Example of the standard unit sphere S n−1 in Rn . We now prove that great circles on S n−1 are geodesics – a result we already announced in Lecture 1.2. In place of a computationally demanding approach based on solving the geodesic equations, we prove this result by taking advantage of symmetries of S n−1 and Theorem 1.2. By making a judicious choice of coordinates in Rn if necessary we may without loss of generality focus only on finding the geodesic through the point P = (1, 0, . . . , 0) in the direction of the vector V = (0, 1, 0, . . . , 0)T . The goal is to show that the geodesic in question is the great circle in the subspace x3 = · · · = xn = 0. Consider the map I : (x1 , x2 , x3 , . . . , xn ) → (x1 , x2 , −x3 , . . . , −xn ). This map is a symmetry of S n−1 : it preserves distances in Rn and so it preserves both S n−1 and the metric on S n−1 . (In circumstances of this sort we say that I is an isometry of a Riemannian manifold.) It follows that I has to take geodesics to geodesics. Since both P and the tangent vector V are mapped to itself under I, the unique geodesic passing through P with velocity vector V must also be mapped to 13

The theorem is attributed to Heinz Hopf and his student Willi Rinow.

1.3. Geodesics

37

itself under I. It follows that this geodesic is (contained within) the great circle in the subspace x3 = · · · = xn = 0. Examples of hyperbolic spaces. This example examines the upper halfspace model of the n-dimensional hyperbolic space Hn . It is given by  Hn = {(x1 , x2 , ..., xn ) ∈ Rn xn > 0}, and the Riemannian metric g = (xn )−2 (d(x1 )2 + · · · + d(xn )2 ). On an intuitive qualitative level, the factor of (xn )−2 enlarges the distances near the boundary xn = 0 and diminishes the distances far away from xn = 0. In some sense, there is a whole lot more “space” in Hn near xn = 0 than it might appear to our Euclidean eyes. Here are some more concrete observations which can help us visualize this new geometry. For example, the dilation of the form I : (x1 , . . . , xn ) → (λx1 , . . . , λxn ), λ > 0 preserves the line element g: (λxn )−2 (d(λx1 )2 + · · · + d(λxn )2 ) = (xn )−2 (d(x1 )2 + · · · + d(xn )2 ). Thus, the map I preserves distances and should be thought of as an isometry. In particular, it maps shapes to congruent shapes. The same is true for “horizontal” translations (x1 , ..., xn−1 , xn ) → (x1 + a1 , ..., xn−1 + an−1 , xn ), boundary preserving reflections (..., xi , ..., xn ) → (..., −xi , ..., xn ), and boundary preserving rotations (..., xi , ..., xj , ...) → (..., cos α xi + sin α xj , ..., − sin α xi + cos α xj , ...). Such an abundance of symmetries and isometries makes it so that no point of hyperbolic space is special and so that all of the shapes in Figure 1.18 are congruent.

38

1. Introduction y

x Figure 1.18. Congruency with respect to y −2 (dx2 + dy 2 )

The question we would like to address is that of geodesics on Hn . Through employment of symmetries listed above it suffices to analyze the geodesic problem in H2 , that is, the upper half-plane y > 0 with the metric g = y −2 (dx2 + dy 2 ). To find the geodesic equation we first compute the Christoffel symbols. The only non-vanishing Christoffel symbols here are: Γyxx = y1 , Γxxy = Γxyx = Γyyy = − y1 . Thus, the geodesic equations are: ⎧ 2 ⎨ d 2x − 2 dx dy = 0, dt y dt dt  2   2 ⎩ d 2y + 1 dx 2 − 1 dy = 0. dt y dt y dt  −2 dx  d y dt vanishes. ConA consequence of the first equation is that dt 2 = cy for some constant c. We can insert this sequently, we have dx dt information into the second geodesic equation but we can also make use of the fact that geodesics have constant speed parametrizations! Without loss of generality we may assume that geodesics are of unit speed. (Compare with Exercises 5 and 6 below.) In other words, we may assume that    dy 2 2 2 = 1 and dy ) + ( ) y −2 ( dx dt dt dt = y 1 − (cy) . The latter can be solved by the method of separation of variables. In the case when c = 0 we obtain y = y0 et and x = x0 . In other words,

1.3. Geodesics

39

vertical rays are geodesics! In the case when c = 0 we get14 y=

1 c cosh(t+t0 )

and consequently

x=

1 c

tanh(t + t0 ) + a,

for some constants t0 and a. To visualize these geodesics observe that  2 (x − a)2 + y 2 = 1c . In other words, these geodesics are upper semi-circles centered along the boundary y = 0. Overall, geodesics in H2 are either (contained in) vertical rays or upper semi-circles centered at the boundary y = 0. Although each pair of distinct points determines such a geodesic uniquely, it is not at this point clear that such a geodesic is necessarily length minimizing. However, our explicit formulae show that each geodesic on H2 is defined on all of R, and that therefore H2 is geodesically complete. It now follows from the Hopf-Rinow Theorem (Theorem 1.3) that each pair of points in H2 can be joined by a unique length-minimizing geodesic: it is either a vertical line segment or an arc of an upper semi-circle centered at the boundary y = 0. The same holds in Hn , due to symmetries of Hn listed above. y

x Figure 1.19. Geodesics in

H2

1.3.6. The broader context. The last example has a big historical significance. After nearly 2,000 years of attempts to clarify the status of Euclid’s Fifth Postulate several mathematicians managed to develop a geometry based on the negations of Euclid’s Fifth Postulate. As a reminder, Euclid’s Fifth Postulate has a somewhat complicated 14 The hyperbolic trigonometric functions are defined as cosh(x) = and sinh(x) = 12 (ex − e−x ).

x 1 2 (e

+ e−x )

40

1. Introduction

statement which was quickly recognized to be equivalent to the statement15 that through a point not on a line  one can draw exactly one line which is coplanar with, but does not intersect, . y P l1

l2 l l ∩ l1 = l ∩ l2 = ∅

x

Figure 1.20. The Fifth Postulate does not hold in H2

For centuries nobody questioned the validity of Euclid’s Fifth Postulate and yet nobody managed to successfully prove that the Fifth Postulate is logically dependent on the remaining four postulates. The intellectual climate of the late 18th and early 19th century surrounding mathematics was to a great extent influenced by the work of the Prussian philosopher Immanuel Kant who in his Critique of Pure Reason (1781) addressed the nature of mathematical statements and knowledge. This work was understood as saying that the laws of Euclidean space are already in the human mind and that without these laws no consistent reasoning is possible. Thus, coming out with a geometrical theory based on a negation of Euclid’s Fifth Postulate was highly controversial. The two mathematicians to do so (independent of one another) were the Russian mathematician Nikolai Ivanovich Lobachevsky (1826) and the Hungarian mathematician Janos Bolyai (1829). It took a long time for their work to reach public acceptance (e.g., Gauss’s lukewarm reception of Janos Bolyai’s work in which he suggested that he already made the same discoveries but has been unwilling to share them publicly). Eventually, models of axiomatic geometries developed by Bolyai and Lobachevsky were discovered and the logical validity of their hyperbolic geometries was established. The 15 This statement is known as Playfair’s Axiom, in honor of the late 18th and early 19th century Scottish mathematician John Playfair who advocated that the Fifth Postulate be replaced by this axiom.

1.3. Geodesics

41

first model goes back to the Italian mathematician Eugenio Beltrami (1868). The example we saw above is the Poincar´e upper half-space model of hyperbolic geometry, in which the term “line” is interpreted as a geodesic. As illustrated in Figure 1.20, hyperbolic geometry satisfies the opposite of Playfair’s axiom. The model we analyzed in this lecture is named in honor of the French mathematician Henri Poincar´e who in his 1882 papers on the theory of functions of complex variables pointed to a remarkable connection between, on one hand, the class of functions of complex variables he was studying and, on the other hand, the line element 2 2 and hyperbolic geometries. ds2 = dx y+dy 2

Figure 1.21. Poincar´ e Disk Model

It should be mentioned that there is another (isomorphic) model of hyperbolic geometry which is attributed to Poincar´e, the so-called disk model. In this model we consider the interior of the Euclidean unit disk  {x ∈ Rn |x| < 1} and the line element ds2 =

2 4 (1−|x|2 )2 |dx| .

The geodesics in this model are line segments through the origin and the circles orthogonal to the unit sphere |x| = 1. It is no coincidence that the form of this hyperbolic line element is so similar to the spherical line element in stereographic coordinates. Interestingly enough, this general form of the line element was foreseen by Riemann who in his Riemann’s Habilitation lecture discussed

42

1. Introduction

line elements of the form ds2 =

1 (1 + α4 |x|2 )

|dx|2 .

1.3.7. Exercises for Lecture 1.3. Energy functional.

b (1) (a) Starting from the fact that a (f (t) + sg(t))2 dt ≥ 0 for all s, conclude that  2     b b b f (t)g(t) dt ≤ f (t)2 dt g(t)2 dt ; a

a

a

this is the Cauchy-Schwarz inequality for integrals. Furthermore, show that the equality holds only when the functions f and g are constant multiples of one another. (b) Show that for any (thus not necessarily constant speed) path γ(t), a ≤ t ≤ b we have L(γ)2 ≤ 2(b − a)E(γ). Furthermore, show that the equality holds if and only if γ is parametrized by constant length. (2) (a) Show that the energy of a parametrized curve depends on its parametrization. (b) Show that for a given geometric curve energy is minimized by parametrizations of constant speed. Calculus of Variations. (3) (a) Let f (t), a ≤ t ≤ b be a continuous function such that b f (t)χ(t) dt = 0 for all continuous functions χ(t) for a which χ(a) = χ(b) = 0. Show that f (t) = 0 for all a ≤ t ≤ b. (b) Justify the claim made in the text that the equation (1.4) implies the system of equations (1.5).

1.3. Geodesics

43

(4) This problem addresses surfaces which for a given boundary minimize the enclosed surface area; such surfaces are higher-dimensional analogues of length minimizers. Critical surfaces for area functionals S → Area(S) are called minimal surfaces. (a) Consider xn+1 = f (x1 , . . . , xn ) for (x1 , . . . , xn ) ∈ Ω; this defines an n-dimensional surface S in Euclidean Rn+1 . Show that the area functional is given by   1 + |gradf |2 dx1 . . . xn . Ω

(b) Show that S is a minimal surface if and only if   1 gradf = 0. div  1 + |gradf |2 Geodesic equations. (5) Let c be a non-zero constant. Show that a curve γ(t) is a geodesic if and only if γ(ct) is. (6) Show that any solution γ(t) = (x1 (t), . . . , xn (t)) of the geodesic equation satisfies d ˙ 2g ) dt (|γ|

= 0.

In particular, show that every geodesic γ(t) is necessarily parametrized in such a way that its tangent vectors γ˙ have constant length |γ| ˙ g. (7) Let P be a point of the Riemannian manifold (M, g) and let Bε denote the set of all tangent vectors V at P of magnitude less than ε: |V |g < ε. Show that there is ε > 0 and a smooth map γ : Bε × (−2, 2) → M such that for each V ∈ Bε the parametrized curve γV (t) := γ(V, t) is a geodesic passing though P with velocity vector V .

44

1. Introduction

Spherical and hyperbolic geometry. (8) (a) Find the Christoffel symbols for the standard unit 2dimensional sphere with respect to the standard spherical coordinates (θ, φ). (b) Express the geodesic equations on the standard unit sphere, and verify that a great circle of your choice is indeed a geodesic. (9) Compute the distance between the following points of the Poincar´e upper half-plane model H2 of hyperbolic geometry. (a) (0, ea ) and (0, eb ); (b) (−1, 1) and (1, 1); (c) (0, 2) and (−1, 1). (10) Investigate circles in hyperbolic plane geometry using the following guidelines. (a) Express the metric of the 2-dimensional Poincar´e disk model in polar coordinates. (b) Find the hyperbolic distance between (0, 0) and (r, 0). (c) Describe circles of radius  centered at (0, 0). (d) What are the circumferences of these circles?

Chapter 2

Differential calculus with tensors

2.1. Introduction to differential calculus 2.1.1. Introduction. The goal of this lecture is to develop a way of measuring change in a function or a vector field over a Riemannian manifold. Being in a multivariable calculus situation it is natural to employ the concept of a directional derivative. While directional differentiation of functions turns out to be relatively straightforward, a surprise is waiting for us in the case of vector fields. It turns out that when differentiating vector fields one has to account for the fact that coordinate systems change with respect to themselves. 2.1.2. Directional derivatives of functions. The directional derivative of a scalar-valued function f (x) of x = (x1 , . . . , xn ) ∈ Rn with respect to a vector field V on Rn , denoted ∇V f , measures the rate of change of f in the direction of V and is given by (∇V f )(x) := lim 1t (f (x + tV ) − f (x)). t→0

An alternative formulation of the directional derivative is as follows: if V = i V i ∂i then (2.1)

∇V f =

 i

Vi

∂f . ∂xi 45

46

2. Tensor calculus

Since x + tV is undefined on general manifolds, the difference quotient definition of ∇V f does not readily generalize. However, the identity (2.1) does carry over to manifolds. Although the expression (2.1) is local in nature and is potentially coordinate dependent, one proves that for a given function f and a given vector field V the ∂f value of i V i ∂x i does not depend on the choice of coordinates. This phenomenon is explored in Exercise 2 following this lecture, and the reader is strongly encouraged to complete this exercise. The overall idea here is that the local coordinate expressions (2.1) for ∇V f do seamlessly overlap to give a globally defined ∇V f . If a vector field V happens to agree with the tangent vector field γ˙ along some curve γ(t) = (x1 (t), . . . , xn (t)) then along γ we have: ∇V f =

 i

Vi

 ∂f dxi ∂f d = = f (γ(t)). i i dt ∂x ∂x dt i

As intuition suggests, the value of ∇V f along γ depends only on the d f (γ(t)) where f is a function values of f along γ. The derivative dt potentially only defined along γ is often denoted by ∇γ˙ f . We would like to emphasize the following features of ∇V f ; note that the last of the properties serves as a de-facto Product Rule. • ∇αV +βW f = α∇V f + β∇W f for all functions α, β; • ∇V (f + g) = ∇V f + ∇V g; • ∇V (αf ) = (∇V α)f + α(∇V f ) for all (differentiable) functions α, β. 2.1.3. Lie bracket. A general warning regarding directional derivatives is in order; it applies over Rn as well. Although ordinary partial derivatives of a (twice  ∂  differentiable) function f always  continuously ∂ ∂ ∂ f = commute, ∂x i ∂xj ∂xj ∂xi f , directional derivatives ∇V and ∇W do not generally commute:   j ∂ j ∂ ∂ ∂ V i ∂x W i ∂x ∇V ∇W f − ∇W ∇V f = i (W ∂xj f ) − i (V ∂xj f ) ij

ij

 j j ∂ ∂ ∂ = (V i ∂x − W i ∂x iW i V ) ∂xj f. ij

.

2.1. Differential calculus

47

The last computation shows very clearly the reason for the lack of commutativity: the fact that ∇V ∇W f − ∇W ∇V f = 0 has very little to do with f itself and is more about the fact that W changes in the ∂ j direction of V (as evidenced by the term V i ∂x i W ) and the fact that ∂ j V changes in the direction of W (as evidenced by the term W i ∂x i V ). In other words, the reason that ∇V ∇W f − ∇W ∇V f = 0 is inherent in the vector fields V and W . Readers who are seeing this particular phenomenon for the first time are strongly encouraged to examine specific instances of it in Rn ; Exercise 1 following this lecture can serve as a good starting point.  ∂  i ∂ j i ∂ j The structure of j i (V ∂xi W − W ∂xi V ) ∂xj f suggests that the commutator ∇V ∇W −∇W ∇V behaves as if it were directional differentiation with respect to a new vector field:  j j ∂ ∂ (V i ∂x − W i ∂x iW i V )∂j . ij

The latter is just a local formula which, prima facie, need not correspond to a globally defined vector field. (Of course, there are no such issues if we are working over Rn .) Thankfully, it can be shown that the last expression is invariant under changes of coordinates and so it does indeed give rise to a globally defined vector field. (See also Exercise 2 below.) This new vector field is denoted by [V, W ] and called the commutator of V and W . The role of the commutator [V, W ] is that it measures the offset between the changes in V and W with respect to each other. It is sometimes also called the Lie bracket in honor of the Norwegian mathematician Sophus Lie, who in the second half of the 19th century developed a whole field of mathematics (Lie algebras) based on objects which behave like the commutators [V, W ]. By its very definition the commutator [V, W ] satisfies [V, W ] = −[W, V ] and ∇V ∇W f = ∇W ∇V f + ∇[V,W ] f. 2.1.4. Subtleties with differentiating vector fields. The discussion of the commutator [V, W ] may have left the reader with the (false!) impression that the rate of change of a vector field W in the di ∂ j rection of V is given by ij V i ∂x i W ∂j . Perhaps the stated formula

48

2. Tensor calculus

makes sense in Rn , assuming we use the standard Cartesian coordinates. However, the next example shows that the stated formula no longer applies if curvilinear coordinate systems are used. In essence, the reason for this is that (generally speaking) coordinate systems change with respect to themselves! The only reason the proposed formula indeed works in Rn is because standard Cartesian coordinate vector fields are somehow constant: ∇∂i ∂j = 0. Ultimately, one has to first understand how a coordinate system changes with respect to itself. Examples. Here is a pictorial investigation of how coordinate systems change with respect to themselves. Arguments we are about to make are informal, but could be backed up by an explicit computation, e.g. Exercise 4. The hope is that pictures will aid in developing an intuitive rather than a purely computational understanding of what is going on. ∂r

(δ θ )·

(δ θ

1 ∂θ r

)∂ θ P O

∂r

r

Figure 2.1. Coordinate systems change with respect to themselves

Consider polar coordinates in the Euclidean plane R2  {(0, 0)}, and the coordinate vector fields ∂r and ∂θ . Let us study the rate of change of ∂r in the direction of ∂θ , i.e., ∇∂θ ∂r . Starting with ∂r based at a point P with coordinates (r, θ) and moving infinitesimally in the ∂θ direction produces ∂r which is rotated by δθ. Consider the infinitesimal triangle with two sides of length |∂r | = 1, enclosing the angle of δθ at P . The change in ∂r can be seen as the side of this triangle across from P . There is a similar (pun intended!) triangle with the infinitesimal angle of δθ at the origin O and with sides of

2.1. Differential calculus

49

−−→ size |OP | = r; the remaining side is the infinitesimal (δθ) multiple of ∂θ . Using the properties of similar triangles we are lead to ∇∂θ ∂r = 1r ∂θ .

(δ r ) |∂ θ| =

1 · r ∂θ

r+δ r

|∂ θ| = r

P δr O

r

Figure 2.2. Illustration of ∇∂r ∂θ =

1 ∂ r θ

Likewise, to find ∇∂r ∂θ move infinitesimally (δr) away from P in the ∂r direction. The two vectors ∂θ are parallel but of different length: one is of length r and the other is of length r + δr. In particular, the change in ∂θ is a multiple of δr and the unit vector 1r ∂θ . It follows that the rate of change of ∂θ in the direction of ∂r is ∇∂r ∂θ = 1r ∂θ . Similar arguments show that ∇∂r ∂r = 0 and ∇∂θ ∂θ = −r∂r . The four directional derivatives we found show how the polar coordinate system changes with respect to itself.

50

2. Tensor calculus

cos(φ) sin(φ) ∂θ

δφ

∂θ

∂θ

Figure 2.3. Tangent vectors based at nearby points of S 2

In principle, analogous “computations” can be executed on the sphere S 2 . For instance, to find ∇∂φ ∂θ we observe that the coordinate vectors ∂θ at two nearby points with coordinates (θ, φ) and (θ, φ+δφ) are parallel and pointing in the same direction while their magnitudes are sin(φ) and sin(φ + δφ). Since ∂φ is unit and ∂θ is of length sin(φ) it follows that

∇∂φ ∂θ =

cos(φ) ∂θ . sin(φ)

However, finding some of the remaining directional derivatives (e.g., ∇∂θ ∂φ ) is much harder to do primarily because it is hard to compare vectors which lie in two different tangent spaces! The issue is illustrated in Figure 2.4: if one were to move ∂φ from P1 to P0 in order to perform a subtraction (or vice versa: see the dashed vector) one would have difficulty staying within the tangent plane to the sphere. The effect is even more pronounced with the base point P2 .

2.1. Differential calculus

P0

51

P1

P2

Figure 2.4. Subtleties of parallelism on S 2 : Difference of indicated vectors at P1 (or P2 ) is no longer tangent to S 2 .

Throughout the above discussion we used the word “parallel” and our primary tool for comparing vectors with different base points has been parallel transport along a coordinate line. This approach can become very confusing very quickly. For example, it is natural for “a Flatlander” on S 2 to perceive the vector field ∂φ along a meridian to be parallel to itself (thus giving us ∇∂φ ∂φ = 0), since ∂φ is always unit and always points “due South”. However, from the perspective of R3 in which S 2 is located the vector field ∂φ changes along the meridian as it bends towards the South Pole. The following question arises: Given two tangent vectors with different base points, how does one know if they are to be treated as “parallel”? Ultimately, one comes to the conclusion that parallelism and differentiation are a little bit like a chicken and an egg. If one knows how to parallel-transport a coordinate vector along a coordinate “axis”, then one can deduce the formulae which describe the rates of change of a coordinate system with respect to itself and – by means of linearity and some sort of Product Rule – the derivative of any vector field with respect to any other vector field. Conversely, if one knows how to differentiate vector fields then one can parallel transport a vector along a curve by requiring that its derivative along the way vanish. It seems that in either case one has to be open to the possibility that an additional piece of mathematical structure, parallelism or the differentiation operation for vector fields, is to be imposed on Riemannian

52

2. Tensor calculus

manifolds; it may be the case that this new piece of mathematical structure is completely independent of the Riemannian metric. 2.1.5. Connections. We expect a differentiation operation on vector fields to have two inputs: one for the direction in which we are differentiating and one for the vector field which is being differentiated. Based on our experience with differentiation operation on functions (scalar fields) we expect the map ∇ : (V, W ) → ∇V W to have the following properties: • ∇αU+βV W = α∇U W + β∇V W for all functions α, β; • ∇V (W1 + W2 ) = ∇V W1 + ∇V W2 ; • ∇V (αW ) = (∇V α)W + α(∇V W ) for all (differentiable) functions α, β. Any operation on vector fields ∇ : (V, W ) → ∇V W which satisfies the three properties stated above is called a connection. The word connection is used because differentiation (through the notion of parallel transport which we briefly discussed earlier) provides a way of relating tangent vectors based at different points. Just like a given manifold can carry many different metrics, a given Riemannian manifold can carry many different connections and consequently: many different ways of relating tangent spaces at different base points. One consequence of the defining properties of connections is that   j ∂ V i ∂x V i W j ∇∂i ∂j ∇V W = i W ∂j + ij

ij



i for all vector fields V = V i ∂i and W = W ∂i . This identity confirms the claim we made earlier that an understanding of ∇∂i ∂j i.e., how coordinate systems change with respect to themselves, together with linearity and the Product Rule, determine ∇V W in general. The coordinates Akij of ∇∂i ∂j ,  Akij ∂k , ∇∂i ∂j = k

are called the connection coefficients. Once again, they are to be thought of as some sort of derivatives of a coordinate system with respect to itself.

2.1. Differential calculus

53

If along a curve γ(t) = (x1 (t), . . . , xn (t)) we have V = γ, ˙ then along γ the vector field ∇V W depends only on W along γ:   j k (∇γ˙ W i )∂i + (W i dx ∇V W = dt Aij )∂k . i

ijk

This fact allows us to define the derivative ∇γ˙ W along γ of any vector field W defined (solely) along γ. 2.1.6. Levi-Civita connection and covariant differentiation. Geodesics on Riemannian manifolds play the role of straight lines in Rn . Tangent vectors to a straight line in Rn are parallel to one another and the tangent vector field is (in some sense) constant. Thus from a geometric standpoint it is natural to expect/require our differentiation/connection/parallelism structure to be such that tangent vectors along any geodesic remain parallel to one another: ∇γ˙ γ˙ = 0. This requirement is equivalent to  i j  2 i k d x dx ( dx dt2 ∂i + dt dt Aij )∂k = 0 i

ijk

or, equivalently, to the system of equations  i d2 xk dxj Akij dx dt2 + dt dt = 0, 1 ≤ k ≤ n. ij

It is geometrically natural to require that all geodesics satisfy this system of equations. By comparing with the geodesic equations from Lecture 1.3 we conclude that the geometrically natural connections must have Christoffel symbols as the connection coefficients:

 ∂glj ∂gil ∂gij k k kl 1 g + − . Aij = Γij = 2 ∂xi ∂xj ∂xl l

This is not to say that connections which are geometrically natural in the sense discussed above actually do exist. In fact, we have only proven that if such a connection exists it is unique because connection coefficients are the same as the Christoffel symbols. The good news is that one can exhibit a de-facto explicit formula for a connection ∇ whose coefficients are equal to Christoffel symbols Γkij : g(∇U V, W ) = 12 {∇U (g(V, W )) + ∇V (g(W, U )) − ∇W (g(U, V )) + g([U, V ], W ) − g([V, W ], U ) + g([W, U ], V )}.

54

2. Tensor calculus

A verification that the above indeed defines a connection, and that its coefficients are the Christoffel symbols Γkij could be done at this moment. However, to make the verification process feel more natural we postpone it until Exercise 5 following Lecture 2.2, the stage at which the reader will be familiar with the basics of tensor calculus. Unless specifically stated otherwise, the geometrically natural connection we just introduced should be assumed throughout these lecture notes. The connection itself is called the Levi-Civita connection in honor of the Italian mathematician Tullio Levi-Civita who, at the turn of the 20th century, participated in the discovery of tensor calculus and the development of covariant differentiation. We study both the concept of tensors and the concept of covariant differentiation in the upcoming two lectures. For now the reader may want to consider the phrase “covariant differentiation” to be synonymous with the phrase “Levi-Civita connection”, even though there is a difference. In our discussion of the Lie bracket we loosely described [V, W ] as a measure of the offset between the changes of V and W with respect to one another. In the context of Levi-Civita connections this claim can be made very precise. Observe that the Christoffel symbols are symmetric Γkij = Γkji , and that ∇∂i ∂j = ∇∂j ∂i as a result. It further follows that ∇V W − ∇W V =

j  ∂W j i ∂V − W Vi ∂j = [V, W ], ∂xi ∂xi ij

exactly as claimed earlier in this lecture. In full generality, connections which satisfy ∇V W −∇W V = [V, W ] are said to be torsion-free. Torsion-free connections include but are not limited to Levi-Civita connections.

2.1.7. Parallel transport. We finally get to go back to the idea of parallel transport mentioned earlier in the lecture. We say that a vector field V which is defined along a curve γ is parallel along γ if

2.1. Differential calculus ∇γ˙ V = 0. Since ∇γ˙ V =



55

⎛ ⎝dVk+



dt

⎞ i⎠ ∂k (Γkij dx dt )V j

ij

k

the condition that a vector field be parallel along a curve is just a system of first order ODE’s:  j k i d V + (Γkij dx dt dt )V = 0, 1 ≤ k ≤ n. ij

This is a linear system, and its (long-term) solutions are well under stood: for a given initial condition V P = V0 there is exactly one long-term solution of the system. In other words, we have the following theorem. Theorem 2.1. Let γ : I → M be a curve passing through γ(t0 ) = P . a unique vector field V For any tangent vector V0 at P there exists  which is parallel along γ and satisfies V P = V0 .  If γ is a curve joining points P and Q then the tangent vector V Q is said to be the parallel transport of V0 along γ. Even though the vector (field) ∇γ˙ V depends on the parametrization  of γ, its vanishing does not. In particular, V Q only depends on V P and a geometric path from P to Q.  Once the  path is fixed, parallel transport can be viewed as a map V P → V Q between tangent spaces at P and Q. In fact, since the parallel transport equations are linear so is this map. As it turns out, more is true: Theorem 2.2. Parallel transport between tangent spaces is an isometry. As a reminder, isometries are linear transformations which preserve the innerproduct – in our case g – on tangent spaces. In other    words, our claim is that parallel transports of vectors V P and W P satisfy:     g(V  , W  ) = g(V  , W  ). Q

Q

P

P

The proof of Theorem 2.2 is a nice little exercise which we leave for the reader. (See Exercise 9 below.) We end the lecture with two quick examples which illustrate parallel transport.

56

2. Tensor calculus

Example on H2 . Consider the Poincar´e upper half-plane model of 2 2 with y > 0. Consider the upper hyperbolic geometry, ds2 = dx y+dy 2 2 2 semi-circle γ of x + y = 1 and the vector V0 = ∂y at P (0, 1). To find the parallel transport of V0 along γ note that γ is a geodesic and that V0 is a unit vector which is orthogonal to the (tangent vector to the) geodesic. Since parallel transport serves as an isometry, and since the tangent vector field to a geodesic is necessarily parallel, we see that the parallel transport of V0 remains unit and orthogonal to γ. Vectors orthogonal to γ at a point Q(x, y) are multiples of x∂x + y∂y . By normalizing we see that the parallel transport of V0 at Q(x, y) is  V Q = y (x∂x + y∂y ) .

Figure 2.5. Parallel transport on H2

Example on S 2 . Consider the following two South-to-North meridians (geodesics): γ1 given by θ = 0 and γ2 given by θ = π2 . Let V0 be tangent to γ1 at the South Pole. Since γ1 is a geodesic, the parallel transport of V0 along γ1 remains tangent to γ1 even at the North Pole. On the other hand, V0 forms an angle of π2 with the tangent vector to γ2 at the South Pole. This means that the parallel transport of V0 remains to form the angle of π2 with the tangent vector to γ2 at the North Pole. In particular, parallel transports of V0 along γ1 and γ2 are not the same: they form the angle of π with one another; see Figure 2.6.

2.1. Differential calculus

57 z

x

γ1

γ2

y

V0 Figure 2.6. Parallel transport is path-dependent

The very important take-away message here is that the parallel transport is highly path-dependent! Prima facie, there is no way to say that a vector at P is parallel to a vector at Q without specifying a path from P to Q relative to which a parallel transport is to be performed. In particular, a parallel transport along a loop typically leads to a vector which is different from the one at the beginning of the loop. Exercises below contain several examples of this kind. 2.1.8. Exercises for Lecture 2.1. Computations in coordinates. (1) Compute the following commutators in R2 : (a) [x∂x , −y∂x + x∂y ]; (b) [y∂y , −y∂x + x∂y ]; (c) [x∂x + y∂y , −y∂x + x∂y ]. (2) Show that for a given function f and a given vector field ∂f V the value of i V i ∂x i does not depend on the choice of coordinates. (3) Let {∂i } and {∂α } be the coordinate vector fields corresponding to the x and the y coordinates, respectively, and

58

2. Tensor calculus let V and W be two vector fields. Prove the following equality.

j β  ∂W β  ∂W j i i ∂V α α ∂V −W −W V ∂j = V ∂β . ∂xi ∂xi ∂y α ∂y α ij αβ

(4) Consider the standard Cartesian coordinates (x, y) and the standard polar coordinates (r, θ) in the Euclidean plane. Recall that ∂r = √ 2x 2 ∂x + √ 2y 2 ∂y and ∂θ = −y∂x + x∂y . x +y

x +y

(a) Compute ∇∂r ∂r , ∇∂θ ∂r , ∇∂r ∂θ and ∇∂θ ∂θ by using the above. Express your answers in terms of ∂r and ∂θ . (b) Compute ∇∂r ∂r , ∇∂θ ∂r , ∇∂r ∂θ and ∇∂θ ∂θ by using Christoffel symbols. Compare with the above. (5) This problem concerns the standard unit sphere. (a) Compute ∇∂θ ∂θ , ∇∂θ ∂φ , ∇∂φ ∂θ and ∇∂φ ∂φ .   (b) Compute ∇∂φ (∇∂θ ∂θ ) and ∇∂θ ∇∂φ ∂θ . Does the order of covariant differentiation matter in this case?   (c) Compute ∇∂φ (∇∂θ ∂φ ) and ∇∂θ ∇∂φ ∂φ . Does the order of covariant differentiation matter in this case? Parallel transport. (6) (a) Find the equations of parallel transport along the lines of constant latitude φ = φ0 on the surface of standard S 2 . (Assume standard spherical coordinates (θ, φ).) (b) Find the parallel transport of a vector ∂φ along a line of constant latitude. Illustrate your solution for φ0 = π4 , φ0 = π3 and φ0 = π2 . (c) Foucault1 pendulum swings freely in a plane orthogonal to the surface of the Earth. The only force impacting the swing is the gravitational force, and no forces tangential to S 2 are impacting the plane in which the pendulum swings. Thus the tangent vector to S 2 determining the plane of the pendulum remains parallel 1 The pendulum is named in honor of French physicist L´ eon Foucault who first publicly displayed it in 1851.

2.1. Differential calculus

59

throughout the day, that is, it remains parallel along the line of constant latitude φ0 . Find the equations describing the motion of the Foucault pendulum. In particular, prove that the period of the Foucault pen1 days. dulum is cos(φ 0) (7) Find the parallel transport of V0 along the curve γ, for the following choices of γ on S 2 . In each case, illustrate what the parallel transport does along the curve. (a) γ is the loop consisting of two meridians forming angle α at the North Pole P ; (b) γ is the contour of a (geodesic) triangle with one vertex at the North Pole P while the other two vertices are on the Equator; (c) γ is the (piece-wise smooth) counterclockwise contour of any other geodesic triangle with one vertex at the North Pole P . In each case assume that V0 to be transported is the unit tangent to the meridian leaving the North Pole P . (8) Consider the Poincar´e upper half-plane model of hyperbolic geometry. (a) Find the equations of parallel transport along γ1 (t) = (t, 1) and γ2 (t) = (0, t). (b) Find the parallel transport of V0 = ∂y at P (0, 1) along γ1 and γ2 . In each case, illustrate what the parallel transport does along the curve. (9) (a) Let V and W be two vector fields which are parallel d g(V, W ) = 0. along a curve γ. Show that dt (b) Show that parallel transport along a curve γ is (i) a linear transformation; (ii) an isometry.

60

2. Tensor calculus (10) Let γ(t) be a curve through a point P = γ(0) on a Riemannian manifold. Let Φt denote parallel transport along γ from the tangent space at P to the tangent space at γ(t). Show that     1 (Φt )−1 V γ(t) − V P (∇γ˙ V ) P = lim t→0 t for all vector fields V along γ.

2.2. Tensors 2.2.1. Introduction. The main goal of this lecture is to develop a framework and a language of tensors which then allows us to work with higher derivatives more efficiently. As an application we introduce gradient, divergence and Laplace operators compatible with a Riemannian metric. Theorem-wise, we work our way to the Divergence Theorem and the Maximum Principles – particularly in the context of compact manifolds (without boundary). 2.2.2. Preliminary linear-algebraic considerations. Thus far we have viewed differentiation as an operation with two inputs: one for the object being differentiated (function, vector field, etc,...) and one for the direction of differentiation. We are about to shift perspectives again, fix the object being differentiated – a function f , for example – and deal with ∇f , a linear map defined by ∇f : V → ∇V f. Although one might be tempted to think of ∇f as the gradient vector field of f this would be incorrect: ∇f is not a vector field. Instead, it is what is known as a covector2 field: a linear map which takes vector fields to functions. A more thorough understanding of the underlying linear algebraic framework pays dividends here! Vector fields form a structure similar to that of a vector space in which the role of scalars is played by functions (scalar fields). Namely, one can add vector fields (V + W ) and multiply them with functions (αV ). These operations result in 2 It is not a coincidence that the phrases co-variant differentiation and co-vector field have the same prefix!

2.2. Tensors

61

vector fields and obey all the usual vector space rules. We use F to denote the set of scalar fields and X to denote the set of vector fields. Unless specifically stated otherwise one may assume that the coordinate representation of any element in X and F is smooth. In some sense X forms a “vector space”3 over F. Covector fields are F-linear maps from X to F; the set of covector fields will be denoted by X∗ . Since ∇αV +βW f = α∇V f + β∇W f for all α, β ∈ F, covariant derivatives ∇f are examples of covector fields. In addition, covariant derivatives of vector fields are also Flinear maps which take vector fields to vector fields: ∇X : X → X, ∇X : V → ∇V X. A natural approach to higher-order differentiation is to proceed inductively, e.g. ∇2 f = ∇(∇f ). To make sense out of this idea we need to first develop a strategy for differentiating covector fields such as ∇f . Whatever ∇2 f may be we expect it to need two vector fields as inputs: one for the direction of differentiation, e.g. ∇W (∇f ), and the other for the vector field, e.g. V , on which the directional derivative is to be evaluated, e.g. ∇2 f (V ; W ) = ∇W (∇f ) (V ). In general, we expect ∇k f to be a mapping of k vector fields. We leave the discussion of higher-order differentiation for Lecture 2.3 and instead pause to address the linear-algebraic framework in more detail. 2.2.3. Tensor fields. In linear algebra one studies multilinear mappings, that is, mappings L(V1 , . . . , Vk ) which are linear in each entry. The concept of a tensor product is designed to identify the set of multilinear maps with the set of linear maps on an appropriately modified 3 There are two issues here. One is that F is not a field but a ring; thus the more appropriate word to use here is that of a module. The other issue is that we have made no commitments about the domains of our functions and vector fields. (Are these defined on all of M , or just on portions of M ?) In the interests of brevity we avoid dealing with this issue in detail. The upshot is that for the most part we are able to treat locally defined fields/functions in essentially the same manner as the fields/functions which are globally defined.

62

2. Tensor calculus

domain. In view of this, our multilinear maps are called tensor fields, or tensors for short. Terminology. We use the phrase (k, 0)-tensor field (resp.(k, 1)-tensor field ) for a F-linear map taking k vector fields to a scalar field (resp. vector field). Alternatively, (k, 0)-tensor fields are called covariant ktensor fields. Note that in this terminology vector fields are the same as (0, 1)-tensor fields and Riemannian metrics are positive definite symmetric covariant 2-tensors. It is interesting to observe that vector fields can themselves be thought of as F-linear mappings of covector fields to scalar fields: V : ω → ω(V ). More generally, a (k, 1)-tensor field can be seen as an F-linear mapping from covector fields to covariant k-tensor fields: T : ω → ω ◦ T. In these lecture notes we stop short of discussing the more general (k, l)-tensor fields and only remark that they could be understood at F-multilinear mappings of l covector fields to the codomain of covariant k-tensor fields. Why F-linearity is so important. If a map L(. . . , V, . . . ) is F-linear in the V -entry then the value of L(. . . , V, . . . ) at a particular point P depends only on the value of V at P , and not on any nearby values of V . For instance, if V, W ∈ X are the same at P then in coordinates V − W = i αi ∂i with each of the functions αi vanishing at P . Since L is F-linear in the V -entry we have: L(. . . , V, . . . ) − L(. . . , W, . . . ) =L(. . . , V − W, . . . )  αi L(. . . , ∂i , . . . ) = 0. = i

Thus, the values of L(. . . , V, . . . ) and L(. . . , W, . . . ) are equal at P . The concept of F-linearity might appear innocent at first, but is in fact quite deep and valuable. Although nominally covector fields act on vector fields, they restrict to a single point and give a welldefined operation on tangential directions/vectors based at that single point. For example, restricting ∇f to a point produces a linear map Rn → R; its rule can be described as “give me a direction based at

2.2. Tensors

63

P and I can find the rate of change of f in that particular direction at P ”. Likewise, restricting ∇X to a point produces a linear map Rn → Rn . Compare this situation with the behavior of ∇f and ∇X in f and X-entries, respectively. In place of F-linearity it is the Product Rule that applies: ∇V (αX) = (∇V α)X + α∇V X = α∇V X. It particular, ∇V f and ∇V X do not restrict to a single point in the sense of the previous paragraph. This is very intuitively clear: knowledge of a vector X at a point alone cannot possibly be sufficient for understanding the outcome of the differentiation ∇V X at that point. Maps L(. . . , V, . . . ) which are F-linear in the V -entry are said to be tensorial in the V -entry. For example, ∇V X is tensorial in the V -entry but it is not tensorial in the X-entry. If L is tensorial in V then, the value of L(. . . , V, . . . ) at a particular point P ∈ M depends only on the value of V at P , and not on any nearby values of V . Consequently, all tensor fields can be restricted to a point! At each point P ∈ M a covariant tensor T restricts to an R-multilinear map  which takes tangent vectors at P to real numbers: that is, T P : Rn × ... × Rn → R. At times it is fruitful to think of tensor fields as ordinary multilinear maps which change from point to point. 2.2.4. Index notation. Another consequence of F-multilinearity of great interest to us is the fact that a tensor T is completely determined by the set  {T (∂i1 , ∂i2 , ..., ∂i )  1 ≤ i1 , ..., ik ≤ n}. k

For reasons of notational efficiency we write Ti1 ,...,ik := T (∂i1 , ..., ∂ik ) in the case of (k, 0)-tensors, and Tii1 ,...,ik in the case of (k, 1)-tensors:  T (∂i1 , ..., ∂ik ) = Tii1 ,...,ik ∂i . i

k i j For example, (2, 1)-tensors T satisfy T (V, W ) = ijk Tij V W ∂k . i The functions Ti1 ,...,ik and Ti1 ,...,ik are called tensor components. It is worth emphasizing that despite the notational similarities connection

64

2. Tensor calculus

coefficients (e.g. the Christoffel symbols Γkij ) are not components of a tensor. Further details on this can be found in Exercise 6 below. To give the reader an opportunity to adapt to these conventions we offer some examples of index notation: • For a covector field ω and a vector field V we have   ω(V ) = V i ω(∂i ) = ωi V i . • By (∇X)ji we mean the jth component of ∇∂i X:  (∇X)ji ∂j . ∇∂i X = j

• It then follows that (∇V X)j =

i

V i (∇X)ji .

We should point out that the locations of upper and lower indices are strategically chosen so that summation formulas always have one upper and one lower index. Careful placement of upper and lower indices makes it very easy to keep track of ways in which tensor components transform under changes of coordinates. For example, components of a (1, 1)-tensor T transform as follows:  ∂y α ∂xj T β. Tij = ∂xi ∂y β α αβ

In linear algebra courses the above corresponds to the claim about similarities of matrix representations of linear transformations in different coordinates: [Tij ] = (dΦyx )−1 [Tαβ ](dΦyx ). The reader is encouraged to complete Exercise 1 for further examples of how tensor components change under the change of coordinates. The following example further illustrates how to make use of index notation:      ∂y α ∂xi i Ti = Tαβ i ∂y β ∂x i i αβ

=

 ∂y α αβ

∂y β

Tαβ =



Tαα .

α

There is more to this example than just gymnastics with indices: it shows that the scalar quantity i Tii is coordinate independent and

2.2. Tensors

65

thus globally defined. In other words, we have proved that to any (1, 1)-tensor field T we can associate a scalar field called the trace of T and defined by  Tr(T ) := Tii . i

The operation of taking the trace is commonly known as index contraction. The contraction of indices is an operation of summing over repeated indices with one being upper and the one being lower. The computation presented above in effect shows that contraction of indices turns a (k, 1)-tensor into a (k − 1, 0)-tensor. Lecture 3.2 below illustrates well the fact that index contraction serves as a form of averaging. 2.2.5. Raising and lowering indices. Those familiar with the concept of a vector space dual will recognize that in some sense covector fields are dual to vector fields. In fact, Riemannian metric gives rise to a natural correspondence between X and X∗ . Let us address this phenomenon in more detail. If V ∈ X, then the map W → g(V, W ) is F-linear and defines a covector field. This covector field is commonly denoted4 by V , and should be thought of as a covector counterpart (dual) to V . Allowing V to vary as well, we arrive at

: X → X∗ , defined by

V → {W → g(V, W )}.

In coordinates we have V (∂i ) = (V )i =



gij V j .

j

Motivated by these notational effects of the operation is also known under the phrase lowering an index . It can be easily checked that is 1-1; see Exercise 2a. More interestingly, the map is onto. That is, for each ω ∈ X∗ there is a vector field V such that g(V, W ) = ω(W ) for all W ∈ X. The existence of such a vector field V is most easily established in coordinates: one 4

We read V as “V -flat”.

66

2. Tensor calculus

can easily verify that   V = V i ∂i where V i = g ij ωj i

j

satisfies the relationship g(V, W ) = ω(W ). In order to verify that V is defined as globally as ω is, one also needs to verify that the expression for V is independent of the choice of coordinates. We leave this task to the reader; see Exercise 2b following this lecture. The vector field V corresponding to the covector field ω in the manner described above is commonly denoted5 by ω  . The fields ω and ω  are to be thought of a being duals of one another. The map 

: X∗ → X, ω → ω 

is the inverse of the map . Notationally  seems to raise an index of a tensor component, as is thus commonly referred to by raising an index . The operations of raising and lowering indices apply more broadly to tensors of all types. For example, if T is a (1, 1)-tensor field then T (V, W ) := g(T (V ), W ) is a (2, 0)-tensor. In components we have  (T )ij := gjk Tik , k

although one should be warned that the roles of V and W need not be symmetric and that as a result we need to be very clear with the order of indices. Likewise, for a (3, 0)-tensor field T  g kl Tilj (T  )kij := l

defines a (2, 1)-tensor field. The notation (T  )kij , however, does not indicate that it is the second index that is being raised. Whenever confusion is possible it is prudent to include some spaces in the notation. For example, (T  )i kj indicates that the second index is raised while (T  )ij k indicates that the third index is raised. Further examples of raising and lowering indices can be found in exercises following 5

We read ω  as “ω-sharp”.

2.2. Tensors

67

this lecture. Finally, we would like to point out that the operations of raising and lowering of indices are F-linear. 2.2.6. The gradient vector field. Motivated by the properties of the gradient vector field for functions on Rn we define gradient vector field grad(f ) ∈ X(M ) to be the unique vector field for which g(grad(f ), V ) = ∇V f. In other words, we define grad(f ) to be the vector field dual to ∇f : grad(f ) = (∇f ) . In coordinates we have grad(f ) =



∂f g ij ∂x j ∂i .

ij

As always, the gradient vector field is orthogonal to the level sets of f and it points in the direction of increase of f . At a point of a local extremum the gradient vanishes. We expect the gradient to spread away from the point of local minimum and to converge towards the point of local maximum. This observation ties to the concept of divergence of a vector field. 2.2.7. Divergence (Theorem). Reasons for introducing the concept of divergence. Divergence of a vector field is a concept designed to measure the amount of outward flux across an (infinitesimal) surface surrounding a point per amount of enclosed volume. More specifically, if Ω1 ⊇ Ω2 ⊆ Ω3 ⊇ ....  P is any chain of compact regions with (“nice”) boundaries ∂Ωn shrinking towards a single point P which they all have in their interior6 then  X, ν dvol∂Ωn . (2.2) div(X) = lim ∂Ωn n→∞ vol(Ωn ) (Here and elsewhere in these lecture notes ν is reserved to denote the outward pointing unit normal to the domain of the indicated volume element.) 6

In point-set topology notation:

 n

Int(Ωn ) = {P }.

68

2. Tensor calculus

Definition of divergence from multivariable calculus. Introductory multivariable calculus courses tend to define the divergence of a vector field X within Rn in terms of its coordinate representation with respect to Cartesian coordinates (x1 , ..., xn ): div(X) =

 ∂X i i

∂xi

.

In hindsight, this is simply the trace of the (1, 1)-tensor ∇X: div(X) = Tr(∇X). The perspective according to which div(X) measures the rate of outward flux is then seen as a consequence of Gauss’s Divergence Theorem   X, ν dvol∂Ω .

div(X)dvolΩ = Ω

∂Ω

While the expression Tr(∇X) is coordinate-invariant and easily extendible to general Riemannian manifolds, it is not at all clear that it would lend itself to a divergence theorem and the perspective outlined in (2.2). For this reason we examine the behavior of the right-hand side in (2.2) under an orientation-preserving mapping Φyx . Formula for divergence in curvilinear coordinates in Rn . The expression X, ν dvol∂Ωn measures the volume of the parallelotope with infinitesimal base on ∂Ωn and X as the remaining side. Upon applying dΦyx this parallelotope maps to the parallelotope with infinitesimal base on ∂(Φyx Ωn ) and dΦyx (X) as the remaining side. The volume of the image parallelotope is det(dΦyx ) times larger. We thus have dΦyx (X), ν dvol∂(Φyx Ωn ) = det(dΦyx ) X, ν dvol∂Ωn = det(dΦyx ) X, ν dvol∂Ωn . It follows that the outward flux of dΦyx (X) across ∂(Φyx Ωn ) is the same as the outward flux of det(dΦyx ) X across ∂Ωn . Since dvolΦyx Ωn = det(dΦyx )dvolΩn , the rate of flux of dΦyx (X) is det(dΦyx ) times smaller than the rate of flux of det(dΦyx ) X: (2.3)

div(dΦyx (X)) =

1 div(det(dΦyx ) X). det(dΦyx )

2.2. Tensors

69

Next, take the perspective that x denotes some (curvilinear) coordinates on Rn while y denotes the standard Cartesian coordinates. Within this context the above simply reads as:    α i ∂ ∂ √1 |g|). ∂y α (X ) = ∂xi (X |g|

α

i

We now see that Tr(∇X) = √1

|g|



i ∂ ∂xi (X

 |g|),

i

regardless of our choice of coordinates on Rn . Defining divergence on Riemannian manifolds. On general Riemannian manifolds the identity (2.3) proves that the expression   i ∂ √1 |g|) ∂xi (X |g|

i

is independent of the choice of coordinates (compare with (1.3)) and thus defines a global function. It is this function that we call the divergence of X. The coordinate expressions for divergence and   i ∂ Tr(∇X) = X + Γiij X j i ∂x i

ij

are both rational expressions in components X i and gij , and their first derivatives ∂X i and ∂gij . The fact that two rational expressions are identical in Rn suggests that       i i j 1 ∂ ∂ √ X + Γ X = X i |g| i i ij ∂x ∂x i

ij

|g|

i

for exclusively algebraic reasons. While this is true it is neither a partcularly pleasant nor an illuminating algebraic computation. Instead one may want to observe that the two expressions are equal because they are both invariant under change of coordinates and because for each location coordinates can be chosen7 so that ∂g = 0, ∂ i i.e., Tr(∇X) = i ∂x i X = div(X) there. 7 We discuss such coordinates in great detail in Lecture 3.1. approach to divergence consult Exercise 4 following Lecture 3.2.

For yet another

70

2. Tensor calculus

Divergence Theorem. Given the motivation behind our definition of divergence the following does not come as much of a surprise. Theorem 2.3 (Divergence Theorem). Let Ω be a region of a Riemannian manifold M with piecewise smooth boundary ∂Ω and outward pointing unit normal ν. We have   div(X) dvolΩ = X, ν dvol∂Ω . Ω

∂Ω

In particular, if M is compact and without boundary we have:  div(X) dvolM = 0. M

We work out the proof of the Divergence Theorem in the case when the region Ω is a coordinate box. The case of general Ω is handled by breaking Ω into box-like regions and addition; note that addition of boundary integrals contributed by two adjacent boxes leaves only the outmost boundary integral. In the case when the region of integration is compact and without boundary all the boundary integrals cancel, leading to the second statement of our theorem. Sketch of a proof: Assume that in some coordinates the region Ω takes the form of B := [−c1 , c1 ] × ... × [−cn , cn ]. We start by examining what the statement of the Divergence Theorem looks like in these coordinates. First observe that the unit normal to the boundary component given by xa = ±ca has the coordinate expres aj a sion ± j √ggaa ∂j . Therefore, X, ν = ± √Xgaa . The volume element for this component of the boundary is determined (pun intended!) through the submatrix of [gij ] obtained by removing the ath row and the ath column; according to the formula for the inverse matrix the absolute value of the determinant of this submatrix is equal to g aa |g|. Overall, we need to argue that:      a ...dxn ∂i (X i |g|)dx1 ...dxn = X a |g| dx1 ...dx B

a

i

−

xa =ca

 a

Xa



a ...dxn , |g| dx1 ...dx

xa =−ca

a signifies that dxa has been omitted. The latter follows where dx  immediately from the Divergence Theorem in Rn .

2.2. Tensors

71

The following is an immediate consequence of the Divergence Theorem and the identity div(f X) = ∇X f + f div(X). (See Exercise 7 following this lecture.) Theorem 2.4 (Integration by Parts). If M is compact without boundary, then    f div(X)dvolM = − gradf, X dvolM = − (∇X f )dvolM . M

M

M

2.2.8. The Laplace operator and Maximum Principles. We define the Laplace operator Δ (also know as the Laplacian) by Δf = div (grad(f )) . The coordinate expression for Δf can easily be found from the coordinate expression for gradient and divergence:  1  ∂i (g ij |g|∂j f ). Δf =  |g| ij The Laplacian carries a certain amount of information about interior local extrema of a function. For example, at a point of interior local maximum the divergence of the gradient (being the rate of outward flux of the gradient) has to be8 non-positive. This proves the following: Theorem 2.5 (Weak Maximum Principle). If Δf > 0 over a region Ω then f cannot attain a local maximum inside Ω. In particular, the maximum of f can only be attained on the boundary ∂Ω. By replacing f by −f one obtains the corresponding Weak Minimum Principle. Oftentimes the Maximum Principle is used in one of the following forms. Theorem 2.6 (Weak Maximum Principle on Compact Manifolds without Boundary). Let M be compact without boundary, let c be a positive function, and let Lf := Δf − cf . (1) If Lf ≥ 0 on M , then f ≤ 0. (2) If Lf ≤ 0 on M , then f ≥ 0. 8 Readers interested in more explicit details behind this claim are referred to Exercise 9 following this lecture.

72

2. Tensor calculus (3) If Lf = 0 on M , then f = 0 everywhere.

Proof. It suffices to prove the first claim. Suppose the opposite: that f achieves a positive interior maximum. At the point where this interior maximum is achieved we must have Δf − cf ≤ −cf < 0, which contradicts our assumptions.  Maximum Principles are one of the most often used tools in geometric analysis. They are commonly used in tandem with the following (immediate) consequence of Integration by Parts (Theorem 2.4): Theorem 2.7. Suppose α and β are functions over a compact man ifold M without boundary. We then have M Δα dvolM = 0 and    α(Δβ)dvolM = − gradα, gradβ dvolM = (Δα)βdvolM . M

M

M

To illustrate the point we just raised let us prove the following. Theorem 2.8. Let M be compact without boundary, let c be a nonnegative function, and let Lf := Δf − cf . If Lf = 0 then f must be constant. If in addition c is not identically zero, then we must have f = 0. Proof. A function f which satisfies Lf = 0 must also satisfy   f · Lf dvolM = f (Δf ) − cf 2 dvolM 0= M M  =− gradf 2 + cf 2 dvolM . M

 Since c ≥ 0 we have M gradf 2 + cf 2 dvolM ≥ 0 with equality only in the case that both gradf = 0 and cf 2 = 0. We now see that f is a constant and unless c = 0 we must have f = 0.  There is a stronger version of the above, applicable to situations when the region under consideration has a boundary or is not compact. We only state it here – its proof goes beyond the scope of these lectures. A proof can be found, for example, in [4]. Theorem 2.9 (Strong Maximum Principle).

2.2. Tensors

73

(1) If Δf ≥ 0 over a region Ω then f cannot attain a local maximum inside Ω unless it is a constant. In particular, the maximum of f can only be attained on the boundary ∂Ω. (2) Let c be a non-negative function, and let Lf := Δf − cf . If Lf ≥ 0 over a region Ω then f cannot attain a nonnegative local maximum inside Ω unless it is a constant. If in addition c is not identically zero, then this constant is zero. 2.2.9. Exercises for Lecture 2.2. Tensors. (1) (a) Let ωi and ωα , respectively, be components of a covector field ω in x- and y-coordinates. Prove that  ∂y α ωα . ωi = ∂xi α (b) Similarly, show that for (2, 1)-tensors T we have Tijk =

 ∂y α ∂y β ∂xk γ T . ∂xi ∂xj ∂y γ αβ

αβγ

(c) Let T be a (2, 1)-tensor field. Show that  ωj = Tiji i

defines a globally defined covector field. (2) (a) Show that the map lecture is 1-1.



: X → X∗ introduced in this

(b) Let ω be a covector field. Show that the vector field  (g ij ωj )∂i ij

is invariant under the change of coordinates and is thus a globally defined vector field.

74

2. Tensor calculus (3) (a) Let ωi be the components of a covector field ω with respect to x-coordinates. Show that, on the domain of x-coordinates, we have  ωi ∇xi . ω= i

(b) Show that the set of covectors based at a point of an n-dimensional manifold forms an n-dimensional vector space. Exhibit one of its bases. (4) Let T be a (2, 0)-tensor. (a) Prove that there is a unique (1, 1)-tensor L such that g(L(V ), W ) = T (V, W ). (b) Show that Tr(L) = ij g ij Tij . Connections. (5) Let T (U, V, W ) := 12 {∇U (g(V, W )) + ∇V (g(W, U )) − ∇W (g(U, V )) − g(U, [V, W ]) + g(V, [W, U ]) − g(W, [U, V ])}. (a) Show that for a fixed vector field U the expression (V, W ) → T (U, V, W ) is a (2, 0)-tensor. (b) Define ∇U to be the (1, 1)-tensor obtained by raising the last index of T : g((∇U )V, W ) = T (U, V, W ) and set ∇V U := (∇U )V. Show that ∇V U satisfies the requirements to be a connection. (See Section 2.1.5.) (c) Show that the connection coefficients of ∇ are equal to the Christoffel symbols Γkij . (6) (a) Let ∇ be some connection. Let Akij and Aγαβ be its coefficients computed with respect to x- and y-coordinates respectively. Find the relationship between Akij and Aγαβ .

2.2. Tensors

75

(b) Let ∇ and D be two connections. Show that (V, W ) → ∇V W − DV W is a (2, 1)-tensor. Vector calculus. (7) (a) Verify the formula div(f X) = ∇X f + f div(X).   (b) Express Ω (Δα)βdvolΩ − Ω α(Δβ)dvolΩ in terms of integrals over the boundary ∂Ω. (8) Let f be a spherically symmetric function on Rn , i.e., let f be a function on Rn which depends only on r = |x|. (a) Show that the Euclidean Laplacian satisfies Δf =

∂2f n − 1 ∂f . + ∂r 2 r ∂r

(b) Use the above to find all spherically symmetric solutions of Δf = 0 in the Euclidean space. (9) Suppose f reaches an interior local maximum at a point P . (a) Consider a geodesic γ(t) passing through P . Prove that the following holds at P : d g(gradf, γ) ˙ = g(∇γ˙ gradf, γ) ˙ ≤ 0. dt (b) Show that Δf = div(gradf ) ≤ 0 at P . (10) Let c be a non-negative function, and let Lf := Δf − cf . Prove the following: (a) If Lf ≥ 0 over a region Ω, and if f ≤ 0 along the boundary ∂Ω, then f ≤ 0 throughout Ω. (b) If Lf1 ≥ Lf2 over a region Ω, and if f1 ≤ f2 along the boundary ∂Ω, then f1 ≤ f2 throughout Ω. (c) If Lf1 = Lf2 over a region Ω, and if f1 = f2 along the boundary ∂Ω, then f1 = f2 throughout Ω.

76

2. Tensor calculus

2.3. Differentiation of tensors 2.3.1. Introduction. The primary goal of this lecture is to make sense out of second and higher derivatives. As discussed in Lecture 2.2 we expect higher covariant derivatives ∇k f and ∇k X to be (k, 0)and (k, 1)-tensor fields, respectively. Thus the main step in the process of understanding higher derivatives, e.g., ∇k+1 f = ∇(∇k f ), is to understand ∇T for general tensor fields T . As we are about to see, there is a somewhat unexpected feature of the second covariant derivative of tensors. Ultimately, this feature leads to the notion of curvature. 2.3.2. Covariant differentiation of tensors. Being bilinear the mapping (ω, V ) → ω(V ) for covector fields ω and vector fields V can be interpreted as a form of multiplication. This perspective is further encouraged by the coor dinate expression ω(V ) = i ωi V i . As a result, it is appropriate to impose a formal Product Rule relationship ∇W (ω(V )) = (∇W ω)(V ) + ω(∇W V ) between the rates of change in the direction of W . Guided by this we define ∇ω by ∇ω(V ; W ) := ∇W (ω(V )) − ω(∇W V ). (Since symmetry of ∇ω is not being taken for granted we choose to use the semi-colon to distinguish the directional derivative entry whenever confusion is possible.) While it is clear from the definition that ∇ω(V ; W ) is tensorial in the W -entry, the presence of ∇W V in the formula for ∇ω(V ; W ) makes it much less clear that ∇ω is tensorial in the V -entry. This is something that should be checked directly: ∇ω(αV ; W ) = ∇W (αω(V )) − ω(∇W (αV )) = (∇W α)ω(V ) + α∇W (ω(V )) − (∇W α)ω(V ) − αω(∇W V ) = α (∇W (ω(V )) − ω(∇W V )) = α∇ω(V ; W ). Thus, as expected, ∇ω is a (2, 0)-tensor.

2.3. Differentiation of tensors

77

i j In general, identities such as T (U, V ) = suggest ij Tij U V some kind of an extended Product Rule formula: ∇W (T (U, V )) = (∇W T )(U, V ) + T (∇W U, V ) + T (U, ∇W V ), which in turn motivates us to define the covariant derivative of a (k, i)-tensor field T , with i = 0, 1, by: ∇T (V1 , . . . , Vk ; W ) :=∇W (T (V1 , . . . , Vk )) −T (∇W V1 , . . . , Vk ) − · · · − T (V1 , . . . , ∇W Vk ). One important point of the story here is that as soon as we know how to differentiate scalar fields (functions) and vector fields we also know how to differentiate covector and more general tensor fields. As above, one can verify directly that ∇T for a (k, i)-tensor T is a (k + 1, i)-tensor. 2.3.3. Examples. Computing in coordinates. It is important to gain some experience in computing coordinate expressions for covariant derivatives. Here we include the example of the covariant derivative of a covector field ω, but the reader should make sure to do at least one more example on their own (e.g. Exercise 1). ∂ ∂xj (ω(∂i )) − ω(∇∂j ∂i ) ∂ ω( k Γkji ∂k ) = ∂x j (ωi ) −

(∇ω)i;j =(∇∂j ω)i = ∂ = ∂x j (ωi ) −



k k Γij ωk .

Differentiating the metric tensor. To compute the covariant derivative of the metric tensor it suffices to find the components of ∇g in coordinates. We have: (∇g)ij;k = = =

∂gij ∂xk ∂gij ∂xk ∂gij ∂xk

− g(∇∂k ∂i , ∂j ) − g(∂i , ∇∂k ∂j ) − l Γlki glj − l Γlkj gli ∂g

− 12 ( ∂xijk +

∂gjk ∂xi

−

∂gik ∂xj )

∂g

− 12 ( ∂xijk +

∂gik ∂xj

In hindsight, we have just proven an important result. Theorem 2.10. We have: (1) ∇g = 0.

−

∂gjk ∂xi )

= 0.

78

2. Tensor calculus (2) ∇W (g(U, V )) = g(∇W U, V ) + g(U, ∇W V ).

In some very loose sense of the word, Levi-Civita connections treats the metric as being constant. Consequently, the covariant derivative with respect to the Levi-Civita connection commutes with operations of raising and lowering indices. For example, one can show that ∇(V ) = (∇V ) ; for further details the reader is encouraged to work through Exercise 6 following this lecture. 2.3.4. Hessian and (the lack of ) its symmetry. The higher covariant derivatives are defined inductively as follows: the kth covariant derivative of a tensor T is defined as ∇k T = ∇(∇k−1 T ). The second covariant derivative is called the Hessian. For example, the Hessian of a function f satisfies ∇2 f (V, W ) = ∇W (∇V f ) − ∇∇W V f and its coordinate expression is given by:  ∂f ∂2f − Γkij k . ∇2 f (∂i , ∂j ) = i j ∂x ∂x ∂x k

(The correspondence between ∇grad(f ) and ∇2 f is investigated in Exercise 6.) Since the coordinate expression for ∇2 f is symmetric in i and j the tensor ∇2 f is itself symmetric: ∇2 f (V, W ) = ∇2 f (W, V ). The symmetry result is perhaps expected as ordinary partial derivatives of a function commute. Before proceeding it is worthwhile to remind ourselves of reasons why ordinary partial derivatives (of twice continuously differentiable scalar-valued functions) commute. Consider the expression f (x1 + t, x2 + s) − f (x1 + t, x2 ) − f (x1 , x2 + s) + f (x1 , x2 ) . st  ∂f  , In the limit as (s, t) → (0, 0) this expression approaches ∂x∂ 2 ∂x 1 (x1 ,x2 ) as is evidenced by the following grouping of terms (and the Mean (2.4)

2.3. Differentiation of tensors

79

Value Theorem):

1 f (x1 + t, x2 + s) − f (x1 + t, x2 ) f (x1 , x2 + s) − f (x1 , x2 ) − t s s

∂f 1 2 ∂ ∂f  1 ∂f 1 2 (x , x + s) − 1 (x∗ , x ) = . =  s ∂x1 ∗ ∂x ∂x2 ∂x1 (x1∗ ,x2∗ ) The same argument executed on the alternative grouping of terms

1 f (x1 + t, x2 + s) − f (x1 , x2 + s) f (x1 + t, x2 ) − f (x1 , x2 ) − s t t  ∂ ∂f  . Thus, proves that the limit as (s, t) → (0, 0) of (2.4) is ∂x 1 ∂x2 (x1 ,x2 ) the two mixed derivatives must be the same. Note just how sensitive the above argument is to f being scalarvalued. The vector field version of the  expression (2.4) has to involve parallel transport. In place of X (x1 +t,x2 ) we would need to  use the parallel transport of X (x1 +t,x2 ) along the x1 -grid-line back  to (x1 , x2 ). Similarly, X (x1 ,x2 +s) would need to be parallel transported along the x2 -grid-line back to (x1 , x2 ). However, it is unclear which path to use to transport X (x1 +t,x2 +s) back to (x1 , x2 ). Parallel transport is path-dependent (cf. Figure 2.6) and it is conceivable that parallel transport back along the x2 -grid-line followed by parallel transport back along the x1 -grid-line would produce a different outcome than the parallel transport with the roles of the grid-lines switched. As the computations in the cases of scalar functions indicate, one route yields ∇∂s ∇∂t X while the other yields ∇∂t ∇∂s X. Readers who have completed Exercise 5 following Lecture 2.1 have already experienced this phenomenon: the result of the exercise is that on the 2-dimensional standard unit sphere we have ∇∂φ (∇∂θ ∂φ ) = −∂θ = 0 = ∇∂θ (∇∂φ ∂φ ). Consequently, one needs to be very careful in situations where one would like to exchange ∇V ∇W X for ∇W ∇V X, even when V and W are coordinate vector fields: ∇∂i (∇∂j X) = ∇∂j (∇∂i X).

80

2. Tensor calculus

In fact, what is true is that (2.5)

∇∂i (∇∂j X) − ∇∂j (∇∂i X) =

lim

(t,s)→(0,0)

Φt,s X − X , ts

where Φt,s denotes the parallel transport of X along the coordinate t × s rectangle whose sides are located along xi and xj grid-lines (in that order). Working out the details of the identity (2.5) (which is assigned in Exercise 5 following this lecture) provides an excellent opportunity for the reader to internalize this very crucial moment. 2.3.5. The definition of the curvature tensor. It follows from the above discussion that we should not expect the Hessian of a vector field to be symmetric! The following computation examines the symmetry (or the lack thereof) of the Hessian of a vector field X: ∇2 X(V ; W ) − ∇2 X(W ; V ) = (∇W (∇V X) − ∇∇W V X) − (∇V (∇W X) − ∇∇V W X) =∇W (∇V X) − ∇V (∇W X) − ∇∇W V −∇V W X =∇W (∇V X) − ∇V (∇W X) − ∇[W,V ] X. The very last equal sign is due to the fact that the Levi-Civita connection is torsion-free: ∇W V − ∇V W = [W, V ]. In the example of the 2-dimensional standard unit sphere we have: (∇2 ∂φ )(∂φ , ∂θ ) − (∇2 ∂φ )(∂θ , ∂φ ) = ∂θ . The example of S 2 shows very clearly that in general the Hessian of a vector field is not symmetric. In this lecture, and throughout the course, we use Riem(V, W )X to denote the curvature operator ∇2 X(W ; V ) − ∇2 X(V ; W ) = ∇V (∇W X) − ∇W (∇V X) − ∇[V,W ] X. For example, on S 2 we have that Riem(∂θ , ∂φ )∂φ = ∂θ . In general, we have the formula ∇V (∇W X) = ∇W (∇V X) + ∇[V,W ] X + Riem(V, W )X. Also note that (2.5) can be rewritten as Riem(∂i , ∂j )X =

Φt,s X − X . ts (t,s)→(0,0) lim

2.3. Differentiation of tensors

81

The last identity reveals that Riem is F-linear, i.e., tensorial in X, in addition to being tensorial in V and W ! This is a hugely important property that is well worthy of an additional verification by hand: Riem(V, W )(αX) =∇V (∇W (αX)) − ∇W (∇V (αX)) − ∇[V,W ] (αX) =∇V ((∇W α)X + α∇W X) − ∇W ((∇V α)X + α∇V X) − ∇[V,W ] (αX)   = ∇V (∇W (α)) − ∇W (∇V (α)) − ∇[V,W ] α X + αRiem(V, W )X =αRiem(V, W )X. Note that the last equality is due to the very definition of the Lie bracket [V, W ] discussed in Lecture 2.1. Overall, the conclusion here is that we need to view Riem(V, W )X as a (3, 1)-tensor. The (3, 1)-tensor defined9 by Riem(V, W )X := ∇V (∇W X) − ∇W (∇V X) − ∇[V,W ] X is called the Riemann curvature tensor . tensor defined by

The “dual” covariant 4-

Riem(V, W, X, Y ) := g(Riem(V, W )X, Y ) is also referred to as the Riemann curvature tensor. It follows from Riem(V, W )X = ∇2 X(W ; V ) − ∇2 X(V ; W ) that Riem(V, W ) is anti-symmetric in V and W : Riem(V, W ) = −Riem(W, V ). In particular, we have Riem(V, V ) = 0 for all V . It is for these reasons (see Exercise 8a below) that the normalized curvature operator Riem(V, W )  |V |2g |W |2g − g(V, W )2 9 It should be emphasized that there are two equally common sign conventions involving curvature. A number of geometry resources have the curvature tensor defined as the negative of our curvature tensor, so one should always familiarize themselves with curvature conventions of the text before citing any results.

82

2. Tensor calculus

does not depend on the choice of an oriented basis {V, W } for the plane Span{V, W }. Going back to (2.5), we see that Riem(∂i , ∂j ) Φ X −X t,s  X = lim (t,s)→(0,0) 2 2 gii gjj − gij ts gii gjj − gij Φt,s X − X . Area This formula leads to a rough geometric interpretation of the normalized curvature operator as a rate: a measure of the “overturning due to parallel transport within an oriented plane, per amount of enclosed area”. A more rigorous formulation of this idea is the subject of the Gauss-Bonnet Theorem, which we address in Lecture 3.2. In fact, we dedicate the following two lectures to studying further geometric/visual/physical interpretations of the curvature tensor. For the remainder of this lecture we study its formal properties. = lim

Area→0

2.3.6. Curvature in coordinates. For practice reasons we now compute the curvature components in coordinates. By definition we have: (2.6) Riem(∂i , ∂j )∂k =∇∂i (∇∂j ∂k ) − ∇∂j (∇∂i ∂k ) = ∇∂i ( l Γljk ∂l ) − ∇∂j ( l Γlik ∂l )   ∂ ∂ l l l l = l ∂x i (Γjk )∂l + Γjk ∇∂i ∂l − ∂xj (Γik )∂l − Γik ∇∂j ∂l   ∂ ∂ l l m l l m = l ∂x i (Γjk )∂l + m Γjk Γil ∂m − ∂xj (Γik )∂l − m Γik Γjl ∂m   ∂ ∂ l l m l m l Γ Γ − Γ Γ = l ∂x i (Γjk ) − ∂xj (Γik ) + m jk im m ik jm ∂l . The exact nature of the above formula for Riemijkl is not as important as its schematic form: ∂Γ − ∂Γ + ΓΓ − ΓΓ. Since the Christoffel symbols are made out of terms which look like ∂ g.. , we see that the curvature components behave as a non-linear g .. ∂. second-order differential operator acting on the components of the metric tensor. Therefore, any sort of requirement placed on curvature becomes a de-facto non-linear second-order differential equation on g.

2.3. Differentiation of tensors

83

Many a problem in geometric analysis boils down to an analysis of (solutions of) such differential equations. The schematic formula for Riemijkl shows that the curvature tensor of the Euclidean Rn vanishes. It also shows that the presence of a non-zero curvature component (such as Riem(∂θ , ∂φ )∂φ in the examples of the 2-dimensional sphere; see above) serves as an obstruction to finding coordinates in which the metric tensor takes the Euclidean form everywhere. In particular, we now have a proof of the fact that one can never find a to-scale accurate map of any – no matter how small – portion of the Earth! Manifolds are often depicted as spaces which locally look like Rn . It is important to understand that this is true only on the level of coordinatizations and not on the level of geoMETRY. 2.3.7. Bianchi identities and other symmetries. At this stage it should not come as a surprise to the reader that the Hessian ∇2 T (·; V, W ) of a general tensor field T is not symmetric in V and W . Since ∇2 T (·; V, W ) = ∇W ∇V T − ∇∇W V T we once again have Riem(V, W )T :=∇2 T (. . . ; W, V ) − ∇2 T (. . . ; V, W ) =∇V ∇W T − ∇W ∇V T − ∇[V,W ] T. For the exact same reasons as in the case of vector fields T , the transformation T → Riem(V, W )T is F-linear in the T entry. To understand Riem(V, W )T in greater detail it is beneficial to perform an explicit computation of Riem(V, W )T (U1 , ..., Uk ) for (k, 0)- or (k, 1)tensor fields T . We state the final results here and encourage the reader to take the time to execute the computations on their own (Exercise 7). In the case of (k, 0)-tensor fields T the outcome of the computation is  T (..., Riem(V, W )Ui , ...), (2.7) Riem(V, W )T (U1 , ..., Uk ) = − i

while in the case of (k, 1)-tensor fields the outcome is (2.8)

Riem(V, W )T (U1 , ..., Uk ) =Riem(V, W ) (T (U1 , ..., Uk ))  T (..., Riem(V, W )Ui , ...). − i

84

2. Tensor calculus

In what follows we examine some significant consequences of the stated expressions for Riem(V, W )T . The First Bianchi Identity. While the symmetry of the Hessian ∇2 f of a scalar function f could be taken as a coordinate-independent counterpart to the claim that the ordinary partial derivatives of scalar functions commute, one ought to be very careful about making any such claims about ∇3 f and higher derivatives of f . In fact, it follows from (2.7) that ∇3 f (U, V, W ) − ∇3 f (U, W, V ) = −∇f (Riem(W, V )U ). There is a delicate balance between the need for ∇3 f (U, V, W ) to be symmetric in U and V but not symmetric in V and W . This situation is reflected in the requirement on the curvature tensor known as the First Bianchi10 Identity. Specifically, symmetry in the first two entries of ∇3 f implies ∇f (Riem(W, V )U ) = − ∇3 f (V, U, W ) + ∇3 f (W, U, V ) = − ∇3 f (V, W, U ) + ∇f (Riem(W, U )V ) + ∇3 f (W, V, U ) − ∇f (Riem(V, U )W ) =∇f (Riem(W, U )V + Riem(U, V )W ). From here one11 sees that Riem(U, V )W + Riem(V, W )U + Riem(W, U )V = 0 holds identically for all U , V and W . The Second Bianchi Identity. Further properties of the curvature tensor can be discovered by performing a very similar analysis of the lack of symmetries of ∇3 X for a vector field X: ∇3 X(U, V, W )−∇3 X(U, W, V ) = Riem(W, V )∇U X −∇Riem(W,V )U X, ∇3 X(U, V, W )−∇3 X(V, U, W ) = ∇W (Riem(V, U )X) − Riem(∇W V, U )X − Riem(V, ∇W U )X. 10 11

The identities are named in honor of an Italian mathematician Luigi Bianchi. The reader may find it helpful to consult Exercise 3 from Lecture 2.2.

2.3. Differentiation of tensors

85

We leave it to the reader to play with the last two equalities until the following is reached: ∇Riem(U, V, X, Y ; W )+∇Riem(V, W, X, Y ; U ) +∇Riem(W, U, X, Y ; V ) = 0. The stated identity is known as the Second Bianchi Identity. An additional symmetry of the curvature tensor. Since ∇g = 0 by Theorem 2.10, we always have Riem(V, W )g = 0. Applying (2.7) leads to 0 = −g(Riem(V, W )U1 , U2 ) − g(U1 , Riem(V, W )U2 ) for all V , W , U1 and U2 . The identity we just stated is more commonly seen in the form of Riem(V, W, U1 , U2 ) = −Riem(V, W, U2 , U1 ). In fact, when combined with the first Bianchi identity one can further obtain Riem(V, W, U1 , U2 ) = Riem(U1 , U2 , V, W ). The derivation of the latter is algebraic/combinatorial in nature and a good exercise for the reader. (See Exercise 8b below.) To summarize, we have proven the following theorem: Theorem 2.11. identities.

Riemann curvature tensor satisfies the following

(1) Riem(V, W, X, Y ) = −Riem(W, V, X, Y ); (2) Riem(V, W, X, Y ) = Riem(X, Y, V, W ); (3) The First Bianchi Identity: Riem(V, W, X, Y ) + Riem(W, X, V, Y ) + Riem(X, V, W, Y ) = 0; (4) The Second Bianchi Identity: ∇Riem(V, W, X, Y ; U )+∇Riem(W, U, X, Y ; V ) +∇Riem(U, V, X, Y ; W ) = 0. 2.3.8. Exercises for Lecture 2.3.

86

2. Tensor calculus

Computations in coordinates. (1) Let T be a (2, 1)-tensor. Prove the following formula. l Tjk;i =

m l ∂ l m l (Tjk ) + m Γlim Tjk − m Γm ij Tmk − m Γik Tjm . i ∂x

(2) Show that ∇ commutes with contractions of indices. (3) Compute the non-vanishing curvature components of (a) the 2-dimensional sphere of radius  in R3 ; (b) the upper half-plane model of the hyperbolic plane. (4) Prove the following schematic formula for the components of ∇Riem.

∂2 ∂2 p p Riemijkl;m = Γ − Γ gpj ∂xk ∂xm il ∂xl ∂xm ik + terms in Γ∂Γ and ΓΓΓ. Further practice. (5) Prove the identity (2.5). (Familiarity with Exercise 10 from Lecture 2.1 is necessary.) (6) (a) Let T be a (k + 1, 0)-tensor field and let T  denote the (k, 1)-tensor field obtained by, say, raising the first index on T . Show that ∇(T  ) = (∇T ) . (b) Show that ∇2 f and ∇gradf correspond to one another under the operations of raising and lowering indices: i.e., show that ∇2 f (V, W ) = g(∇W grad(f ), V ) for all V and W . (7) Prove the identities (2.7) and (2.8).

2.3. Differentiation of tensors

87

Curvature identities. (8) (a) Let T (X, Y ) be anti-symmetric in X and Y : T (X, Y ) = −T (Y, X), and let g be a dot-product. Suppose that {X, Y } and {V, W } are two equally oriented bases for the plane Span{X, Y } = Span{V, W }. Show that T (V, W ) T (X, Y )  = . |X|2g |Y |2g − g(X, Y )2 |V |2g |W |2g − g(V, W )2 (b) Suppose T is a (4, 0)-tensor for which T (V, W, X, Y ) = −T (W, V, X, Y ) = −T (V, W, Y, X) along with T (V, W, X, Y ) + T (W, X, V, Y ) + T (X, V, W, Y ) = 0. Show that T also satisfies T (V, W, X, Y ) = T (X, Y, V, W ). (9) Prove the following identity. The derivatives are to be evaluated at (t, s) = (0, 0). 1 ∂2 Riem(V + tX, W + sY, W + sY, V + tX) 6 ∂t∂s 1 ∂2 Riem(V + tY, W + sX, W + sX, V + tY ). − 6 ∂s∂t (10) Complete the proof of the Second Bianchi Identity outlined in the lecture.

Riem(W, V, X, Y ) =

Chapter 3

Curvature

3.1. Intuiting curvature via Jacobi equation 3.1.1. Introduction. The concept of curvature was formally introduced in Lecture 2.3. The goal for this lecture is to develop a more intuitive understanding of curvature. The plan is to do so by means of a differential equation (Jacobi equation) which controls the behavior of nearby geodesics. This differential equation features a particular curvature operator whose sign (or, rather, the sign of its eigenvalues) determines whether nearby geodesics are ultimately attracted to (e.g. sphere) or forced away from one another (e.g. hyperbolic space). At the end of the lecture we introduce the concept of sectional curvature which completely determines the curvature tensors on S n and Hn .

3.1.2. Radial geodesics and the geodesic normal coordinates. Consider a point P of an n-dimensional Riemannian manifold (M, g) n of tangent vectors V based at P for which |V |g < R. and the ball BR n the geodesic γV passing If R is sufficiently small then for all V ∈ BR through P with velocity vector V is defined at least over the interval (−2, 2). (Compare with Exercises 5 and 7 from Lecture 1.3.) We now switch perspectives and view V → γV (1) 89

90

3. Curvature

n as a mapping of the domain BR . The map is called the exponential map and is denoted by expP . A depiction of it is offered in Figure 3.1.

V3 P V1 V

expP (V )

expP (V3 )

V2 expP (V1 )

expP (V2 ) t

expP (tV2 )

Figure 3.1. The exponential map

The tangent space at P can be put into an isometric correspondence with Euclidean Rn . Specifically, each basis of tangent vectors at P which is orthonormal with respect g can be placed into a bijective correspondence with the standard basis of Rn , leading to an n can thus isometry between the tangent space and Rn . The ball BR n be put into a bijective correspondence with a subset of R and n →M expP : BR

leads to a parametrization1 of a neighborhood of P . At the very n centered at least this parametrization is defined on a small ball BR n the origin of R . In the case of unit spheres one has to restrict to R ≤ π due to the fact that geodesics emanating from the North Pole all converge at the South Pole thus making the exponential mapping no longer injective past R = π. In the case of the Euclidean or n = hyperbolic space it turns out that one may take R = ∞, i.e., BR n R . Coordinates (on a neighborhood of P ∈ M ) defined by expP are called geodesic normal coordinates centered at P . 1 Rigorous verification of the details of this claim is a technical undertaking beyond the scope of these lecture notes.

3.1. Jacobi equation

91

An examination of Figure 3.1 suggests that t → expP (tV ) is the parametrization of the geodesic γV passing though P with the velocity vector V ; this geodesic is parametrized by constant speed |V |g . It follows that in geodesic normal coordinates centered at P the geodesics passing through P take the form of “straight lines” (x1 (t), . . . , xn (t)) = (tV 1 , . . . , tV n ). Inserting the straight line expressions into the geodesic equations allows us to conclude that Γkij (0, ..., 0) = 0 for all i, j, k. In fact, one can further show that it is not only the Christoffel symbols at P that vanish but also all the first derivatives of all the metric components: ∂gij (0, ..., 0) = 0 for all i, j, k. ∂xk The reader is strongly encouraged to complete the details of what we just described; necessary steps are outlined in Exercise 5 following this lecture. Since the equations of geodesics passing through the center of geodesic normal coordinates are the straight line equations one cannot in these coordinates, informally speaking, detect any fictitious forces (Christoffel symbols) which would make geodesics “curve”. Thus, in certain respects geodesic normal coordinates centered at P behave as if they were what physicists would call an inertial reference frame (at P ). Even more generally, any coordinates in which

gij (P ) =

 1

if i = j,

0

if i = j,

and

∂gij (P ) = 0, ∂xk

for all i, j and k are called inertial coordinates at P . Having inertial coordinates at hand is extremely valuable when one has to perform demanding computations in coordinates. For example, if necessary, one can perform computations of curvature components at a fixed point P using coordinates which are inertial at P ; this suffices due to the tensorial nature of Riem. In such coordinates

92

3. Curvature

we have ∂Γm ∂Γm jk ik gml − gml i ∂x ∂xj

2 ∂ gkl 1 ∂ 2 gjl ∂ 2 gjk = + i k − i l 2 ∂xi ∂xj ∂x ∂x ∂x ∂x

2 ∂ gkl ∂ 2 gil ∂ 2 gik 1 + j k − j l − 2 ∂xi ∂xj ∂x ∂x ∂x ∂x

2 ∂ gjl ∂ 2 gik ∂ 2 gjk ∂ 2 gil 1 + − − = . 2 ∂xi ∂xk ∂xj ∂xl ∂xi ∂xl ∂xj ∂xk

Riemijkl =

One nice aspect of this formula is that it reveals (anti-)symmetries of the curvature tensor. Identities such as Riemijkl = −Riemjikl , Riemijkl = Riemklij and Riemijkl + Riemjkil + Riemkijl = 0 imply the first three claims of Theorem 2.11. A similar line of reasoning proves the second Bianchi identity as well; see Exercise 6 following this lecture. 3.1.3. Gauss’ Lemma. Unit speed geodesics expP (tV ) for t sufficiently small also serve as length-minimizing paths. The verification of this claim once again exceeds the level of these notes, and the reader is referred to [8]. Ultimately, the point it that under the exponential mapping the Euclidean sphere S n−1 of very small radius  centered at the origin quite literally corresponds to the sphere on (M, g) with radius  and center P , i.e., the set of points Q ∈ M such that dg (P, Q) = . A similar statement can be made about the ball B n . An employment of spherical coordinates (r, u) ∈ (0, ∞) × S n−1 composed with the exponential mapping, leads to the parametrization (r, u) → expP (ru) of a neighborhood of P . The corresponding coordinates are called the geodesic polar coordinates centered at P . In this context the vector field ∂r serves as the outward-pointing unit normal vector field to the sphere of radius r centered at P . A verification of this claim is well within the scope of these notes and is included as Exercise 4 following this lecture. Consequently, there can be no cross-terms (e.g. dr dθ) in the expression for the metric in geodesic polar coordinates. In other

3.1. Jacobi equation

93

words, in geodesic polar coordinates the metric g can be expressed as g = dr 2 + h(r), where h(r) is a smooth family of metrics on S n−1 . We have already seen such expressions for the metric: the Euclidean metric in polar and spherical coordinates takes the form of ds2 = dr 2 + r 2 dθ 2 ,

ds2 = dr 2 + r 2 ds2S n−1 ,

where ds2S n−1 denotes the line element on the standard unit sphere. The formula g = dr 2 +h(r) for the metric in geodesic polar coordinates is often found under the name of Gauss’s Lemma. In particular, in dimension 2, Gauss’s Lemma ensures that the metric g can always be expressed in the form of ds2 = dr 2 + f (r, θ)2 dθ 2 . 3.1.4. Examples of radial geodesics and the Jacobi equation. One at first glance subtle but in fact very crucial feature shown in Figure 3.1 is that geodesics expP (tV1 ) and expP (tV2 ) in some ways diverge from one another while expP (tV2 ) and expP (tV3 ) converge slightly towards one another. It is time to examine this behavior in more detail. Assume γ0 (t) is a geodesic belonging to a family of geodesics γs (t) = γ(s, t) = (x1 (s, t), x2 (s, t), . . . , xn (s, t)) which is smooth in the parameter s. The vector field  ∂xi Y := ∂s = ∂i ∂s i defined along γ0 provides a measure of separation between geodesics which are infinitesimally close to γ0 . The vector field Y is commonly referred to as a geodesic deviation vector field . The following examples provide further insight into its nature. Example of Rn : Let γ0 be a straight line through a point P with Cartesian coordinates x, in the direction of the unit vector u. Let v be a unit vector which is orthogonal to u,

94

3. Curvature and consider the following family of lines through P : γ(s, t) =x + t(cos s u + sin s v) = expP (t(cos s u + sin s v)) . We then have Y = tv or, more rigorously, we have Y = tV where V is the vector field obtained through parallel transport of v along γ0 . For future purposes we note that Y¨ = 0. Example of S n : Let γ0 be a great circle through the point P ∈ S n of the standard unit sphere, initially going in the direction of the unit tangent vector u based at P . Taking advantage of the ambient Rn+1 we may express P by its Cartesian coordinates x ∈ Rn+1 and γ0 as γ0 (t) = cos t x + sin t u = expP (tu). Let v be a tangent vector at P which is orthogonal to u. (This makes x, u and v pairwise orthogonal.) Consider the following family of great circles through P : γ(s, t) = cos t x + sin t (cos s u + sin s v) = expP (t(cos s u + sin s v)) . We then have Y = sin t v. Note that Y is tangent to S n along γ0 and that it is more correct to express it as Y = sin t V where V is the vector field obtained through parallel transport of v along γ0 . This vector field describes the rate at the which the geodesics through P are spreading apart. At first, due to sin(t) ≈ t, i.e., Y ≈ t V , the geodesics spread at the rate similar to that in Euclidean space. As t increases the rate of spreading out diminishes, and the geodesics are spread out the most at t = π2 . (Intuitively speaking: Meridians are spread out the most at the equator.) After t = π2 the geodesics start approaching one another. The whole story culminates at t = π when all the geodesics meet at a point.

3.1. Jacobi equation

95

Another way to look at this story is through the differential equation Y¨ + Y = 0, which the deviation vector field Y satisfies. There is an obvious structural similarity with the equation for a harmonic oscillator; the role of the spring constant is played by +1. In contrast, the corresponding equation in Rn is Y¨ = 0 and its spring constant is 0. Relative to Rn , therefore, the geometry of S n is such that spreading geodesics are being pulled back by a spring-like mechanism. Example of Hn : Here we invoke the Poincar´e disk model of hyperbolic space discussed in Lecture 1.3. Recall that this model entails the unit disk |x| < 1 and the metric ds2 =

2 4 (1−|x|2 )2 |dx| .

In this model geodesics passing through the origin P are usual (Euclidean) line segments. The parametrization of these geodesics takes the form of t → f (t)u where u denotes the tangent vector at the origin. (One should be cautious not to rush to the conclusion that these geodesics are parametrized as t → tu.) Since geodesics necessarily have constant (without loss of generality unit) speed we see that the function f is a solution to the initial value problem  f˙ 2 1−f 2 = 1, f (0) = 0. The unique solution of this problem (as the reader can easily verify) is f (t) = tanh(t/2). Overall, the conclusion is that geodesics discussed above are given by γ0 (t) = tanh(t/2)u, with the Euclidean magnitude of u equal to 1. Also note that the vector field γ˙0 = 2 cosh12 (t/2) u is parallel and unit along γ0 . Next, consider the family of geodesics γ(s, t) = tanh(t/2)(cos s u + sin s v) = expP (t(cos s u + sin s v)) ,

96

3. Curvature where v is orthogonal to u and also of Euclidean magnitude 1. The corresponding deviation vector field along γ0 is Y = tanh(t/2)v. To be able to make a fair comparison with the previous two examples we need to rewrite Y as a multiple of a parallel unit vector field, in this case V = 2 cosh12 (t/2) v. (The reader may want to verify that V indeed is parallel along γ0 .) Using the identity sinh(t) = 2 sinh(t/2) cosh(t/2) we get:

1 v = sinh(t) V. Y = sinh(t) 2 cosh2 (t/2) As sinh(t) is increasing and asymptotic to 12 et we see that geodesics on Hn repel away from each other. In fact, since d2 dt2 sinh(t) = sinh(t) the vector field Y satisfies Y¨ − Y = 0. This again resembles the harmonic oscillator equation, but its “spring constant” is negative. One way to intuit this is to think of a hypothetical scenario in which the “spring” in a “mass-spring system” is not pulling back when the “spring” is stretched out of equilibrium but is instead pushing further away with the force proportional to the displacement from the equilibrium. Compounding such influences yields the displacement which increases exponentially. Geodesics of hyperbolic space are subjected to such a mechanism.

The analogy with the harmonic oscillator exists in general. It is provided by the Jacobi equation. Solutions of the Jacobi equation are called Jacobi vector fields. Theorem 3.1 (Jacobi Equation). The geodesic deviation vector field Y satisfies Y¨ + Riem(Y, γ˙ 0 )γ˙ 0 = 0. Proof. On the basis of [∂t , ∂s ] = 0 we have: Y¨ = ∇∂t (∇∂t ∂s ) = ∇∂t (∇∂s ∂t ) = ∇∂s (∇∂t ∂t ) + Riem(∂t , ∂s )∂t .

3.1. Jacobi equation

97

Since each ∂t is a geodesic, we have ∇∂t ∂t = 0. It now follows that Y¨ = Riem(∂t , Y )∂t . The antisymmetry of the curvature tensor in the first two entries now proves the Jacobi equation.  We are lead to the following interpretation of the operator Y → Riem(Y, γ˙ 0 )γ˙0 . Very loosely speaking, this operator is analogous to the “spring constant” of a harmonic oscillator and it describes the mechanism by which a geometry forces its geodesics to converge towards or repel away from one another. The more “positive” Y → Riem(Y, γ˙ 0 )γ˙0 is, the more the geodesics are attracted to one another creating a more tightly curved geometry. (By the end of the following lecture we can make this statement very precise.) On the flip side, the more “negative” Y → Riem(Y, γ˙ 0 )γ˙0 becomes the more spread apart are the geodesics. 3.1.5. Asymptotic expansion formula for Jacobi vector fields corresponding to radial families of geodesics. The following result generalizes the investigations performed above in the cases of Rn , S n and Hn . In essence, the result states the following: Consider radial geodesics emanating from a point P in directions determined by tangent vectors u and v. To first order, they spread much as the straight lines do in Euclidean spaces. However, in the presence of curvature there is a deviation from the Euclidean spreading behavior. It is observable at third order and it is equal to − 16 Riem(v, u)u. Theorem 3.2. Let γ0 (t) = expP (tu) be a unit speed geodesic through P . Let v be a unit vector at P which is orthogonal to u, and let U and V be vector fields formed by parallel transports of u and v, respectively, along γ0 . Consider the radial family of geodesics γ(s, t) = expP (t(cos s u + sin s v)) emanating from P . Finally, consider the corresponding Jacobi vector field Y along γ0 . We have (3.1)

Y = tV −

t3 3! Riem(V, U )U

+ O(t4 ),

where O(t4 ) signifies a term whose size grows no faster than some constant multiple of t4 when t ≈ 0.

98

3. Curvature

Proof. Let {u, v, e3 , ..., en } be an orthonormal basis of tangent vectors at P , and let {U, V, E3 , ..., En } be their parallel transports along γ0 . At each point along γ0 the vectors {U, V, E3 , ..., En } still form an orthonormal basis for the corresponding tangent space. They define (coordinate) functions c1 (t), c2 (t), ..., cn (t) for which Y = c1 U + c2 V + c3 E3 + ... + cn En . These functions are smooth because they are simply dot-products of Y with the corresponding basis vector fields. Each function ci permits a Taylor expansion of its own: ci (t) =

K−1  k=0

(k)

ci (0) k t + O(tK ); k!

K

here O(t ) signifies a term whose size grows no faster than some constant multiple of tK when t ≈ 0. Inserting these Taylor expansions into the expression for Y yields a Taylor-like expansion Y = Y0 + tY1 + t2 Y2 + t3 Y3 + O(t4 ) in which vector fields Yi are constant linear combinations of vector fields {U, V, E3 , ..., En }. In particular, the vector fields Yi are parallel and we have Y˙ =Y1 + 2tY2 + 3t2 Y3 + O(t3 ), Y¨ =2Y2 + 6tY3 + O(t2 ). On the other hand, it follows from the very definition of the family γ(s, t) that Y (0) = 0, Y˙ (0) = v. We may now conclude that Y0 (0) = 0 and Y1 (0) = v = V (0). Since Y0 , Y1 and V are parallel we must have Y0 = 0, Y1 = V from which we further see that: Y = tV + t2 Y2 + t3 Y3 + O(t4 ). Next, we use the Jacobi equation Y¨ = −Riem(Y, γ˙ 0 )γ˙ 0 = −Riem(Y, U )U.

3.1. Jacobi equation

99

We see that 2Y2 + 6tY3 + O(t2 ) = −tRiem(V, U )U − O(t2 ). Evaluation at t = 0 yields Y2 (0) = 0, meaning that Y2 = 0, and 6Y3 + O(t) = −Riem(V, U )U − O(t). We may now conclude that Y3 = − 16 Riem(V, U )U + O(t). 

This observation completes our proof.

3.1.6. The concept of sectional curvature. We conclude this lecture by reflecting upon the analyses of radial geodesics on S n and Hn . A comparison of our explicit computation on S n with the general asymptotic formula of Theorem 3.2 leads to Y = sin t V = tV −

t3 3! V

+ O(t5 ) = tV −

t3 3! Riem(V, U )U

+ O(t4 ).

The conclusion here is that Riem(V, U )U = V + O(t) when t ≈ 0. In the case of Hn we have Y = sinh t V = tV +

t3 3! V

+ O(t5 ) = tV −

t3 3! Riem(V, U )U

+ O(t4 )

and so it follows that Riem(V, U )U = −V + O(t) when t ≈ 0. Specializing to t = 0 we obtain  Riem(v, u)u =

v

in the case of S n ,

−v

in the case of Hn .

The formula for Riem(v, u)u we arrived at completely determines the Riemann curvature tensors on S n and Hn . To see this, recall that the point P in our considerations is arbitrary and that the only requirements on the tangent vectors u and v are that they are mutually orthogonal and unit. Also recall that Riem(u, u)u = 0 due to anti-symmetry of Riem in the first two entries. It follows that   Riem(X, Y )Y = ± |Y |2g X − g(X, Y )Y ,

100

3. Curvature

or equivalently Riem(X, Y, Y, X) = ±1, |X|2g |Y |2g − g(X, Y )2 for all vector fields X and Y on S n and Hn . The values of ±1 which appear here are the same as the values of ±1 which appear as “spring constants” in the Jacobi equation on S n and Hn , respectively. In general, on any Riemannian (M, g) of any dimension one can show that (see Exercise 7 following this lecture) the value of Riem(X, Y, Y, X) |X|2g |Y |2g − g(X, Y )2 is common across all {X, Y } so long as their span is the same. In other words, each 2-dimensional subspace π of a tangent space at point P generates its own value of SecP (π) =

Riem(X, Y, Y, X) . |X|2g |Y |2g − g(X, Y )2

The function π → SecP (π) is called the sectional curvature of (M, g). When a point P is clear from context or not relevant we do not explicitly include it in the notation. The examples of Rn , S n and Hn we investigated are examples of constant sectional curvatures 0, 1 and −1, respectively. Changing the radius of a sphere away from being unit alters the value of its sectional curvature. For example, the sectional curvature of a sphere of radius  is Sec = −2 . For more details on constant sectional curvature examples the reader is directed to Exercise 10. As was explored in Exercise 9 following Lecture 2.3, multilinearity and symmetries of the curvature tensor are responsible for the identity 1 ∂2 Riem(V + tX, W + sY, W + sY, V + tX) 6 ∂t∂s 1 ∂2 − Riem(V + tY, W + sX, W + sX, V + tY ), 6 ∂t∂s

Riem(W, V, X, Y ) =

3.1. Jacobi equation

101

where all the derivatives are evaluated at (t, s) = (0, 0). The ultimate point of this identity is that sectional curvature determines the curvature tensor. In other words, having a complete knowledge of sectional curvature enables us to get an explicit formula for the curvature tensor. In the case of constant sectional curvature Sec we have that all (not necessarily linearly independent) tangent vectors X and Y satisfy   Riem(X, Y, Y, X) = Sec · |X|2g |Y |2g − g(X, Y )2 . In combination with the formula for Riem(W, V, X, Y ) in terms of sectional curvature one arrives at the following theorem. The computational details are left to the reader (Exercise 8.) Theorem 3.3. The standard Riemannian manifolds Rn , S n and Hn have constant sectional curvatures Sec = 0, Sec = 1 and Sec = −1, respectively. Their curvature tensors take the form Riem(V, W, X, Y ) = Sec · (g(V, Y )g(W, X) − g(V, X)g(W, Y )) . The above formula applies to all manifolds of constant sectional curvature. 3.1.7. Exercises for Lecture 3.1. The exponential mapping. (1) Consider the standard unit sphere S 2 in R3 , and its North Pole P = (0, 0, 1). Note that in this context the domain of the exponential mapping expP is the entire R2 .     (a) Find expP (1, 0, 0)T and expP (4, 0, 0)T .     (b) Find expP (π, 0, 0)T and expP (0, π, 0)T . (c) Is expP a 1-1 mapping? If not, can you restrict the domain of expP to make it a 1-1 mapping? (d) Describe the images expP (S 1 ) and expP (B 2 ) of spheres and balls of radius  in R2 , for  increasing from 0 to ∞. Then describe the spheres and balls (circles and 2-dimensional disks, respectively) of radius  centered at P ∈ S 2 .

102

3. Curvature (e) Compute the volumes2 of spheres and balls of radius  centered at the North Pole. (2) Consider the 2-dimensional Poincar´e disk |x| < 1, equipped 4 2 with the metric ds2 = (1−|x| Furthermore, let P 2 )2 |dx| . denote the center of the disk.       (a) Find expP (1, 0)T , expP (2, 0)T and expP (, 0)T for general . (b) Is expP a 1-1 mapping? If not, can you restrict the domain of expP to make it a 1-1 mapping? (c) Describe the images expP (S 1 ) and expP (B 2 ) of spheres and balls of radius  in R2 for  increasing from 0 to ∞. Then describe the spheres and balls of radius  within the Poincar´e disk. (d) Compute the volumes of spheres and balls of radius  centered at the origin of the Poincar´e disk. (3) Suppose that (M, g) is a 2-dimensional Riemannian manifold which is rotationally symmetric about its point P . In geodesic polar coordinates centered at P the metric g can, by Gauss’s Lemma, be expressed as ds2 = dr 2 + f (r)2 dθ 2 . (a) Reflect on Exercises 3 and 4 from Lecture 1.1 from the current standpoint, and discuss their connection to Gauss’s Lemma. (b) Compute the volumes of spheres of radius  on M . (Your answer should be in terms of the function f .) (c) Is the standard unit sphere S 2 rotationally symmetric about the North Pole? If so, what is the function f (r)? (d) Is the 2-dimensional Poincar´e disk rotationally symmetric about the origin? If so, what is the function f (r)? 2

What we really mean here is circumferences of circles and areas of disks.

3.1. Jacobi equation

103

(4) Let (r, u) → expP (ru) be the parametrization associated with the geodesic polar coordinates near P . (a) What are |∂r | and ∇∂r ∂r ? Why? (b) Let θ denote any coordinate function for S n−1 . What can you say about ∂θ at P , and why? ∂ (c) Compute ∂r g(∂r , ∂θ ). Hint: use ∇g = 0 and the fact that ∇ is torsion-free.

(d) Show that g(∂r , ∂θ ) = 0. Inertial coordinates. (5) (a) Show that t → expP (tV ) is the geodesic passing though P with the velocity vector V . Conclude that in geodesic normal coordinates centered at P geodesics passing through P take the form of (x1 (t), . . . , xn (t)) = (tV 1 , . . . , tV n ) where V = i V i ∂i serves as the tangent vector to the geodesic as it passes through P . (b) Show that the Christoffel symbols with respect to geodesic normal coordinates centered at P satisfy  Γkij (0, ..., 0)V i V j = 0 ij

for all V =

i

V i ∂i .

(c) Show that Γkij (0, ..., 0) = 0 for all i, j, k. Hint: use the above with judicious choices of V such as V = ∂i + λ∂j for fixed i, j but variable λ. (d) Show that Γkij (0, ..., 0) = 0 for all i, j, k is equivalent to ∂ g (0, .., 0) = 0 for all i, j, k. ∂xk ij (6) Prove Theorem 2.11 from Lecture 2.3 using a direct computation in inertial coordinates. (For the purposes of proving the second Bianchi identity the reader may want to consult Exercise 4 from Lecture 2.3.)

104

3. Curvature

Sectional curvature. (7) Let π be a 2-dimensional subspace of the tangent space at a point of an n-dimensional manifold (M, g). Show that the value of R(X, Y, Y, X) |X|2g |Y |2g − g(X, Y )2 is independent of the choice of a basis {X, Y } of π. (Also compare with Exercise 10 following Lecture 2.3.) (8) Prove Theorem 3.3 following the directions given in the lecture. (9) Compute the covariant derivative ∇R for Riemannian manifolds of constant sectional curvature. (10) In this lecture we performed analyses of radial geodesics on S n and Hn , which in turn allowed us to compute the curvature tensor and the section curvature of the standard S n and Hn . Adjust the narrative as needed in order to compute: (a) The curvature tensor and the sectional curvature of the sphere of radius  in Rn+1 . (b) The curvature tensor and the sectional curvature of the 4k2 2 unit disk |x| < 1 with the metric ds2 = (1−|x| 2 )2 |dx| .

3.2. Ricci and scalar curvature 3.2.1. Introduction. Geometric information carried within the Riemann curvature tensor is often best accessed through objects obtained from Riem by means of algebraic manipulations. We have already seen one such object in Lecture 3.1: sectional curvature Sec. In this lecture we discuss two additional “curvatures”: Ricci curvature Ricci and scalar curvature Scal . To develop a good intuition for these new curvature concepts we first do an analysis of the 2-dimensional situation where geometry is much more visually accessible. Upon introducing the Ricci curvature we provide an introduction to comparison geometry—itself a rich area within the field of Riemannian geometry.

3.2. Ricci and scalar curvature

105

For reasons of historical completeness we conclude the lecture with a reflection on Gauss’s work in dimension 2. 3.2.2. Curvature of 2-dimensional manifolds. The plan here is to apply Theorem 3.2 within the context of 2-dimensional manifolds. In what follows we assume geodesic polar coordinates (r, θ) centered at a point P . The metric in these coordinates, by Gauss’s Lemma, takes the form of ds2 = dr 2 + f (r, θ)2 dθ 2 . Our goal is to establish an explicit relationship between the sectional curvature of dr 2 + f (r, θ)2 dθ 2 and the function f . Along a radial geodesic γ0 = γ0 (r), where θ = θ0 is fixed, the Jacobi vector field Y can be identified with ∂θ . It follows from Gauss’s Lemma that U = γ˙ 0 = ∂r is orthogonal to ∂θ . By properties of parallel transport, we in addition have that U is orthogonal to the vector field V of Theorem 3.2. Since we are working in dimension 2, it must be that Y and V are proportional. In fact, since |∂θ | = f and since V is unit we must have Y = f V. The asymptotic formula (3.1) shows that f (r, θ0 ) = g(Y, V ) = r −

r3 ˙ 0 , γ˙ 0 , V 3! Riem(V, γ

) + O(r 4 ).

Along γ0 we have that Riem(V, γ˙ 0 , γ˙ 0 , V ) = SecP +O(r) which further 3 implies f (r, θ0 ) = r − SecP r3! + O(r 4 ). In other words, in geodesic polar coordinates centered at a point P of a 2-dimensional manifold the metric takes the form

2 r3 + O(r 4 ) dθ 2 . (3.2) ds2 = dr 2 + r − SecP 3! Readers who have completed Exercises 3 and 4 from Lecture 1.1 and Exercise 3 from Lecture 3.1 certainly see the power behind such an asymptotic formula for the metric. Integrating the expressions   2π  2π 3 3  − SecP 3! + O(4 ) dθ and 0 0 r − SecP r3! + O(r 4 ) dθ dr leads 0 to the following theorem.

106

3. Curvature

Theorem 3.4. On 2-dimensional Riemannian manifolds the circumference C ,P of  S M (P ) = {Q ∈ M dg (P, Q) = } and the surface area A ,P of  B M (P ) = {Q ∈ M dg (P, Q) ≤ } for  ≈ 0 satisfy π3 + O(4 ), 3 π4 + O(5 ). =π2 − SecP 12

C ,P =2π − SecP A ,P

Theorem 3.4 shows that sectional curvature is related to higherorder corrections to C ,P ≈ 2π and A ,P ≈ π2 for small . More precisely, we have SecP = lim

→0

2π − C ,P π 3 3

= lim

→0

π2 − A ,P π 4 12

.

It is in this sense of the word that the sectional curvature of a 2dimensional manifold can be viewed as a relative failure of the circumference/surface area of small circles/disks to be what they are in Euclidean space. In positive sectional curvature small circles/disks have consistently smaller circumference/surface area than the corresponding circles/disks in Euclidean space. On the other hand, negative sectional curvature has the opposite effect: circumferences and surface areas are bigger than what one would expect from the Euclidean experience. It is important to understand that, as such, sectional curvature can be measured by a “Flatlander” – from a purely intrinsic perspective. Any result of this type in dimension 3 would allow us to detect if our 3-dimensional space is curved! To fully appreciate what Theorem 3.4 means from the visual standpoint the reader is encouraged to do Exercises3 1 and 2 below. 3 I am indebted to Jeffrey Weeks’s “The Shape of Space” [11] for introducing me and generations of my students to these exercises.

3.2. Ricci and scalar curvature

107

3.2.3. Ricci and scalar curvature. The proof of Theorem 3.4 is centered around an asymptotic formula for the 2-dimensional volume (surface area) element in geodesic normal coordinates:

r3 4 dvolg = r − Sec + O(r ) dr dθ. 3! Our next goal is to develop an asymptotic formula for the volume element which applies in any dimension. The guiding idea in what follows is that the amount of space available (volume) depends on the overall attractiveness of geodesics towards or away from each other. To quantify the amount of available space/volume around a geodesic γ0 (r) = expP (ru) passing through a point P we use geodesic normal coordinates at P and Jacobi vector fields along γ0 . Without loss of generality we may assume that the geodesic normal coordinates are “turned around” so that γ˙ 0 = ∂n along γ0 . Applying Theorem 3.2 to vectors v = ∂i and geodesic deviations expP (r(cos s ∂n + sin s ∂i ) with 1 ≤ i ≤ n − 1 leads to n − 1 Jacobi vector fields Yi = r∂i , 1 ≤ i ≤ n − 1 which in turn satisfy Yi = rVi −

r3 ˙ 0 )γ˙ 0 3! Riem(Vi , γ

+ O(r 4 );

as in Theorem 3.2 the vector fields Vi denote the parallel transports of ∂i P along γ0 . For future purposes note that {Vi } are everywhere orthonormal, and that as a result the matrix of dot-products [ Yi , Yj ] satisfies [ Yi , Yj ] = r 2 Id −

r4 ˙ 0 , γ˙ 0 , Vj )] 3 [Riem(Vi , γ

+ O(r 5 ).

The vector fields ∂1 , . . . , ∂n−1 are tangential to the coordinate spheres and are thus orthogonal to ∂n along γ0 . (Compare with the decomposition g = dr 2 + h(r) claimed by Gauss’s Lemma.) It follows that

108

3. Curvature

along γ0 (r) = expP (ru) the volume element can be expressed as:   dvolg (r,u) = det([ ∂i , ∂j ]1≤i,j≤n ) dx1 . . . dxn   (3.3) = det([ ∂i , ∂j ]1≤i,j≤n−1 ) r n−1 dr(dvolS n−1 )u   = det[ Yi , Yj ] dr(dvolS n−1 )u . As seen above, the matrix [ Yi , Yj ] takes the schematic form of   [ Yi , Yj ] = r 2 Id + r 2 A where A = − 13 Riem(Vi , γ˙ 0 , γ˙ 0 , Vj ) + O(r). Note that, due to the fact that {Vi } are orthonormal, the matrix A is identical to the matrix representation of the linear operator V → − 13 Riem(V, γ˙ 0 )γ˙ 0 acting on the orthogonal complement of γ˙ 0 . (The matrix representation on the full tangent space would contain a row and a column of zeros due to Riem(γ˙ 0 , γ˙ 0 , γ˙ 0 , Vi ) = Riem(Vi , γ˙ 0 , γ˙ 0 , γ˙ 0 ) = 0.) Next, we claim that in general det(Id + r 2 A) = 1 + r 2 Tr(A) + O(r 4 ). To see this, observe that repeated expansions of the determinant along a first row yield the product of the diagonal entries of Id + r 2 A and a term of size O(r 4 ). Distributing the product of diagonal entries yields 1 + r 2 Tr(A) and terms of size O(r 4 ). Therefore, we have   2 det[ Yi , Yj ] = r 2(n−1) 1 − r3 Tr [Riem(·, γ˙ 0 )γ˙ 0 ] + O(r 3 ) . At this moment it is important to pause and reflect on the quantity Tr [Riem(·, γ˙0 )γ˙0 ]. The discussion which brought us to it suggests that it serves as a measure of overall attractiveness of the geodesic γ˙ 0 . While Riem(V, γ˙ 0 )γ˙ 0 conveys the attraction towards γ˙ 0 experienced by V , and is a vector, it is the scalar  g ij Riem(∂i , γ˙0 , γ˙0 , ∂j ) Tr [Riem(·, γ˙0 )γ˙0 ] = ij

3.2. Ricci and scalar curvature

109

which conveys the net attraction. For this reason we are motivated to introduce a covariant 2-tensor:  Ricci(V, W ) := Tr [Riem(·, V )W ] = g ij Riem(∂i , V, W, ∂j ). ij

Since Riem(∂i , V, W, ∂j ) = Riem(∂j , W, V, ∂i ), due to curvature symmetries, this tensor is symmetric: Ricci(V, W ) = Ricci(W, V ). The tensor we just introduced is called Ricci curvature tensor , in honor of an Italian mathematician Gregorio Ricci-Curbastro who (together with Levi-Civita) developed tensor calculus. In summary, it is Ricci(u, u) for unit u that measures the strength of overall attraction of the geodesic γ˙ 0 (r) = expP (ru) as it leaves P . Based on Theorem 3.3, Ricci curvature of an n-dimensional sphere of radius  is found4 to be  g ij (gij g(V, W ) − g(V, ∂i )g(W, ∂j )) RicciS n (V, W ) =−2 ij

=(n − 1)−2 g(V, W ) where g = 2 gS n denotes the metric on the sphere of radius . Likewise, the Ricci curvature of the hyperbolic space Hn is RicciHn = −(n − 1)gHn . In hindsight one might think that, being some kind of a net object, the Ricci tensor would be defined by means of an integral. That kind of definition is in fact possible! In exercises following this lecture the reader is guided towards proving that  n Riem(X, V, W, X) dvolS n−1 , (3.4) Ricci(V, W ) = vol(S n−1 ) |X|=1 where dvolS n−1 denotes the standard volume element on the unit sphere. By the same token, the scalar curvature   n g ij Ricciij = Ricci(X, X) dvolS n−1 (3.5) Scal := vol(S n−1 ) |X|=1 ij 4

The computation is particularly easy when inertial coordinates are used.

110

3. Curvature

measures the net attractiveness (of all geodesics starting) at a point P . From this standpoint is it quite natural to think that an asymptotic formula for volumes of small balls is tied to the concept of scalar curvature. We proceed by establishing a rigorous result along such lines. 3.2.4. Asymptotic formula for volumes of small balls in arbitrary dimension. Let us complete the computation we started in (3.3):    2 dvolg (r,u) = r n−1 1 − r3 Ricci(γ˙ 0 , γ˙ 0 ) + O(r 3 ) dr(dvolS n−1 )u . √ Recall that 1 + x = 1 + 12 x + O(x2 ) when x ≈ 0. In combination with  Ricci(γ˙0 , γ˙0 ) = Ricci(u, u) + O(r), the asymptotic expansion for 1−

r2 3 Ricci(γ˙0 , γ˙0 )

+ O(r 3 ) produces

 n+1 dvolg (r,u) = r n−1 − Ricci(u, u) r 6 + O(r n+2 ).

Inserting this expression into (3.3) proves that     n+1 dvolg (r,u) = r n−1 − Ricci(u, u) r 6 + O(r n+2 ) dr(dvolS n−1 )u . We are now ready to prove our main result regarding the volumes of small geodesic balls. A related result about geodesic spheres is left to the reader; see Exercise 7 following this lecture. Theorem 3.5. The volume vol(B M (P )) of the geodesic ball  B M (P ) = {Q ∈ M dg (P, Q) ≤ } ⊆ M of radius  centered at P satisfies n

 Scal (P ) n+2 M n+3 −  + O( ) vol(S n−1 ) vol(B (P )) = n 6n(n + 2) when  ≈ 0. In particular, if vol(B n ) denotes the volume of the standard ball of radius  in Rn then vol(B M (P )) Scal (P ) 2 =1−  + O(3 ). vol(B n ) 6(n + 2) This theorem provides a very geometric understanding of scalar curvature at P : it determines higher-order corrections to vol(B M (P )) ≈ vol(B n ),   1.

3.2. Ricci and scalar curvature

111

In positive scalar curvature small balls have consistently smaller volume than the corresponding balls in Euclidean space. On the other hand, negative scalar curvature has the opposite effect: volumes are bigger than what one would expect from the Euclidean experience. 3.2.5. Comparison theorems. Ricci curvature is the protagonist of several so-called comparison theorems: theorems which claim that a geometric quantity on a manifold which curves at least (or at most) as much as a sphere of radius  has to be at most (or at least) what it is on the sphere. We are about to prove one such theorem (Myers’s Theorem), and state a couple of others (e.g., Bishop-Gromov Comparison Theorem). Theorem 3.6 (Myers’s Theorem). Suppose that the Ricci curvature of a complete n-dimensional manifold (M, g) satisfies Ricci ≥ (n − 1)−2 g in the sense that Ricci(V, V ) ≥ (n − 1)−2 g(V, V ) for all tangent vectors V . Then for all points P, Q ∈ M we have that dg (P, Q) ≤ π. Furthermore, the manifold M is compact. Let us paraphrase the theorem in non-technical terms: if a manifold curves at least as much as a sphere of radius  then the manifold has to “close up” at least as “quickly” as a sphere of radius . Proof. Let P, Q ∈ M and let dg (P, Q) = L. Since (M, g) is complete there is a minimizing geodesic γ0 : [0, L] → M joining P and Q; without loss of generality we may assume that γ0 is of unit speed. Our task now is to prove L ≤ π. To this end consider a vector field V which is parallel, unit and fields exist in abundance as they are all orthogonal to γ0 ; such vector  parallel transports of V P . In fact, in what follows we make use of a family {V (1) , ..., V (n−1) } of such vector fields V all obtained through parallel transport of an orthonormal basis for the orthogonal complement γ˙ 0⊥ of γ˙ 0 at P . Let Y = sin πt L V . The consideration of Y is inspired by the form of the Jacobi fields on spheres; note that Y vanishes at both P and

112

3. Curvature

Q. Finally, let γ s be any5 (not necessarily geodesic) perturbation of γ0 for which ∂s s=0 = Y . Since minimizing geodesics also minimize the energy functional6 , we see that d2 E(γs )  ≥ 0.  ds2 s=0 A general computation which is structurally very similar to the one in the proof of the Jacobi equation for geodesic deviations, and is an excellent exercise for the reader, shows that  L d2 E(γs )  = − Y¨ + Riem(Y, γ˙ 0 )γ˙ 0 , Y dt. (3.6)  ds2 s=0 0 In our case we have that Y¨ =

d2 dt2

  π 2 sin πt L V = −( L ) Y,

because V is parallel along γ0 . Thus, we have that  L  π 2  d2 E(γs )  = sin( πt )2 ( L ) − Riem(γ˙ 0 , V, V, γ˙ 0 ) dt ≥ 0.  L 2 ds s=0 0 We now let V range over the orthonormal set {V (i) |1 ≤ i ≤ n − 1} mentioned at the beginning of the proof, and perform the summation of all the inequalities with respect to i. We obtain  L   2 π 2 (n − 1)( L sin( πt ) − Ricci(γ˙ 0 , γ˙ 0 ) dt ≥ 0. L) 0

Since Ricci ≥ (n − 1)−2 g, and since γ˙ 0 is unit, we may further conclude that  L   π 2 (n − 1) ( L ) − −2 sin2 πt L dt ≥ 0. It follows that

0 π 2 (L)

≥ −2 , i.e., that L ≤ π.



Those familiar with the concept of the fundamental group might be interested to know that the fundamental group of a manifold with strictly positive Ricci curvature is necessarily finite. This is basically a consequence of the fact that the above argument applies to the universal cover of the manifold and that – as a result – the universal cover is compact. 5 6

For example, we can use γs (t) = expγ0 (t) sY .  E(γ) = 12 ab |γ| ˙ 2g dt; see Lecture 1.3 for details.

3.2. Ricci and scalar curvature

113

It should be pointed out that a weaker version of the theorem, which used sectional curvature in place of Ricci curvature, was originally developed by the 19th-century French mathematician Pierre Ossian Bonnet. The theorem was subsequently improved by Hopf and Rinow, and by an American mathematician Sumner Byron Myers. The version of the result we have just gone over goes back to the early 1940’s. While we are discussing this subject we should mention a related rigidity result. It was proven by Shiu-Yuen Cheng in the 1970’s. The proof of this theorem is too advanced for us right now. Theorem 3.7. If (M, g) is a complete n-dimensional Riemannian manifold with Ricci ≥ (n − 1)−2 g and

max dg (P, Q) = π

P,Q∈M

then M is (isometric to) the standard sphere S n of radius . Recall that Ricci(u, u) for unit u measures the strength of overall attraction of the geodesic γ˙ 0 (r) = expP (ru) as it leaves P , and that it thus dictates how much available space/volume there is around expP (ru). With this in mind it is not surprising that Ricci curvature also plays a role in volume comparison theorems. The idea behind the following theorem, also known as the Bishop-Gromov Inequality 7 , is that volumes of balls of radius R on a manifold which curves at least as much as a sphere of radius  cannot be more than the volume of a ball of radius R on a sphere of radius . In fact, one could expect that the discrepancy between the two volumes only gets worse as R increases. We state the Bishop-Gromov Inequality in a more general form, so that it applies to manifolds of negative curvature as well. In our statement we use M n (k) to denote the complete (simply connected) n-dimensional Riemannian manifold of constant sectional curvature k. For k > 0 this simply means a sphere of radius k−1/2 ; for k = 0 this means the standard Euclidean space Rn ; for k < 0 this means the Poincar´e disk with the metric 4|k| |dx|2 . (1 − |x|2 )2 7 The inequality is named in honor of Richard L. Bishop and Mikhail Gromov. In the form presented here the inequality goes back to the work of R. Bishop in the early 1960’s.

114

3. Curvature

Theorem 3.8 (Bishop-Gromov Inequality). Let (M, g) be a complete n-dimensional Riemannian manifold whose Ricci curvature satisfies Ricci ≥ (n − 1)k · g M for some k ∈ R. Let vol(BR (P )) denote the volume of the geodesic M (k) ball of radius R centered at P ∈ M , and let vol(BR ) denote the n volume of the geodesic ball of radius R in M (k). The function

R →

M (P )) vol(BR M (k)

vol(BR

)

is non-increasing. In particular, we have M (k)

M vol(BR (P )) ≤ vol(BR

).

The proof of the inequality can be found in [3]. 3.2.6. A reflection on Gauss’s work in dimension 2. Classically trained Riemannian geometers are likely to begin their journey with a course in curves and surfaces which closely follows Gauss’s original work “Disquisitiones generales circa superficies curvas”. In fact, no text in Riemannian geometry is complete without at least some reflection on two of Gauss’s theorems: Theorema Egregium and the Gauss-Bonnet Theorem. Before concluding the curvature portion of the course we give the reader an opportunity to familiarize themselves with the two theorems. Gauss’s Theorem, a precursor to the Gauss-Bonnet Theorem, is discussed below in Theorem 3.10. For further information about the Gauss-Bonnet Theorem and Theorema Egregium the reader is directed to Exercises 9 and 10. As seen in Lecture 2.3, the normalized curvature operator √1 Riem(∂1 , ∂2 ) |g|

captures the difference between parallel transport Φt,s around a small coordinate t × s rectangle and the identity mapping Id. Since we are in dimension 2 the isometry Φt,s must be a rotation for an angle8 8 The concept of angle is, as always, only defined up to integer multiples of 2π. In addition, it is sensitive to the choice of orientation. We always assume that the tangent vector pointing in the direction of traversal forms positive angles with tangent vectors pointing inward into the coordinate rectangle.

3.2. Ricci and scalar curvature

115

αt,s . Therefore, for a positively oriented orthonormal basis {X, Y } of tangent vectors we must have Φt,s X = cos(αt,s )X + sin(αt,s )Y, Φt,s Y = − sin(αt,s )X + cos(αt,s )Y. At the same time, due to invariance of the normalized curvature operator under oriented changes of bases, we must have  Φt,s = Id + Riem(X, Y ) |g| ts + higher-order terms. Comparison between the two expressions for Φt,s leads to  sin(αt,s ) = −Riem(X, Y, Y, X) |g| ts + higher-order terms. It follows that (3.7)

lim

Area→0

αt,s sin(αt,s ) = lim  = −Riem(X, Y, Y, X) = −Sec, Area ts→0 |g| ts

where Area denotes the area of the coordinate rectangle. Overall, in dimension 2 sectional curvature can be seen as a rate: “angle of overturning due to parallel transport, per amount of enclosed area”. Do note just how heavily (3.7) relies on the fact that we are in dimension 2: it is only in dimension 2 that parallel transport around a (small) loop is a rotation which can be expressed through a single numerical value – the angle. Since angles behave additively under composition we in addition (pun intended) also have an integral version of (3.7): Theorems 3.9 and 3.10 below. It is for reasons of high dependency on dimension 2 that we chose not to center our treatment of curvature around the traditional “Gaussian” approach. Theorem 3.9. Suppose Ω is a region within a 2-dimensional Riemannian manifold whose boundary is a single loop ∂Ω. Parallel transport around ∂Ω is a rotation for the angle9  − Sec dvolg . Ω

A special case of this result in which Ω is a geodesic triangle, that is a triangle whose sides are geodesics, is attributed to Gauss. 9 See the previous footnote regarding consistency between the direction of traverse along the boundary and the sign of the angle.

116

3. Curvature

Theorem 3.10 (Gauss’s Theorem). If ΔP QR is a geodesic triangle on a 2-dimensional Riemannian manifold then  ∠P + ∠Q + ∠R − π = Sec dvolg . ΔP QR

An exercise following this lecture guides the reader through a proof of the celebrated Gauss-Bonnet Theorem, a result which re lates total sectional curvature M Sec dvol of a 2-dimensional compact manifold without boundary to its overall shape (topology). One of the consequences of the Gauss-Bonnet Theorem is that the total sectional curvature of compact Riemannian manifold is the same for all metrics. In the case of the sphere this common value is always 4π! 3.2.7. Exercises for Lecture 3.2. Curvature in dimension 2. (1) You will need a ruler, lots of paper, scissors and a fair amount of tape for this exercise. • Tile four or five sheets of paper with regular triangles. • Cut your sheets into individual triangles. Note that taping the triangles together so that at each vertex exactly six of them meet reproduces our (Euclidean) sheet of paper. • Now tape the triangles together so that at each vertex exactly five of the triangles meet. What surface are you reproducing? Comment on how this relates to Theorem 3.4. • Now tape the triangles together so that at each vertex exactly seven of the triangles meet. The extremely ruffly surface you are creating is a discretized version of a particular geometry. Which geometry is that? (2) Discuss the sectional curvature for each of the following (2dimensional) surfaces: a small orange, a large grapefruit, an egg shell, a cucumber, a celery stick, a banana, a donut, a round loaf of bread, an hour glass. Specifically, indicate

3.2. Ricci and scalar curvature

117

where sectional curvature is (highly) positive, (highly) negative, or zero. Further exercises on trace and contraction of indices. (3) The purpose of this exercise is to guide the reader towards proving (3.4) and (3.5). (a) Let x1 , ..., xn be the standard Cartesian (orthonormal) coordinates for Euclidean Rn , and let S n−1 denote the standard unit sphere. (i) Suppose i = j. Prove that  xi xj dvolS n−1 = 0. S n−1

Hint: consider the fact that the reflection Ri satisfying Ri (∂i ) → −∂i and Ri (∂j ) → ∂j otherwise is a symmetry of S n−1 . (ii) Prove that  vol(S n−1 ) . (xi )2 dvolS n−1 = n S n−1 Hint: make use of the symmetry Sij : ∂i ↔ ∂j . (b) Prove (3.4) and (3.5). Hint: fix a point P and make use of coordinates which are inertial at P . (4) The purpose of this problem is to guide the reader towards proving the expression  n−1 X, ν dvol n−1 Sr (3.8) Tr(∇X) = lim Sr r→0 vol(Br ) for div(X) directly from Taylor’s Theorem, i.e., without reference to the Divergence Theorem. For convenience the reader may want to do the exercise in Euclidean space and note that the strategy does carry over to any manifold as stated, all provided geodesic polar coordinates are used.

118

3. Curvature

(a) Use the Taylor Theorem to show that flux across the sphere of radius r centered at P satisfies 

   X, ν dvolSrn−1 = X P , ν dvolS n−1 r n−1 Srn−1 ν∈S n−1 

  + (∇ν X) P , ν dvolS n−1 r n + ... ν∈S n−1

where, assuming sufficient differentiability of (components of) X, the suppressed terms are on the order of r n+1 as r → 0. (b) Now use Exercise 3 to prove (3.8). (Small) spheres and balls. (5) The standard unit n-dimensional sphere S n is rotationally symmetric about the North Pole P . In geodesic polar coordinates centered at P the spherical metric can, by Gauss’s Lemma, be expressed as ds2S n = dr 2 + f (r)2 ds2S n−1 . (a) Use your knowledge of Jacobi vector fields on spheres to prove that f (r) = sin r. (b) Compute the spherical volume element in geodesic polar coordinates centered at P . (c) Prove that vol(S 2n ) =

(4π)n (n − 1)! , (2n − 1)!

vol(S 2n+1 ) = 2

π n+1 . n!

(6) The hyperbolic space Hn is rotationally symmetric about any of its points. In geodesic polar coordinates the hyperbolic metric can be expressed as ds2Hn = dr 2 + f (r)2 ds2S n−1 . (a) Use your knowledge of Jacobi vector fields on spheres to prove that f (r) = sinh r. (b) Compute the hyperbolic volume element in geodesic polar coordinates.

3.2. Ricci and scalar curvature

119

(c) Prove that vol(B H ) = 2π(cosh() − 1) 2

in dimension 2. (d) Show that vol(B H ) ≈ cn e(n−1) if   1. n

for some sequence of constants cn . (7) Compute the asymptotic formula for the “surface area” of the geodesic sphere S M (P ) of small radius  centered at the point P ∈ M . Gauss’s work. (8) Prove Theorem 3.10 based on Theorem 3.9. (9) This problem guides you towards the Gauss-Bonnet Theorem. The theorem involves a topological invariant of a manifold, which for 2-dimensional compact manifolds can be defined in terms of its (geodesic) polygonal decompositions. More precisely, for a given 2-dimensional compact manifold M the value χ=V −E+F where V denotes the number of vertices, E denotes the number of edges and F denotes the number of faces of a polygonal decomposition turns out to be independent of the choice of the polygonal decomposition. Thus, this value constitutes an invariant of M . This invariant is called the Euler10 characteristic of M and is denoted by χ(M ). It is a topological invariant in the sense that it remains the same under “elastic deformations” and is not affected by the change of metric. 10 Leonhard Euler’s discovery of the formula V − E + F = 2 for convex polyhedra marked the beginning of the investigations of this invariant.

120

3. Curvature

Figure 3.2. A possible triangularization of a sphere (V = 5, E = 9, F = 6) and a possible decomposition of a torus into quadrilaterals (V = 4, E = 8, F = 4).

(a) Consider a triangulation of a 2-dimensional compact manifold M without boundary using small geodesic triangles. Apply Gauss’s Theorem (Theorem 3.10) to each geodesic triangle, and add up the equations you got. Convince yourself that  Sec dvol = (2π)V − πF. M

(b) Argue that, in the case of triangulations, we must have F = 32 E and consequently V − 12 F = V − E + F .  (c) Show that M Sec dvol = 2πχ(M ). This is the celebrated Gauss-Bonnet Theorem. (d) What is the total sectional curvature of any metric on S2? (e) Does there exist a metric of everywhere non-positive sectional curvature on S 2 ? (f) What is the total sectional curvature of any metric on a torus (donut)? What about tori with multiple holes? (g) If a torus carries a constant curvature metric, what sign does it have to be? What about tori with multiple holes? (10) This is the exercise on Theorema Egregium promised in the lecture. Suppose a situation in which an (n+1)-dimensional Riemannian manifold M is foliated by, that is, formed by layering of, n-dimensional manifolds M . For example: R3

3.2. Ricci and scalar curvature

121

is foliated by parallel planes, punctured R3 is foliated by concentric 2-dimensional spheres of increasing radius, while R3 with an axis removed is foliated by circular cylinders of increasing radius. Suppose next that the metric g on M can be expressed as g = dr 2 + g, where the coordinate r enumerates the leaves of the foliation and where g = g(r) denotes the metric on the r-th leaf of the foliation. This is the situation on the punctured R3 where the Euclidean metric takes the form of dr 2 + ds2Sr2 = dr 2 + r 2 ds2S 2 in spherical coordinates. Likewise, the Euclidean metric on R3 with an axis removed takes the form of dr 2 + r 2 dθ 2 + dz 2 in cylindrical coordinates; here r 2 dθ 2 + dz 2 is the metric on the cylinder of radius r. In contrast to these two examples, in the case of the foliation of Euclidean R3 with planes we have dr 2 + dx2 + dy 2 , with dx2 + dy 2 being independent of r. The tensor  d 1 g(V, W ) K : (V, W ) → dr 2 on M is thus interpreted as a measure of whether (or not) the rth leaf of the foliation sits flatly inside M, i.e., if it sits inside M like a plane would sit inside Euclidean R3 . The tensor is called the second fundamental form or alternatively: the extrinsic curvature of M . The latter terminology is more insightful: K detects whether M appears curved from an external, M-perspective. In particular, it is not at all obvious how (if at all) K might be related to intrinsic concepts of curvature such as Riem, Sec, Ricci and Scal . (a) Find the second fundamental forms of planes, spheres and cylinders as they sit inside R3 .

122

3. Curvature (b) Let Riemg and Riemg denote the Riemann curvature tensors on M and (a fixed rth leaf) M . Use the expression (2.6) for Riemann curvature in coordinates to show Riemg (V, W, X, Y ) = Riemg (V, W, X, Y ) + K(V, X)K(W, Y ) − K(V, Y )K(W, X) for all vectors V , W , X and Y on M . (This is the celebrated Theorema Egregium or in translation from Latin: The Remarkable Theorem. It makes the connection between the extrinsic and the intrinsic concepts of curvature.) (c) Apply Theorema Egregium to spheres and cylinders in Euclidean R3 . The example of a cylinder is particularly interesting. Why?

Chapter 4

General relativity

4.1. The framework of special relativity 4.1.1. Introduction. The goal of this lecture is to motivate and present a geometric treatment of the theory of special relativity. Even readers who have had some experience with special relativity are encouraged to work through this lecture and the corresponding exercises. The terminology and the notation used here is integrated with the terminology and the notation developed thus far; they set the stage for the remainder of the course. 4.1.2. Principles of relativity. From a physics standpoint, there are two fundamental principles of special relativity: the Principle of Relativity (which dates back to Galileo) and the Universality of the Speed of Light (which is what makes special relativity “special”). The Principle of Relativity encompasses the idea that no one is at absolute rest and that all unaccelerated (also known as inertial) observers are “created equal”. A typical statement of the Principle of Relativity goes along the lines of “all inertial observers observe the same laws of physics”, but a more careful investigation reveals that such a statement relies on some further (often implicit) assumptions. For example, it is assumed that each inertial observer has their own way of assigning spatial (x, y, z) and temporal t-coordinates to an event, that the spatial geometry is always Euclidean and independent of t, etc. 123

124

4. General relativity

A careful treatment of such implicit assumptions can, for example, be found in [10]. The principle of the Universality of the Speed of Light was postulated by Albert Einstein based on earlier experimental evidence. It states that all inertial observers measure the same speed of light1 . Experimental evidence suggests that this speed, denoted c, is about 299, 792, 458 meters per second, although in mathematical relativity it is common to choose the units of space measurement based on the units of time measurement so that the speed of light is always equal to 1. (Note that in particular this makes speed dimensionless.) 4.1.3. Inertial observers. For an inertial observer the set of all events makes up a space-time, that is, the set R1+3 = {(t, x, y, z) | t, x, y, z ∈ R}. The t-axis {(t, 0, 0, 0) | t ∈ R} represents the observer’s own trajectory. The space-time R1+3 under component-wise operations, much like R4 , has the structure of a vector space. However, it pays huge dividends to turn this perspective upside-down and think of space-time as an abstract vector space and inertial observer as a manifestation of choosing a basis (∂t , ∂x , ∂y , ∂z ) and coordinates (t, x, y, z). Next, we make an additional assumption that the spatial geometry at each t = const moment is Euclidean with (x, y, z) serving as Cartesian coordinates. Under such an assumption the Universality of the Speed of Light states that  (Δx)2 + (Δy)2 + (Δz)2 =c=1 Δt because the units (calibration of the axes) are chosen so that the speed of light is equal to 1. Consequently, the trajectory of light is located along light-cones −(Δt)2 + (Δx)2 + (Δy)2 + (Δz)2 = 0. 1 Built into this is the assumption that light moves along straight paths with time-independent speed which is equal in each direction.

4.1. Special relativity

125

Now suppose a different inertial observer: this means a different choice of basis (∂t˜, ∂x˜ , ∂y˜, ∂z˜) and coordinates (t˜, x ˜, y˜, z˜). Since we are in a linear-algebraic framework, the connection between the two coordinatizations is linear. The Galilean Principle of Relativity and the Universality of the Speed of Light imply that x)2 + (Δ˜ y )2 + (Δ˜ z )2 = 0 −(Δt˜)2 + (Δ˜ along the trajectories of light, i.e., along −(Δt)2 + (Δx)2 + (Δy)2 + (Δz)2 = 0. In other words, for two different inertial observers the nullsets of the expressions  I = −(Δt)2 + (Δx)2 + (Δy)2 + (Δz)2 , I˜ = −(Δt˜)2 + (Δ˜ x)2 + (Δ˜ y )2 + (Δ˜ z )2 , are identical. A mathematical argument2 can now be made here showing that I and I˜ are scalar multiples of one another. The most natural postulate one can make at this point is that the calibration (e.g., choice of units) of the (∂t , ∂x , ∂y , ∂z ) and (∂t˜, ∂x˜ , ∂y˜, ∂z˜) coordinate ˜ are identically systems is such that the two expressions, I and I, equal to one another. In other words we assume x)2 + (Δ˜ y )2 + (Δ˜ z )2 −(Δt)2 + (Δx)2 + (Δy)2 + (Δz)2 = −(Δt˜)2 + (Δ˜ for every pair of points in the space-time. This invariant of special relativity is called the interval. In general, when the units of measurement are not chosen so that the speed of light is 1, the expression for the interval is given by −c2 (Δt)2 + (Δx)2 + (Δy)2 + (Δz)2 . 4.1.4. Minkowski space-time. The concept of interval motivates the use of an inner-product (4.1)

m(U1 , U2 ) = −t1 t2 + x1 x2 + y1 y2 + z1 z2

2 The linear relationship between (Δt, Δx, Δy, Δz) and (Δt˜, Δ˜ x, Δ˜ y , Δ˜ z ) converts the expression for I˜ into a homogeneous quadratic polynomial in (Δt, Δx, Δy, Δz). As a polynomial in (Δt, Δx, Δy, Δz) the expression I is irreducible. By Hilbert’s ˜ Both polynomials are quadratic Nullstellensazt, for example, it follows that I divides I. and thus the polynomials are scalar multiples of one another.

126

4. General relativity

on the vector space R1+3 = {(t, x, y, z)T }, and an articulation of laws of special relativity in terms of geometry of this inner-product space. To a large extent this idea goes back to the German mathematician Hermann Minkowski who in 1907/8 reformulated Albert Einstein’s 1905 work on special relativity (combined with an even earlier work of the Dutch physicist Hendrik Antoon Lorentz) in this new geometric language. In his honor the inner-product space defined by (4.1) is called the Minkowski space-time. As different inertial observers “measure” the same value of interval, different observers “measure” the same value of m(U, U ). In other words, while the individual values of t, x, y, z change with the change of observers the values of m(U, U ) have to remain the same. It then follows that the values of m(U1 , U2 ) =

1 2

(m(U1 + U2 , U1 + U2 ) − m(U1 , U1 ) − m(U2 , U2 ))

when U1 = U2 also do not change with the change of observers. In particular, one gets to think about the Minkowski inner-product as providing an “objective”, observer-independent framework into which the basic relativity postulates are already built in. The laws of special relativity are thus barely more then consequences of the Minkowski geometry, i.e., geometry of R1+3 equipped with the inner-product m. A careful reader has certainly noticed that the Minkowski innerproduct m is not an inner-product in a classical, positive-definite sense of the word. It is an example of a non-degenerate inner-product, i.e., R-valued symmetric bilinear map ., . for which the zero-vector is the only vector U such that U, V = 0 for all V . The concepts of magnitude, angle and distance are at least somewhat problematic in this more general framework. This is due to the fact that non-degenerate inner-products permit (non-zero) vectors U = 0 with U, U < 0 or even U, U = 0. For example, the vector ∂t in Minkowski space-time is such that m(∂t , ∂t ) = −1 while the vector ∂t + ∂x lies on the light cone and satisfies m(∂t + ∂x , ∂t + ∂x ) = 0. In analogy with these special-relativistic interpretations vectors U such that U, U < 0 are called time-like while those for which U, U = 0 are called light-like (or null ) vectors. The phrase space-like vectors is reserved for vectors U with U, U ≥ 0.

4.1. Special relativity

127

An argument involving diagonalization of symmetric matrices shows that each non-degenerate inner-product permits what is sometimes called a pseudo-orthonormal basis: a basis (..., ei , ...) of vectors for which  ±1 if i = j, ei , ej = 0 if i = j. For example, each inertial observer gives rise to a pseudo-orthonormal basis (∂t , ∂x , ∂y , ∂z ) of Minkowski space-time. In general, one shows that the number r of time-like vectors of a pseudo-orthonormal basis does not depend on the choice of the pseudo-orthonormal basis. Assuming the underlying vector space is n-dimensional, the ordered pair (r, n − r) is called the signature of the inner-product. The signature of the Minkowski inner-product m is (1, 3). In the literature signature (1, 3) is commonly referred to as the Lorentz signature. t

t˜ 1

x ˜ 1 x

Figure 4.1. Space-time diagram for Minkowski geometry

Minkowski geometry is often depicted using so-called space-time diagrams, i.e., diagrams such as the one in Figure 4.1. Note that the dashed lines in the diagram depict the light cone. The interior of the light cone corresponds to the set of all time-like vectors and is called a time-cone. The time-cone has two connected components: one future- and one past-pointing. The set of space-like vectors corresponds to the exterior of the time- and light-cones. The hyperbolas in the diagram indicate the calibration of the axes. Specifically, the intersection of the t˜-axis with m(U, U ) = −1, also called the unit

128

4. General relativity

time-like sphere3 , determines the unit of time measurement for the observer ∂t˜. The situation is completely analogous with the unit space-like sphere m(U, U ) = 1. To summarize: the main mathematical idea of special relativity is that inertial observers correspond to different choices of forwardpointing unit time-like vectors ∂t . They give rise to different choices of pseudo-orthonormal bases (∂t , ∂x , ∂y , ∂z ) of the Minkowski space-time R1+3 . There is no absolute, preferred observer-independent choice of time coordinate t just as there is no preferred set of spatial coordinates (x, y, z). This is a familiar situation: as geometers we are taught to be careful and not confuse coordinate-based identities for actual geometric phenomena. Likewise, a relativist is careful not to confuse coordinate-based perspections with objective physical phenomena. Special relativity is a study of consequences of the Minkowski geometry, i.e., geometry of R1+3 equipped with the inner-product m, and not a study of coordinate expressions. t B

t˜ A x ˜ D

C L

1 x

Figure 4.2. According to the observer ∂t the event A appears to be simultaneous to the event B. However, according to the observer ∂t˜ the events A and C appear to be simultaneous.

4.1.5. Lorentz transformations, time dilation and length contraction. For a given observer ∂t events whose time coordinates t are the same are seen as simultaneous with respect to ∂t . Events which are simultaneous with respect to ∂t are located within the 3 This is not literally a sphere; it is a hyperbola/hyperboloid depending on the dimension.

4.1. Special relativity

129

m-orthogonal complement to ∂t within R1+3 . These 3-dimensional subspaces serve as the rest spaces, i.e., simultaneity spaces for ∂t . Do note that the concept of rest space is highly observer-dependent. In the diagram in Figure 4.2 lines parallel to x- and x ˜-axes are simultaneity lines according their respective observers. The geometry in each rest space is Euclidean. Relative to the observer ∂t , the observer ∂t˜ defines a unique vector v in the rest space of ∂t such that ∂t˜ is collinear with ∂t + v. Since m(∂t˜, ∂t˜) = −1 it follows that ∂t˜ =

√ 1 1−v 2

(∂t + v)

where v = |v|. The vector v is often called the spatial velocity of ∂t˜ with respect to the observer ∂t ; meanwhile, v = |v| is called the spatial speed 4 . Contrary to the corresponding classical Gallilean concepts the ! of ∂t with respect to ∂t˜ is not −v. The reason for spatial velocity v this is very simple: −v is not in the rest space of ∂t˜. Instead, the ! is the (appropriately scaled) orthogonal projection relative velocity v of ∂t onto the rest space of ∂t˜. However, it is true that " 1 |v| = |! v| = v = 1 − . m(∂t , ∂t˜) The situation with relativistic addition of velocities and the like is addressed in the exercises following this lecture. The reader should note that the concept of 4-velocity, which is discussed below, is much more suitable for relativistic purposes than the concept of spatial velocity. To compare observations made by ∂t and ∂t˜ it is customary to choose the coordinate axes ∂x and ∂x˜ so that they are collinear with !, v and v ! = −v∂x˜ . v = v∂x and v Under such choices we have ∂t˜ =

√ 1 1−v 2

(∂t + v∂x ) ,

∂x˜ =

√ 1 1−v 2

(v∂t + ∂x ) .

4 Hidden in here is the fact that in the present framework spatial speed is always less than 1, i.e., always less than the speed of light. In our narrative this fact is a consequence of being able to “carry a clock” i.e., have a coordinate trajectory of the form (t˜, 0, 0, 0).

130

4. General relativity

In matrix notation this change of basis/inertial observers can be recorded as   (∂t˜ ∂x˜ ) = (∂t ∂x ) (4.2)

√ 1 t 1−v 2 = √ v x 1−v 2

√ 1 1−v 2 √ v 1−v 2

√ v 1−v 2 √ 1 1−v 2

√ v 1−v 2 √ 1 1−v 2

,

 t˜ . x ˜

Just as the phrase orthogonal is used for transformations/matrices which map orthonormal bases to orthonormal bases, the phrase Lorentz transformations is used for transformations/matrices which map pseudo-orthonormal bases to pseudo-orthonormal bases. Specifically, changes of inertial observers correspond to changes of pseudoorthonormal bases and amount to Lorentz transformations. The matrix in (4.2) is oftentimes called a boost; alternatively we say that it corresponds to a boosted observer. In the diagram in Figure 4.2 the t-coordinate of A is strictly bigger than its t˜-coordinate, as evidenced by the fact that the unit time-like sphere intersects the t-axis “below” the event B. This phenomenon is known as time dilation. The exact relationship follows from (4.2) and is often represented in the form of Δt˜ . Δt = √ 1 − v2 The situation is similar with taking length measurements. The endpoints of a line segment L in Figure 4.2 can be interpreted as the endpoints of a meter stick in the rest space of the observer ∂t . However, the observer ∂t˜ does not perceive the endpoints of the line segment L as the endpoints of a “meter stick” if for no other reason than because the far end of the L is not simultaneous with the close end. The event D when the far end of the meter stick finally “arrives” is marked by an x ˜-coordinate which is most certainly less than 1. This phenomenon is known as length contraction. It follows from (4.2), applied to the event D, i.e.

1 1 v 0 t , = √ 2 v 1 x ˜ 1 1−v √ that the exact value here is x ˜ = 1 − v2 .

4.1. Special relativity

131

Time dilation and length contraction under boost are relativis1 tic effects which, due to √1−v ≈ 1 + 12 v 2 , are not as observable 2 when v  1. Another way to see this is by taking what is called the Newtonian limit. A quick way to take a Newtonian limit of a relativistic expression is to restore the value of c in the expression and then schematically take c → ∞. For example, after restoring c the expression for boost takes the form of ⎞ ⎛ √ 1 2 √ v/c 2 ˜ t 1−(v/c) ⎠ t . = ⎝ 1−(v/c) √ v 2 √ 1 2 x ˜ x 1−(v/c)

1−(v/c)

The Newtonian limit here yields the Gallilean transformations t = t˜, x = x ˜ + v t˜ in which neither time dilations nor length contractions exist. 4.1.6. On causality and proper time. Observers and objects (inertial or not) trace out curves γ through space-time. A curve can be parametrized in many different ways but, in Riemannian geometry at least, it is often natural to employ a unit speed parametrization; up to an additive constant the parameter involved is the same as the arclength. The arc-length approach does not easily extend to Minkowski space-time because of the difficulty with the concept of (vector) magnitude. We now discuss certain genres of curves in Minkowski spacetime for which the broad idea of arc-length does carry through. Reparametrizations lead to tangent vectors γ˙ which are scalar multiples of one another and thus the time-like, space-like or null nature of γ˙ is independent of the choice of the parametrization of γ. A curve γ is called time-like if γ˙ is time-like. For example, trajectories of (inertial) observers (capable of measuring time) are by definition time-like. The concept of space-like curves is defined analogously. It can be shown (see Exercises 2 and 9 following Lecture 1.2) that the value of  uFinal  −m(γ, ˙ γ) ˙ du τ= uInitial

is independent of the choice of the parametrization u. (In the case of space-like curves the minus sign is omitted.) For inertial observers where γ˙ = ∂t is unit time-like, the value of τ is simply the amount

132

4. General relativity

of elapsed time, as measured by ∂t . It is a geometrically/physically meaningful quantity associated with an observer and not an artifact of a choice of time coordinate. This particular observation leads to the following interpretation of τ associated to time-like γ: it is the amount of time measured by γ’s clock. In literature τ is often called proper time. The concept of proper time for a time-like curve γ is analogous to the concept of arc-length for a space-like or Riemannian curve. To find proper time, one, just as in Riemannian geometry, integrates dτ where dτ 2 = −dt2 + dx2 + dy 2 + dz 2 . In Riemannian geometry we frequently employ unit-speed, i.e., arclength parametrizations. Unless specifically stated otherwise, every time-like curve γ throughout these lecture notes should be assumed to be parametrized by its proper time τ ; we then have m(γ, ˙ γ) ˙ = m(∂τ , ∂τ ) = −1. So far the concepts of tangential and velocity vector fields have been used interchangeably. Henceforth all velocities should be understood as 4-velocities ∂τ . In addition to time-like curves we also often restrict our attention to causal curves, that is, curves whose tangent vector field is time-like or null. The idea here is that a physical influence can only “travel” along causal curves. For an event A the set of events B for which there exists a (future-pointing) causal curve from A to B is called a (future) causal cone at A. Rephrased in plain words, the future causal cone at A is the set of events which are in theory influenceable by A. It can be shown that it is precisely the union of (future-pointing) time- and light-cones with vertex at A. The future time-cone at A can also be seen as a set of endpoints of future-pointing time-like curves starting at A. 4.1.7. Matter in special relativity. 4.1.7.1. An example from classical mechanics. Classical mechanics is built around concepts such as mass, energy and momentum. (Linear) momentum of a point object of mass m moving with velocity v is

4.1. Special relativity

133

defined as p = mv. It is a subject of a conservation law, which in turn is commonly seen in the form of Newton’s second law “F = ma”. Many an introductory physics exercise is a variation on the theme of conservation of mass and conservation momentum upon collision of point bodies5 : (4.3)

nin  i=1

mi,in =

n out  j=1

mj,out ,

nin 

pi,in =

i=1

n out 

pj,out .

j=1

Point objects are just a convenient idealization. A more “realistic” object such as a cloud of dust is perceived as being made out of individual small objects. Typically, the number of such objects is too numerous to make keeping track of each individual object useful or even possible. Instead, we keep track of average quantities, attach them to a concept of a point-object and then compute with pointobjects using ideas of calculus. For example, in place of mass m of a point object we frequently make use of the concept of mass density ρ(x) of a dust cloud. The idea here is that the infinitesimal volume element at x has mass ρ(x) dvolx , a quantity which can then be integrated as needed. The stated description of a dust cloud applies to the classical, Euclidean situation in R3 ; we use x as a shorthand for spatial variables (x, y, z) and ρ(x) is assumed to be a non-negative function on R3 . To be fair, the cloud of dust is likely to evolve with time and the function ρ(x) should be viewed not only as a function of spatial position but also time: ρ(t, x). The idea here is that point objects within the dust cloud are likely to have non-zero relative momentum, making the individual objects cluster towards or move away from certain locations. This makes it clear that it is of interest not only to keep track of energy/mass density but also of momentum density ρ(t, x)u(t, x); 5 Here (mi,in , pi,in ) are mass and momentum of nin particles going into the collision, and (mi,in , pi,in ) are mass and momentum of nout particles leaving the collision.

134

4. General relativity

here u(t, x) denotes the classical spatial velocity at time t of the point object located at x. There is a conservation law known as the continuity equation. It is derived from conservation of mass and it informs us about the dynamics of the function ρ(t, x). It relies on the observation that the quantity −div(ρu) computes the amount (in terms of mass density) of objects which cluster at a location in a unit of time. The continuity equation states that ∂ρ + div(ρu) = 0. ∂t Another conservation law addresses the dynamics of the vector field u(t, x). It comes from the conservation of momentum, i.e., Newton’s second law. This is neither the time nor place to go into details of fluid mechanics, but the broad idea is that the motion of dust particles is due to variation in pressure and that particles move towards zones where the pressure is lower. In other words, it is assumed that there is a potential function P called the pressure whose negative gradient −grad P provides the force under which dust particles move. By comparing the velocity u(t, x) a point object has at a moment t to the velocity u(t+δt, x+u·δt) it has δt later we see that the acceleration a(t, x) of the point object located at x at time t is a=

∂ ∂t u

+ ∇u u.

From here we obtain what is known as Euler’s equation: ∂  ρ · ∂t u + ∇u u = −grad P. In many situations of interest P is presumed to be a function of ρ and the exact relation between them is called the equation of state. To summarize: The partnership of the continuity and the Euler’s equation (along with the equation of state) is what governs the evolution of a dust cloud. The two equations stem from the conservation laws for mass and for momentum, and can be seen as continuous versions of (4.3). 4.1.7.2. Relativistic version. We now examine the kind of adjustments needed to be made to the above in order to account for (special) relativistic effects.

4.1. Special relativity

135

Point objects in special relativity are assumed to have a conserved quantity associated to them, the 4-momentum P . For a point-object with 4-velocity U the 4-momentum P is assumed to be a constant scalar multiple of U . Its mass m is then defined by −m2 = m(P, P ), i.e., P = mU. The 4-momentum P described above is often referred to as energymomentum. The reason for this nomenclature is as follows: Suppose a particle is moving with spatial velocity v relative to the observer (∂t , ∂x , ∂y , ∂z ); then U=

√ 1 (∂t 1−v 2

+ v) and P =

√ m ∂t 1−v 2

+

√ m v. 1−v 2

To second order in v the latter expands as P = (m + 12 mv 2 )∂t + mv; the latter suggests that the time component of the 4-momentum P can be thought of as energy while the spatial components constitute spatial (linear) momentum. The said formula for energy is more recognizable in the form in which the value of c is appropriately restored: mc2 + 12 mv 2 . What we are seeing here is the much-celebrated equivalence of mass and energy. For this reason the ∂t -component of P is called massenergy; we interpret it as mass-energy measured by the observer ∂t . There are good reasons, relativistic and non-relativistic alike, to view momentum as a covector . For example, in our context here each given inertial observer ∂t arrives at their own measurement of massenergy: m = m(P, ∂t ). ∂t → √1−v 2 Likewise, for each given unit direction w in their rest space the observer ∂t could measure the component of the momentum in the direction w: m u → √1−v v, w = m(P, w). 2 For us here, therefore, it makes a lot of sense to consider the covector field P instead of P with the operation of lowering indices defined using the Minkowski metric m: P = m U .

136

4. General relativity

We adopt this perspective henceforth. Any attempt at introducing mass-energy-momentum density for a relativistic dust cloud is immediately faced with the following issue: density expresses the amount per unit of spatial volume and from a relativistic standpoint there is no absolute concept of space! In other words, different inertial observers would not only come up with different measurements for energy-momentum due to the reasons discussed in the previous paragraph but also because of phenomena such as length contraction. Different observers observing the same collection of dust particles would agree on their number but not on the amount of space they occupy and not on their energy-momenta. The observations of the sheer number density would change from observer to observer. An observer ∂t moving at relative speed v relative √ to the dust cloud would observe the same cloud as occupying only 1 − v 2 the volume. This, for example, means that mass-energy-density as measured by ∂t satisfies ρ 1 ρ √ ·√ = . 1 − v2 1 − v2 1 − v2 When evaluated at the observer ∂t the straightforward generalization ρ ρ m(U, ·) of the “Euclidean” ρu leads to √1−v instead of the correct 2 ρ mass-energy density 1−v2 . In particular, ρ m(U, ·) cannot possibly be the expression for relativistic energy-momentum density. However, the evaluation T (∂t , ∂t ) of the covariant 2-tensor (4.4)

T = ρ m(U, ·) m(U, ·)

does lead to the correct mass-energy density:  2 1 = T (∂t , ∂t ) = ρ m( √1−v 2 (∂t + v), ∂t )

ρ . 1 − v2 In fact, one can show (see Exercise 1 following Lecture 4.2) that the tensor T in (4.4) is the only symmetric 2-tensor for which ρ for all observers ∂t . T (∂t , ∂t ) = 1 − v2 Overall, we are brought to the idea that a continuum of relativistic matter is best expressed in the language of symmetric 2-tensors. Symmetric 2-tensors T such as the one above are called stress-energy tensors. The terminology here is partly borrowed from classical fluid

4.1. Special relativity

137

dynamics where one makes use of symmetric 2-tensors called stress tensors. The expressions for the stress-energy tensor T vary depending on the physical theory under consideration. The expression for T in (4.4) is the simplest form of the relativistic perfect fluid. Discussion of stress-energy tensors in general or of other physical theories (e.g. electromagnetism) is beyond the scope of these notes. Conservation laws for physical quantities, such as those for mass and momentum, are expected to have a relativistic counterpart in terms of the stress-energy tensor. To gain an insight into how this might work recall the continuity equation ∂ρ ∂t + div(ρu) = 0 discussed earlier in this lecture. It schematically resembles a statement that the space-time 4-vector field ρ∂t + ρu is divergence-free in some Minkowski space-time sense of the word. This resemblance might inspire one to study divergence of stressenergy tensor, i.e., contracted ∇T : (div T )i =

(4.5)

3  j,k=0

0

1

2

∂  jk  m Tik , ∂xj

3

where (x , x , x , x ) denote the coordinates (t, x) of some inertial observer. Note that div T in this situation is a covector field. An ambitious reader could verify that for T as in (4.4) the requirement that div T = 0 is equivalent to a pair of a scalar and a vector equation: ∇U ρ + ρ div U = 0 and ρ ∇U U = 0. Furthermore, the Newtonian limits (taken as in Section 4.1.5 above) of the obtained equations correspond to the continuity and the Euler equation with P = 0. It is for these reasons that we see (4.4) as the stress-energy tensor of the pressure-less relativistic perfect fluid . The stress-energy tensor for a relativistic perfect fluid with nonzero pressure is found to have the form (4.6)

T = Pm + (ρ +

P c2 ) m(U, ·) m(U, ·).

Recovering the continuity and the Euler equations is a doable but a substantially more involved exercise. We shall continue our discussion of stress-energy tensors when we discuss general-relativistic matter in Lecture 4.2.

138

4. General relativity

4.1.8. Exercises for Lecture 4.1. Non-degenerate inner-product spaces. (1) Let ., . be a non-degenerate inner-product on a vector space V of dimension n. Let U be a fixed (non-zero) space-like or time-like vector. Investigate the orthogonal complement of U following the guidelines given below. (a) Show that V → U, V is a linear map of rank 1 and nullity n − 1.  (b) Show that U ⊥ = {V  V, U = 0} is a subspace of dimension n − 1. (c) Show that each vector V in V can uniquely be decomposed as V = U1 + U2 , with U1 in Span(U ) and U2 in U ⊥ . (d) Show that the restriction of the inner-product ., . to U ⊥ is still non-degenerate. (e) How many of the above claims still hold if U is non-zero null? (2) Let ., . be a non-degenerate inner-product on a vector space V of dimension n. (a) Iterate the arguments of the previous problem to show the existence of a pseudo-orthonormal basis − + + (e− 1 , . . . , er , e1 , . . . , es ) − + + with e− i , ei = −1 and ei , ei = 1 for all i.

(b) Let (f1− , . . . , fp− , f1+ , . . . , fq+ ), with fi− , fi− = −1 and fi+ , fi+ = 1 for all i, be another pseudo-orthonormal basis for V . Show that (r, s) = (p, q).

4.1. Special relativity

139

Geometry in Minkowski space-time. (3) (a) Show that the future (top) sheet of the unit time-like sphere −t2 + x2 + y 2 + z 2 = −1 in R1+3 can be parametrized by ⎧ ⎪ t = cosh σ, ⎪ ⎪ ⎪ ⎨ x = sinh σ cos θ sin φ, ⎪ y = sinh σ sin θ sin φ, ⎪ ⎪ ⎪ ⎩ z = sinh σ cos φ with σ ∈ R and the usual domains for θ and φ. (b) In analogy with spherical coordinates for Euclidean Rn , future time-cones can be parametrized by (r, u) where u is future pointing and unit time-like. Show that within the time-cone the Minkowski metric can be expressed as   dτ 2 = −dr 2 + r 2 dσ 2 + sinh2 σ ds2S 2 , where ds2S 2 denotes the metric on the standard unit sphere. (Gauss’s Lemma for Minkowski space-time). (4) Let γ0 (r) = ru for unit time-like u join event A to event B. Let γ be any time-like curve joining A to B. Show that the proper time of γ is no bigger than the proper time of γ0 : τ (γ) ≤ τ (γ0 ). In other words, show that inertial trajectories maximize proper time. (This is analogous to geodesics minimizing arc-length, at least within the domain of geodesic normal coordinates.) (5) Let M denote the future (top) sheet of the unit time-like sphere in Minkowski space-time R1+n . (a) Analyze the geometry of M by following the prompts below:

140

4. General relativity • Suppose γ(u) is a curve on M . Show that γ˙ is orthogonal, in the Minkowski sense of the word, to M . Use this observation to describe the tangent spaces to M . • Is the Minkowski dot-product, when restricted to the tangent spaces to M , non-degenerate? If so, what is the inherited signature? • Use a symmetry argument similar to the one we used on S n in Lecture 1.3 to identify geodesics on M . • Find the expression for the metric on M in coordinates from Exercise 3 above. (Is it a familiar expression? What can you say about the curvature of M ?) (b) Discuss what would change if the radius of the time-like sphere were changed from 1 to some . (6) Let M , as above, denote the future (top) sheet of the unit time-like sphere in R1+n and let L = (−1, 0, . . . , 0). For a point P on M consider the point (0, u1 , . . . , un ) at which the line LP intersects the x0 = 0-coordinate plane. (This is analogous to the stereographic projection for spheres seen in Lecture 1.1.) (a) Convince yourself that the (u1 , . . . , un ) parametrize M . What region of Rn is occupied by (u1 , . . . , un )? (b) Find the rules which relate (x0 , . . . , xn ) to (u1 , . . . , un ) and vice versa. (c) Compute an explicit formula for the metric on M in (u1 , . . . , un ) coordinates. Once again, you should get a familiar result.

Special relativity.

6

6 Exercises and examples from relativity textbooks are heavily coated in the language of clocks, trains, rockets, spaceships etc., in addition to being decorated with some explicit (yet questionably realistic) numerical values. While some readers might find such rhetorical tools a motivation for learning, they are by no means necessary nor enlightening for everyone. While we have chosen to avoid such presentation styles thus

4.1. Special relativity

141

(7) This exercise is on the so-called Twin Paradox. While Albert Einstein in 1905 only described it as “a peculiar consequence” regarding clocks, the formulations of the “paradox” now commonly found in textbooks are believed to go back to the 1911 lecture “L’evolution de l’espace et du temps” (“The Evolution of Space and Time”) by the French physicist Paul Langevin. We quote his formulation here7 . ... our traveler would need only to agree to being shut up inside a projectile that the Earth would launch at a velocity sufficiently close to that of light, but still less than it, which is physically possible, arranging for an encounter with, say, a star to take place at the end of one year in the lifetime of the traveler and to send him back toward the Earth at the same velocity. Having returned to Earth two years older, he will emerge from his ark to find that our globe has aged two hundred years.... (a) Find the velocity of the traveler from Langevin’s formulation. (b) What is the time on Earth just before the traveler turns around? How much time does the traveler, just before he turns around, think passed on Earth since his departure? (c) What is the time on Earth just after the traveler turns around? How much time does the traveler, just after he turns around, think passed on Earth since his departure? (d) From the traveler’s standpoint, how much did Earth age in the moments of his U -turn? (e) Is Earth an inertial observer in this situation? Is the traveller an inertial observer? far, it would be misleading to completely deprive the readers of exposure to traditional relativity folklore. It is for that reason, and that reason alone, that the formulation styles of the remaining exercises are so varied. 7 The translation used is by J. Sykes; see AMS Historica 108:285-300, 1973.

142

4. General relativity (f) The Earth could equally “say” that it was moving away from the traveler for 200 years, and then turned around to reunite with him. If such a perspective was valid the traveler would have aged 20, 000 years during his travels. Why is it that the traveler has not aged more than the Earth? (g) The phenomenon known as the Twin Paradox is tapping into the subject addressed in Exercise 4 above. Discuss this relationship. (8) Leaving his twin Peter at rest in their freely falling spaceship, Paul departs at constant relative speed 0.8 on a trip around a circle of radius of three light years in the rest space of their spaceship. What is the age difference between the brothers when they reunite? (Be sure to use correct units.) (9) This is a problem on addition of spatial velocities or, rather, on relating relative speeds between three inertial observers. Suppose ∂t1 and ∂t2 are two inertial observers whose spatial velocities with respect to a third inertial observer ∂t are v1 and v2 , respectively. (a) Compute m(∂t1 , ∂t2 ) and relate the obtained expression with the relative spatial speed v between the observers ∂t1 and ∂t2 . (b) Show that the relative spatial speed v between the observers ∂t1 and ∂t2 is  1 v= |v2 − v1 |2 − |v1 × v2 |2 . 1 − v1 , v2

(c) Suppose the vectors ∂t1 , ∂t2 and ∂t are coplanar, i.e., suppose that the spatial velocities v1 and v2 are collinear. Show that8 |v2 − v1 | v= . 1 − v1 v2 v+v

8 The identity is often seen in the form of addition of spatial velocities v2 = 1+vv1 . 1 Do note that, similar to the situation in classical Newtonian mechanics, this particular expression only applies when the motions of the three observers are aligned.

4.1. Special relativity

143

Is this speed more or less than the speed we would obtain in classical Gallilean mechanics? (d) This time suppose that the spatial velocities v1 and v2 are orthogonal. Show that  v = v12 + v22 − v12 v22 . Is this speed more or less than the speed we would obtain in classical Gallilean mechanics? (10) Suppose 1-dimensional motion of a relativistic train leaving a train station after stopping there. The conductor of the train has the train accelerate with constant acceleration a. Let v(τ ) denote the speed of the train relative to the stationmaster when the conductor’s clock reads τ . (a) Show that dv 1 = a. · 1 − v 2 dτ Hint: consult Exercise 9. (b) Find v(τ ), under the assumption that the train left the station at time τ = 0. (c) Suppose t denotes the time on the stationmaster’s clock. Find t(τ ) assuming that the train left the station at time t = 0. (d) Find the expression for the relative velocity v as a function of the station master’s clock. What is the acceleration of the train as measured by the stationmaster? (e) According to the stationmaster, how far away is the conductor from the stationmaster at time t? (f) According to the conductor, how far away is the stationmaster from the conductor at time τ ?

144

4. General relativity

4.2. Gravity and general relativity 4.2.1. Introduction. Special relativity relies on the concept of inertial, i.e., unaccelerated observers. Experimental evidence suggests that there is no observable difference between effects of gravity and acceleration. In other words, there is no reason to believe that in the presence of gravity any observer could be unaccelerated, i.e., inertial. For this reason the theories of special relativity and Newtonian gravity are incompatible with one another. The goal of this lecture is to motivate and formulate the basic principles of general relativity, a theory which manages to build gravity into the ideas of special relativity. It does so through employment of geometric methods described in Lectures 1.1 – 3.2. 4.2.2. Review of Newtonian gravity. By Newton’s Law of Gravity the magnitude of the gravitational force between two point objects with masses M and m is proportional to both M and m but inversely proportional9 to the square of their distance r: mM . r2 Per unit of mass, therefore, the point mass M makes a gravitational contribution of GM r12 . More specifically, if the point mass M is located at x ∈ R3 then its gravitational field , i.e., force per unit mass which point mass M exerts at the location y ∈ R3 , is: y−x F(y) = −GM . |y − x|3 G

It is important to notice that the gravitational field is the gradient of the gravitational potential Φ(y) = GM

1 . |y − x|

The gravitational field lines point towards the source at x and are orthogonal to the level sets of the gravitational potential. What if we are not dealing with point masses but, say, with a cloud of dust particles described by mass density ρ(x)? In that case 9 Experimentally the measured value of the proportionality constant G is approximately 7 · 10−11 m3 /(kg s2 ).

4.2. General relativity

145

the contributions of an infinitesimal body of volume dvolx and mass density ρ(x) to the gravitational field and potential are −Gρ(x)

y−x 1 dvolx and Gρ(x) dvolx , |y − x|3 |y − x|

respectively. It is therefore reasonable to suspect that the gravitational field F generated by the dust cloud is10 equal to   y−x x (4.7) F(y) = −G ρ(x) dvolx = G ρ(y + x) 3 dvolx 3 |y − x| |x| while the gravitational potential is given by   1 1 (4.8) Φ(y) = G ρ(x) dvolx = −G ρ(y + x) dvolx ; |y − x| |x| here each integral sign refers to integration over x ∈ R3 . One might want to be at least a little bit concerned about the last expressions, because the integrands are unbounded and the integrals are improper. Through an employment of spherical coordinates on R3 one sees that the integrals in question are absolutely convergent as long as R3 ρ(x)dvolx is. The latter is a very reasonable assumption as the stated integral computes the total mass of the cloud. For simplicity reasons, however, we are going to place an additional assumption that ρ is compactly supported and smooth. In other words, we assume that the dust cloud is bounded and that its mass density is an infinitely differentiable function. Under such a convention the dust cloud, that is, the domain where ρ = 0, is referred to as the support of ρ. The theory we are about to present holds under more general assumptions; see the remark after the statement of Theorem 4.2. Perhaps the feature of the expressions (4.7) and (4.8) for F and Φ one should be concerned about the most is the implicit claim that F = grad Φ. The equality seems plausible from the standpoint of differentiation of  1 dvolx ρ(x) |y − x| x∈R3 10 The second equal sign is due to the change of variable x ↔ x − y within the integral.

146

4. General relativity

with respect to y under the integral sign. Yet, the following nonexample clearly shows that differentiation of improper integrals under the integral sign can very easily lead one astray. A direct computation11 shows that



y−x y 1 (4.9) divy = div = −Δ =0 y |y − x|3 |y|3 |y| for all x. Differentiation with respect to y of  y−x ρ(x) dvolx |y − x|3 x∈R3 under the integral sign would lead us to believe that divF vanishes. However, this cannot possibly be true! The gravitational field lines are collecting at matter, and it is thus expected that the divergence of the gravitational field be negative and in proportion to the amount of matter present, i.e., ρ. In fact, the following can be proven to be true. Theorem 4.1. Suppose that ρ is a non-negative, compactly supported and smooth function on R3 . The vector field F and the function Φ defined by (4.7) and (4.8), respectively, are smooth and satisfy: (1) F = grad Φ; (2) divF = −4πGρ; (3) ΔΦ = −4πGρ. The equation ΔΦ = −4πGρ is called the Poisson equation. In connection to Theorem 4.1 we have to mention a related result strictly from the area of partial differential equations (PDEs). It addresses the existence and uniqueness of solutions of the Poisson equation under Dirichlet-like asymptotic boundary conditions. It is commonly referred to under the name of Green’s Representation Formula12 . In 11 For example, apply the formula for the Laplacian in spherical coordinates as in Exercise 8 from Lecture 2.2. 12 The formula is named in honor of the British mathematician George Green who, in the early 1800’s, contributed to the development of mathematical foundations of electricity and magnetism.

4.2. General relativity

147

its statement we make use of the multi-index notational convention in which α = (α1 , α2 , α3 ) denotes a triple of non-negative integers and ∂xα =

∂ |α| (∂x1 )α1 (∂x2 )α2 (∂x3 )α3

for |α| = α1 + α2 + α3 . When the variable x is clear from context (or not important) we drop x from the subscript and simply write ∂ α . One might also encounter hα for a vector h; it is a shorthand for (h1 )α1 (h2 )α2 (h3 )α3 . Theorem 4.2 (Green’s Representation Formula). Suppose that ρ is a compactly supported and smooth function on R3 . There is a unique solution to the asymptotic boundary value problem ΔΦ = −4πρ, and it is given by

 Φ(y) =

lim Φ(y) = 0

|y|→∞

ρ(x) dvolx . |y − x|

Furthermore, for each multi-index α the solution Φ satisfies the asymptotic conditions of the form |∂ α Φ(y)| ≤ C|y|−1−|α| as |y| → ∞. Not to disrupt the narrative of the course with a digression into analysis of PDEs we postpone the proofs of Theorems 4.1 and 4.2 until Lecture 5.1 in the geometric analysis portion of the course. A reader with taste for and experience in analysis would probably be interested to know that Theorems 4.1 and 4.2 do hold under weaker conditions on ρ. For example, they hold under the assumptions that ρ(x) be a smooth absolutely integrable function satisfying the asymptotic conditions of the form |∂ α ρ(x)| ≤ C|x|−3−|α| as |x| → ∞ for each multi-index α. The proofs we provide in Lecture 5.1 can be adapted to this more general setting.

148

4. General relativity

4.2.3. Tidal forces. The acceleration of a particle in free-fall is determined by the gravitational field F = grad Φ. More precisely, if the trajectory of a particle in free-fall is given by y(t) then  d2  dt2 y = F y(t) . In the basic example of the gravitational field of a point mass M centered at the origin we have: d2 dt2 y

y = −GM |y| 3.

Analysis of this differential equation gives us Kepler’s laws. (We do certain aspects of this in Lecture 4.3.) If y1 (t) and y2 (t) are two nearby particles in free-fall then by the Mean Value Theorem we have    d2    dt2 (y2 − y1 ) = grad Φ y2 − grad Φ y1 ≈ Hess(Φ) y1 (y2 − y1 ),  where Hess(Φ)y1 denotes the Hessian of Φ viewed as a matrix acting on the separation vector y2 − y1 . This in particular means that the separation vector field Y(t) for infinitesimally close particles in freed2 fall (or, rather, their relative acceleration dt 2 Y) satisfies d2 dt2 Y

= Hess(Φ)Y.

4.2.4. General relativity. Acceleration due to Earth’s gravity is independent of the nature of the falling object; this was first shown by Galileo at the Leaning Tower of Pisa and has since been confirmed to a very high accuracy by modern experiments. Abstracted into a general principle, this is to say that (at least when working on a small scale with respect to which any variation in the gravitation field is negligible) there is no observable difference between effects of gravity and acceleration. There is no reason to believe that there are “gravitationally neutral observers” and, thus, in the presence of gravity no observer is truly unaccelerated. The theory of special relativity relies on the idea that there is no absolute rest and no such thing as an absolute velocity of an observer. All inertial (unaccelerated) observers are “created equal”, and each one of them is at rest with respect to their own inertial coordinates. Moreover, the relativistic effects due to a small-scale relative velocity v (i.e., |v|  c) between inertial observers are negligible and in that

4.2. General relativity

149

sense of the word Newtonian physics serves as an approximate to special relativity. If we want to include gravity in a relativistic framework then everything said in the previous paragraph needs to be taken one step beyond. There is no such thing as absolute acceleration; all observers in free-fall (that is, observers which are exposed to no forces other than gravity) are “created equal”; each one of them can view themselves as unaccelerated with respect to some appropriate coordinate system; the effects of a small scale relative acceleration between observers in free-fall are negligible and in such a framework laws of special relativity apply (at least approximately). The emphasis placed on inertial (unaccelerated) observers in special relativity needs to be shifted to observers in free-fall, who with respect to themselves appear to be like inertial observers of special relativity. Thus we expect that on small-scale space-time can be coordinatized and be given a metric like that of Minkowski space-time. From the perspective of observers in free-fall laws of special relativity apply over small distances and short times. The perfect backdrop for uniting relativity and gravity is that of a 4-dimensional manifold whose tangent spaces (think “linearization”) are equipped with Minkowski geometry and theory of special relativity. The most basic premise of general relativity is that space-time is a 4-dimensional Lorentz manifold : a 4-dimensional manifold M endowed with a pseudo-Riemannian metric g of Lorentz signature. By this we mean a symmetric covariant 2-tensor g which on each tangent space induces a non-degenerate inner-product of Lorentz signature.

4.2.5. Brief overview of pseudo-Riemannian geometry. The geometry of pseudo-Riemannian manifolds can be developed in parallel to what was done thus far in the lectures. The Christoffel symbols, the Levi-Civita connection, the concepts of parallelism and curvature can be introduced in the exact same manner. Geodesics are defined by the requirement that their tangent vector fields be parallel and they satisfy the exact same geodesic equations. The Jacobi equation for geodesic deviation still holds. Admittedly, there are some new features to take note of. For example, it follows from geodesic equations

150

4. General relativity

(e.g. see Exercise 6 following Lecture 1.3) that the tangent vector field γ˙ is such that g(γ, ˙ γ) ˙ = const. In particular, if γ˙ is initially time-like (space-like or null, respectively) it remains time-like (space-like or null) throughout its existence. We thus distinguish time-like, space-like or null geodesics. Much as in the Twin Paradox, time-like geodesics locally maximize proper time. (Compare with Exercises 4 and 7 following Lecture 4.1.) To be honest, one does need to be cautious at places. Due to the sign difference the volume element in Lorentz framework is  dvolg = − det[gij ] dx1 . . . dxn , the Laplace operator changes its nature and becomes more of a wave equation operator, the spheres are not compact, comparison theorems no longer apply as stated. Nevertheless, the framework of pseudoRiemannian (or semi-Riemannian) geometry is remarkably similar to that of Riemannian geometry. Readers interested in seeing the details should consult [9]. In accordance with the idea that inertial observers in special relativity move along geodesics whose tangent vectors are unit timelike, in general relativity we think of free-falling observers as moving along time-like geodesics. As in special relativity we assume that such geodesics γ are parametrized by proper time, g(γ, ˙ γ) ˙ = −1 = −c2 . 4.2.6. From Poisson to the Einstein equation. In the presence of a non-constant gravitational field (and when working on a larger scale) relative accelerations of free-falling observers do come into play. The equation which describes the relative acceleration of nearby particles in free-fall in Newtonian gravity is d2 dt2 Y

= Hess(Φ)Y.

The equation describing the relative acceleration of nearby geodesics in pseudo-Riemannain geometry is the Jacobi equation d2 dt2 Y

= −Riem(Y, γ) ˙ γ. ˙

The structural similarity is very striking! Up to a negative sign, which some sources in the literature absorb into the gravitational potential

4.2. General relativity

151

Φ, the general relativistic version of the role of the Hessian of the gravitational potential  Y → Hess(Φ)γ˙ Y is played by the curvature-based Jacobi operator Y → Riem(Y, γ) ˙ γ. ˙ The gravitational potential Φ is related to the matter distribution by the Poisson equation ΔΦ = −4πGρ. The Laplace operator is the trace of the Hessian, and in general relativity the Hessian of the gravitational potential corresponds to the Jacobi operator. Thus, we expect there to be a relativistic version of the Poisson equation which relates the trace of the Jacobi operator to the matter distribution. The trace of the Jacobi operator, of course, is the Ricci curvature. So, schematically the relativistic version of the Poisson equation takes the form of Ricci(γ, ˙ γ) ˙ = “matter” along γ. ˙ The most straight-forward case of the above occurs in vacuum regions of space-time where the matter content can be set to zero. Despite how this may sound it does not mean that there is absolutely nothing going on in those vacuum regions. To convince yourself of this consider the fact that the gravitational potential Φ of a point mass 1 and that it satisfies the M in Newtonian gravity is Φ(y) = GM |y−x| Poisson equation ΔΦ = 0 = −4πGρ everywhere except at x, the location of the point mass. Thus understanding spherically symmetric solutions of the relativistic versions of the vacuum Poisson equation is akin to understanding the geometry surrounding relativistic point masses, e.g. relativistic stars, in an otherwise empty space-time. In any case, the vacuum version of the Poisson equation states that Ricci(γ, ˙ γ) ˙ vanishes for all observers γ in free-fall. It can be shown (see Exercise 1 below) that this is equivalent to Ricci = 0. The latter is known as the vacuum Einstein equation. The components of the Ricci tensor are highly non-linear expressions involving up to second-order derivatives of the metric g, and so the Einstein equation is one though non-linear system of PDEs in

152

4. General relativity

which the unknowns are the metric components. We can solve this system in spherical symmetry (see Lecture 4.3) or in the presence of other simplifying assumptions. The overall point is that in general relativity the unknown is the geometry of the space-time itself, and that Einstein equations are non-linear PDEs which place restrictions on the geometry in terms of the matter content. We examine this point in greater detail in Section 4.2.8 below. 4.2.7. On non-vacuum Einstein equations. Recall from Lecture 4.1 that matter in special relativity is best expressed by means of a stress-energy tensor. We carry this idea over to general relativity directly. For example, the expression (4.6) for the stress-energy tensor of the perfect fluid becomes the expression   T = Pg + ρ + P c2 g(U, ·) g(U, ·) for the stress-energy tensor of the perfect fluid in general relativity. Divergence of a (symmetric) 2-tensor T is defined as a contraction of ∇(T  ). It is a covector field whose components are  (divT )i = gjk (∇T )ij;k , and as such it is an extension of the concept of divergence we saw in Minkowski space in (4.5). Conservation laws for the matter once again translate into vanishing of divergence of the stress-energy tensor T . In general stress-energy tensors are assumed to be symmetric and divergence-free. At this point it is probably very tempting to assume that in nonvacuum situations the Ricci tensor is at least proportional if not equal to the stress-energy tensor T . That would be great, but there is an obstacle: while T is divergence-free, Ricci is (generally speaking) not! To compute the divergence of the Ricci tensor we make use of the Second Bianchi Identity (Theorem 2.11): ∇Riemijkl;m + ∇Riemjmkl;i + ∇Riemmikl;j = 0. Raising k and l indices, and then contracting the j-index with the k-index and the l-index with the m-index yields: div Ricci − ∇Scal + div Ricci = 0, i.e.,

div Ricci = 12 ∇Scal .

4.2. General relativity

153

Since div(Scal · g) = ∇Scal the fact that div Ricci = 12 ∇Scal implies that div(Ricci − 12 Scal · g) = 0. In other words, the tensor Ricci − 12 Scal · g is divergence-free. This tensor is called the Einstein tensor . At this point it is conceivable that the non-vacuum Einstein equation would take the form Ricci − 12 Scal · g = k · T where k is some suitable constant which plays the role of −4πG from the Poisson equation. Before we go into what the appropriate value of the proportionality constant k might be, let us observe that the stated equality between the Einstein and the stress-energy tensors can be expressed in terms of Ricci alone. A consequence of the proposed equality, obtained by raising and contracting indices, is that Scal − 12 Scal · 4 = k · Tr T, i.e., Scal = −k · Tr T for Tr T = gij Tij . Thus, we propose Ricci = k(T − 12 Tr T · g). for the non-vacuum Einstein equation. Note that for T = 0 our proposal does reduce to the already-established vacuum Einstein equation Ricci = 0. To determine the value of k we examine the proposed equation on the stress-energy tensor T = ρ g(U, ·)g(U, ·) for dust. Recalling that U is a unit time-like vector field g(U, U ) = −1 = −c2 , we compute that T (U, U ) − 12 Tr T · g(U, U ) = − 12 ρc4 . Our proposed equation thus leads to Ricci(U, U ) = −k · 12 ρc4 . Motivating parallels between tidal forces and Jacobi vector fields amount to the correspondence Ricci(U, U ) ←→ −ΔΦ. It now follows that k 12 ρc4 = 4πGρ, i.e., k =

8πG c4 .

154

4. General relativity

Overall, we have arrived at the non-vacuum Einstein equation Ricci − 12 Scal · g =

8πG c4 T.

The Einstein tensor is not the only symmetric, divergence-free 2tensor made out of Ricci. This is not a place to present the argument, but it is possible to prove that the most general divergence-free geometric (made out of g and Riem) symmetric covariant 2-tensor takes the form of Ricci − 12 Scal · g + Λ · g for some constant Λ. The most general Einstein equation states that Ricci − 12 Scal · g + Λ · g =

8πG c4 T.

Whether or not the constant Λ should be included is still subject to debate. It appears that one might want to include Λ when studying phenomena on a yet larger scale (and not, say, laws of geometry near a system of bodies in an otherwise empty space). The constant Λ, if included, is called the cosmological constant. 4.2.8. Initial value formulation. In many situations of interest we assume an existence of a global coordinate function t. Unlike the situation in special relativity, the function t is not assumed to be the proper time of any observer – it is literally just a function whose level sets are identifiable with a fixed manifold M : M = I × M. (Here I denotes an interval of values for t, and it is possible to have I = R.) The point of view represented here is that for each moment in time we have a copy of “space” M . Alternatively, “space” M is seen as evolving in time thus giving us space-time M. As natural as this point of view may feel it does reduce generality. The premise that M evolves in time refers to the spatial, Riemannian metric g on M : it is assumed that g is a function of t. For example, we could have  gij (t, x)dxi dxj . g = −dt2 + g(t) = −dt2 + ij

Such a situation is perhaps very special as it assumes that the vector field ∂t is unit time-like and everywhere orthogonal to M . In full

4.2. General relativity

155

generality one would have to consider a possibility that ∂t is neither unit nor orthogonal to M . (See Exercise 7 for further details.) To keep the narrative in the lecture itself as simple as possible we do make the assumption that g = −dt2 + g(t). The idea behind the initial value formulation in general relativity is that the Einstein equations in g yield evolution equations for g. In other words, the Einstein equations govern the evolution of the Riemannian metric g. The goal for the remainder of the lecture is to give the reader an insight into what the evolution equations for g are like. Einstein equations involve curvature of an unknown metric, i.e., the second derivatives of the unknown. A standard technique in the field of differential equations is to rewrite a second-order equation as a first-order system. This is done by introducing a variable which is the first derivative of the unknown. For example, second-order equations in position are routinely rewritten as first-order systems in which the unknowns are position and velocity. We employ the same strategy here: in place of working with the second derivatives of the unknown metric g we work with the first derivatives of the pair (g, K) where d dt g

= 2K.

Reasons for including the factor of 2 are purely cosmetic and are clear from the observation that (4.10)

Γtij = Kij , Γjit = Γjti = Kij

for all spatial indices i and j. Before proceeding much further it should be pointed out that K is a deeply meaningful tensor on M called the second fundamental form or alternatively: the extrinsic curvature of M . This tensor (in the Riemannian context) was briefly mentioned in relation to Gauss’s Theorema Egregium in exercises for Lecture 3.2. The reader should know that there are other, equivalent definitions of extrinsic curvature and that in fact such definitions are far more common in the literature. For example one definition relies on the relationship ∇gV W = K(V, W )∂t + ∇gV W

156

4. General relativity

for (spatial) vector fields V and W on M . This relationship is a consequence of the first of the equalities in (4.10). According to this relationship K captures the inability of ∇gV W to lay flat in M , i.e., the extent to which M curves as a subset of M. The latter explains the source of the term extrinsic curvature. Recall that we are aiming to see what Einstein equations on g have to say about (g, K). Thus, we need to compute the Ricci tensor Riccig of g. Assuming that coordinates {xi } are inertial at some point P of M , the Christoffel symbols (4.10) are the only non-vanishing Christoffel symbols of g. This observation makes it relatively easy to compare the components of Riemg to the components Riemg . The following comparison13 is obtained by evaluation of the formula (2.6): Riemgijkl = Riemgijkl + Kil Kjk − Kjl Kik , Riemgijkt = −(∇K)jk;i + (∇K)ik;j , d Riemgtjkt = − dt Kjk + (K 2 )jk .

Here and for the remainder of this lecture latin indices refer to spatial variables, while K 2 denotes the tensor K 2 (V, W ) = K(V, K(W, ·) ) on M . Note that the components of K 2 are (K 2 )jk =



Kji Kkl g il .

Our next step is to (raise and) contractindices. To simplify the  jk d d Kjk , as notation we write Tr K = g Kjk and Tr dt K = g jk dt well as  |K|2 = Tr K 2 = g jk g il Kji Kkl . We obtain Riccigjk = Riccigjk +

d dt Kjk

2 − 2Kjk + (Tr K)Kjk ,

Riccigjt = (divK)j − ∇∂j (Tr K), d  Riccigtt = −Tr dt K + |K|2 .

d  K can be eliminated. It is worthy of noticing that the term Tr dt Upon (raising and) contracting indices the first of the three identities 13 The first of the three identities is attributed to Gauss and is known under the name Theorema Egregium. In general, identities relating intrinsic curvature Riemg , extrinsic curvature K and the ambient intrinsic curvature Riemg are called GaussCodazzi-Mainardi equations. This name honors the highly related work by Italian mathematicians Gaspare Mainardi and Delfino Codazzi from the 1850’s and 1860’s.

4.2. General relativity

157

for Ricci curvature gives us d  K − 3|K|2 + Scal g + (Tr K)2 , Scal g = 2Tr dt d  which can be easily solved for Tr dt K . For example, the last of the three identities for Ricci curvature now becomes: Riccigtt + 12 Scal g = 12 Scal g − 12 |K|2 + 12 (Tr K)2 . As an equality between two symmetric 2-tensors over a manifold M of dimension 4, the Einstein equation amounts to 10 equations. The tt- and ti-components amount to a scalar and a vector equation. Recalling that ∂t under our assumptions is a unit normal vector field ν, these four equations can be stated as follows:   Scal g − |K|2 + (Tr K)2 = 2 Riccig (ν, ν) + 12 Scal g = 16πG c4 T (ν, ν), divK − ∇(Tr K) = Riccig (ν, ·) =

8πG c4 T (ν, ·).

It is interesting to observe that these equations place restrictions on the tensors (g, K) on the manifold M in terms of Riccig and – consequently – the stress-energy tensor T . In fact, the best way to think of these equations is in terms of necessary conditions that two tensors (g, K) have to satisfy in order to be the metric and the extrinsic curvature of M as it sits inside of a space-time with stress-energy tensor T . These equations are known as the Einstein constraint equations. Although our derivation nominally depended on the Ansatz g = −dt2 + g(t), the constraint equations (4.11)

Scal g − |K|2 + (Tr K)2 = divK − ∇(Tr K)

16πG c4 T (ν, ν), = 8πG c4 T (ν, ·)

hold in general. The remaining six of the ten Einstein equations are second-order evolution equations in which the unknown is the symmetric 2-tensor g over the 3-dimensional manifold M . In their first-order form the evolution equations are  d dt g = 2K, (4.12) g g d 2 dt K = −Ricci + 2K − (Tr K)K + Ricci .

158

4. General relativity

The reader should note that this particular form is highly dependent on the Ansatz g = −dt2 + g(t) and that it is far more involved in situations when ∂t is not a unit time-like normal vector field to M . The idea now is to take (g, K) which solves the constraints (4.11) on M as initial data, and then evolve (g, K) according to (4.12). The theorem which guarantees the existence and uniqueness of solutions of such a problem was proven in the early 1950’s by Yvonne ChoquetBruhat. A precise statement of the theorem is too advanced for these lecture notes. It should be mentioned that the evolution equations are nonlinear wave equations. The argument proving this is beyond the scope of the course, but the basic idea is to fix some (unrelated) metric gˆ on M and arrange coordinates where Riccig = − 12 Δgˆ g + non-linear lower order terms. Here the Laplacian Δgˆ is the trace of the Hessian, all computed with respect to gˆ. In combination with (4.12) we get a wave-type equation d2 dt2 g

= Δgˆ g + non-linear lower order terms

for the unknown g. The wave-like nature of this equation is, for example, responsible for the wave-like propagation of disturbances in g. This is the phenomenon known as the gravitational waves. Experimental evidence of gravitational waves was publicly announced on February 11, 2016, based on findings from September 2015. 4.2.9. Exercises for Lecture 4.2. Pseudo-Riemannian geometry. (1) (a) Suppose that T1 and T2 are two symmetric covariant 2-tensors on a Lorentz manifold such that T1 (U, U ) = T2 (U, U ) for all unit time-like vectors U . Show that T1 = T2 . (b) Suppose that the Ricci tensor of a Lorentz manifold is such that Ricci(U, U ) = 0 for all unit time-like vectors U . Show that Ricci = 0.

4.2. General relativity

159

(c) Do the above results hold on pseudo-Riemannian manifolds in general? (2) Let M denote the unit space-like sphere −t2 + x2 + y 2 + z 2 = 1 in Minkowski geometry R1+3 . (a) Show that ⎧ ⎪ t = sinh σ, ⎪ ⎪ ⎪ ⎨x = cosh σ cos θ sin φ, ⎪ y = cosh σ sin θ sin φ, ⎪ ⎪ ⎪ ⎩ z = cosh σ cos φ, with σ ∈ R and usual domains for θ and φ parametrize M. (It may be helpful to review Exercise 3 from Lecture 4.1.) (b) Show that the metric g on M inherited from the ambient Minkowski metric takes the form of ds2 = −dσ 2 + cosh2 σ ds2S 2 , where ds2S 2 denotes the metric on the standard unit sphere. What is the signature of this metric? (c) Use a symmetry argument to address the geodesics of (M, g). (It may be helpful to review Exercise 5 from Lecture 4.1.) Distinguish between time-like, space-like and null geodesics. (d) Fixing (θ, φ) and letting σ vary produces a unit timelike geodesic γ. Compute the Jacobi vector fields along γ. (e) Show that Ricci(∂σ , ∂σ ) = −2 and thus Ricci = 2g. (f) What if any changes to the above do you anticipate if R1+3 were to be changed to R1+n ?

160

4. General relativity (3) Consider Rr+s = {(t1 , .., tr , x1 , ..., xs )} equipped with the metric ds2 = −(dt1 )2 − ... − (dtr )2 + (dx1 )2 + ... + (dxs )2 for general values of r and s. (a) What is the signature of the metric inherited on the unit time-like sphere in Rr+s ? For example, what can you say about a unit time-like sphere in R2+3 ? Do the expressions for the metric and the curvature (Exercise 2) generalize? (b) How about the unit space-like sphere in Rr+s ? What if anything can you say about its (Ricci) curvature? (c) Is the metric inherited on the null-cone −(t1 )2 − ... − (tr )2 + (x1 )2 + ... + (xs )2 = 0 non-degenerate? Does the null-cone inherit a pseudoRiemannian metric from Rr+s ?

Stress-energy tensor. (4) Consider the stress-energy tensor T = (ρ + P)dt2 + Pg where g = −dt2 + a(t)2 gE . Compute div T and show that d dt ρ

+ 3(ρ + P) aa˙ = 0

is the continuity equation for T . What does the Euler equation state? Einstein equations. (5) (a) Consider the unit space-like sphere M in R1+4 . Show that it solves the vacuum (T = 0) Einstein equation with a cosmological constant. What is the value of the constant? (b) Consider the unit time-like sphere in R2+3 . Show that it solves the Einstein equation with a cosmological constant. What is the value of the constant?

4.2. General relativity

161

Initial value formulation. (6) Express the constraint and the evolution equations in the presence of a cosmological constant. For the evolution equations feel free to assume the Ansatz g = −dt2 + g(t). (7) As discussed in the lecture the more general evolution setting would have us assume that ∂t = N ν + X, where ν is a forward pointing unit time-like normal vector field to M , where X is tangential to M , and where N is some function14 . (a) Show that the expression for the metric takes the form  g = −N 2 dt2 + gij (dxi + X i dt)(dxj + X j dt). ij



(b) Suppose X = 0. Show that in this situation the evolution equations are to be replaced by d dt g = 2N K, g d dt K = −Ricci

+ 2K 2 − (Tr K)K +

1 2 N∇ N

+ Riccig .

Warning: Deriving these equations when X = 0 requires Lie derivatives of tensors, a concept well beyond these lecture notes. (8) Let M denote the future time-cone of the Minkowski spacetime R1+3 . Recall from Exercise 3 of Lecture 4.1 that it can be seen as a union of time-like spheres “centered” at the origin with radii r increasing from 0 to ∞. For the purposes of this exercise relabel r by t and let it serve as a time coordinate. Recall (Exercise 3) that the Minkowski metric g on M can be written as g = −dt2 + g(t) = −dt2 + t2 gH3 , where gH3 denotes the hyperbolic metric dσ 2 + sinh2 σgS 2 on M = R3 . 14

In literature N is called the lapse function and X is called the shift vector.

162

4. General relativity (a) Compute the extrinsic curvature K of (M, g) inside (M, g). (b) Verify that for each t the pair (g, K) satisfies the constraints. (c) Verify that (g, K) solves the evolution equations. (9) In Exercises 2 and 5 we have seen that g = −dt2 + cosh2 t ds2S 2 solves the Einstein equations with a cosmological constant. Analyze this example from the standpoint of the initial value formulation. Follow the guidelines presented in Exercise 8 above. (10) Use the initial value formulation as presented in this lecture to verify that the metric g = −c2 dt2 + a(t)2 gE where gE = dx2 + dy 2 + dz 2 solves the Einstein equation for a perfect fluid

P 4 T = c ρ + 2 dt2 + Pg c if and only if the function a(t) solves the differential equations 2

a˙ P a ¨ = 8πGρ and 3 = −4πG ρ + 3 2 . 3 a a c

4.3. Geometry of Schwarzschild space-time 4.3.1. Introduction. In this lecture we find our first solution of the Einstein vacuum equation and describe the geometry surrounding a relativistic object in an otherwise empty space-time. We investigate particle orbits in such a relativistic setting and compare them to predictions of Newtonian celestial mechanics. The story behind this lecture is commonly found in other relativity literature, e.g. [12].

4.3. Schwarzschild space-time

163

4.3.2. Finding spherically symmetric static solutions of the vacuum Einstein equation. We start by making some simplifying assumptions. We are looking for a solution which is static in the sense that the Lorentzian metric: a) does not change with time, i.e., is invariant under t → t+t∗ ; b) is time reversible, i.e., is invariant under t → −t. In particular, we are looking for a metric which can be expressed in the form −E dt2 + h where E is a positive function of spatial variables only and where h is some fixed metric on R3 (or at least a portion thereof). In addition, we are going to make an assumption that the metric is spherically symmetric, i.e., that symmetries of the standard sphere S 2 preserve our metric as well. This puts the unknown metric into the schematic form of g = −E(r)dt2 + F (r)dr 2 + G(r)gS 2 , where E, F, G > 0 and where gS 2 denotes the standard metric on the unit 2-sphere. The function 4πG(r) describes the surface area of spatial coordinate spheres |x| = r. (Careful: these need not actually be spheres of radius r; it all depends on what F is!) Yet another simplifying assumption is that these volumes increase with r. Under such an assumption one can make the change of variables r ↔ G(r)1/2 . Ultimately, we arrive at the Ansatz g = −A(r)dt2 + B(r)dr 2 + r 2 gS 2 . It is possible to conclude that many of the components of the Ricci tensor of g vanish purely based on the symmetries assumed. For example, recall that we are assuming symmetry under t → t + t∗ and under t → −t. At each tangent space, therefore, there exists a symmetry which maps ∂t to −∂t while preserving the spatial directions. It now follows that Ricci(∂t , ∂r ) = Ricci(−∂t , ∂r ) = 0.

164

4. General relativity

Likewise, we have Ricci(∂t , ∂θ ) = Ricci(∂t , ∂φ ) = 0. Spherical symmetry implies invariance under θ → θ + θ∗ and θ → −θ. Repeating the same argument as the one presented in the case of ∂t proves that Ricci(∂θ , ∂r ) = Ricci(∂θ , ∂φ ) = 0. Spheres can always be reparametrized so that any given (non-zero) tangent vector serves as (a multiple of) ∂θ and any tensorial information about ∂θ is “translatable” into tensorial information about ∂φ . Thus, we must also have Ricci(∂φ , ∂r ) = 0. In fact, we now see that the only non-vanishing components of the Ricci tensor of g are: Ricci(∂t , ∂t ), Ricci(∂r , ∂r ) and Ricci(∂φ , ∂φ ) =

1 Ricci(∂θ , ∂θ ). sin2 φ

Given that {∂t , ∂r , ∂θ , ∂φ } in this situation are orthogonal and given the symmetries of the curvature tensor itself, the burden of computing the non-vanishing components of the Ricci tensor falls on computing the following components of the Riemann tensor: Riem(∂t , ∂r , ∂r , ∂t ), Riem(∂t , ∂φ , ∂φ , ∂t ) = Riem(∂r , ∂φ , ∂φ , ∂r ) =

1 Riem(∂t , ∂θ , ∂θ , ∂t ), sin2 φ

1 Riem(∂r , ∂θ , ∂θ , ∂r ), sin2 φ

Riem(∂θ , ∂φ , ∂φ , ∂θ ).

To compute the first few curvature components it is clear that we need to know more about ∇∂t ∂r and ∇∂t ∂θ (or ∇∂t ∂φ ). Due to the symmetry t → t + t∗ none of the components of these vectors depend on t. The symmetry argument based on ∂t → −∂t now also shows that these vectors cannot have non-vanishing components in any of the spatial directions. The same type of argument shows that ∇∂θ ∂t , ∇∂θ ∂r and ∇∂θ ∂φ are collinear with ∂θ . Since ∇ is a covariant derivative, the reparametrization argument also shows that ∇∂φ ∂t , ∇∂φ ∂r are collinear with ∂φ . Overall, we have A 2A ∂t , 1 θ 1 2 Γθr ∂θ = r ∂θ , 1 φ 1 2 Γφr ∂φ = r ∂φ , and cos φ 1 θ 2 Γθφ ∂θ = sin φ ∂θ .

∇∂t ∂r = ∇∂r ∂t = 12 Γttr ∂t = (4.13)

∇∂θ ∂r = ∇∂r ∂θ = ∇∂φ ∂r = ∇∂r ∂φ = ∇∂φ ∂θ = ∇∂θ ∂φ =

4.3. Schwarzschild space-time

165

while ∇∂t ∂φ = ∇∂t ∂θ = 0. The last remaining covariant derivatives are best computed directly: (4.14)

∇∂t ∂t =

A 2B ∂r ,

∇∂r ∂r =

B 2B ∂r ,

∇∂φ ∂φ = − Br ∂r , ∇∂θ ∂θ = −r∂r −

cos φ sin φ ∂φ .

A direct computation now yields     B A Riem(∂t , ∂r , ∂r , ∂t ) =g ∇∂t ( 2B ∂r ), ∂t − g ∇∂r ( 2A ∂t ), ∂t     2   B A + A 2A + (A4A) . = − A4B Similarly, we obtain Riem(∂t , ∂φ , ∂φ , ∂t ) = (4.15)

rA 2B , rB  2B , 2 2

Riem(∂r , ∂φ , ∂φ , ∂r ) =

Riem(∂θ , ∂φ , ∂φ , ∂θ ) = r sin φ(− B1 + 1). Overall we arrive at Ricci(∂t , ∂t ) = (4.16)

A 2B

− 

A B  4B 2

Ricci(∂r , ∂r ) = − A 2A + Ricci(∂φ , ∂φ ) =

−

A B  4AB

(A )2 4AB

+

(A )2 4A2

1 Ricci(∂θ , ∂θ ) sin2 φ

A rB ,

+

B rB ,

+



rA = − 2AB +

rB  2B 2

−

1 B

+ 1.

The majority of the terms in the expressions above stem from the curvature component Riem(∂t , ∂r , ∂r , ∂t ). With that in mind a judicious choice of coefficients of a linear combination of Ricci(∂t , ∂t ) and Ricci(∂r , ∂r ) can be made to remove the terms stemming from Riem(∂t , ∂r , ∂r , ∂t ). Specifically, we have B Ricci(∂t , ∂t ) + A Ricci(∂r , ∂r ) =

 1 rB (A B

+ AB  ) =

 1 rB (AB) .

This algebraic observation is extremely convenient as the vacuum Einstein equation Ricci = 0 now implies that (AB) = 0, i.e., B =

C A

for some positive constant C. Given such a relationship between A and B, the non-vanishing components of Ricci are:    A 1 2rC (rA + 2A ), Ricci(∂r , ∂r ) = − 2rA (rA = sin12 φ Ricci(∂θ , ∂θ ) = − C1 (rA + A) + 1,

Ricci(∂t , ∂t ) = Ricci(∂φ , ∂φ )

+ 2A ),

166

4. General relativity

and the vacuum Einstein equation is equivalent to rA + 2A = 0 and −

 1 C (rA

+ A) + 1 = 0.

Despite being a system of two equations in one unknown, this system of equations in A does permit a family of solutions! The second equation can be rewritten as (rA) = C, which gives us A = C(1 + D r ) for some constant D. It can then be checked that expressions of this type also solve rA + 2A = 0. The overall conclusion here is that spherically symmetric static vacuum space-time metrics can be expressed in the form g = −C(1 +

2 D r )dt

+ (1 +

D −1 2 dr r )

+ r 2 gS 2 .

Physical interpretation of the constant C has to do with units of measurement of time. By making the change of variables √ t↔t C the constant C can be absorbed into the dt2 -factor to produce the metric 2 D −1 2 dr + r 2 gS 2 . g = −(1 + D r )dt + (1 + r ) As r → ∞ the latter is asymptotic to the Minkowski metric −dt2 + dr 2 + r 2 gS 2 = −dt2 + dx2 + dy 2 + dz 2 . Likewise, we could rescale t so that g takes the form of g = −c2 (1 +

2 D r )dt

+ (1 +

D −1 2 dr r)

+ r 2 gS 2

which is asymptotic to −c2 dt2 + dx2 + dy 2 + dz 2 . Reinstating c in our expressions is handy for when we need to quickly understand the Newtonian limit. It is for this exact reason that we choose to keep c in our expressions throughout this lecture. It is too early to provide a physical interpretation of the constant D. For reasons we are going to see later in this lecture it is fruitful to for some “other” constant M . express the constant D as D = − 2GM c2 The metric we just discovered is called the Schwarzschild metric, and the resulting space-time is called the Schwarzschild spacetime. The two are named in honor of the German astrophysicist Karl Schwarzschild who discovered the metric in this exact form in late

4.3. Schwarzschild space-time

167

191515 . The remainder of the lecture is dedicated to understanding geodesics of Schwarzschild space-time as they provide an insight into relativistic celestial mechanics. To provide some context for the relativistic results we first review basic Newtonian celestial mechanics. 4.3.3. Newtonian celestial mechanics. In Lecture 4.2 we mentioned that the equations of motion in the gravitational field generated by a Newtonian point mass M located at the origin are d2 dt2 y

y = −GM |y| 3.

Note that this equation is invariant under the reflection φ → π − φ about the equatorial plane φ = π2 . The basic uniqueness result for the initial value problems now implies that any solution of this equation corresponding to initial conditions within the equatorial plane has to remain in the equatorial plane. Thus, without loss of generality we may assume that the orbit of the body under investigation lies within the equatorial plane: φ = π2 . From this point on it is most convenient to express everything in polar d2 ˙ coordinates. Start by observing that dt 2 y measures the change in y ˙ Since in the direction of y. ˙ θ y˙ = r∂ ˙ r + θ∂ a direct computation based on ∇∂r ∂r = 0, ∇∂r ∂θ = 1r ∂θ = ∇∂θ ∂r and ∇∂θ ∂θ = −r∂r can be performed to show d2 dt2 y

   = ∇y˙ y˙ = r¨ − r θ˙ 2 ∂r + θ¨ +

2 r

 r˙ θ˙ ∂θ .

y GM Combining with −GM |y| 3 = − r 2 ∂r we obtain the following equations of motion:  r¨ − r θ˙ 2 = − GM r2 , ¨ ˙ r θ + 2r˙ θ = 0.

15 Karl Schwarzschild made this discovery while serving on the Russian front during World War I. Unfortunately, he died shortly thereafter due to a painful disease he developed while at the front.

168

4. General relativity

Clearly, the last equation can be rewritten as that we have a conserved quantity ˙ J = r 2 θ.

d 2˙ dt (r θ)

= 0, meaning

This quantity is called the angular momentum. Inserting angular momentum into the equation for r¨ produces GM r2

r¨ +

−

J2 r3

= 0.

Equations of such a form have a conserved quantity – see Exercise 6 following this lecture. In this case this conserved quantity is E =

(4.17)

1 2

r˙ 2 −

GM r

+

J2 2r 2 .

The choice of letter E has to do with the fact that the terms in (4.17) resemble energy. In fact, they are energy per unit of mass of the test J2 body in motion: the terms 12 r˙ 2 and 2r 2 correspond to the kinetic energy while the term − GM is of gravitational nature16 . r From a purely algebraic standpoint, the form of the energy suggests the use of a new variable u = GM r . Since  du  r2 J u˙ J r˙ = − GM u˙ = − GM θ˙ = − GM dθ the energy is equal to E =

J2 2(GM )2

 du 2 dθ

−u+

2 J2 2(GM )2 u .

On one hand, the conservation of E (i.e., dE dθ = 0) implies the differential equation  GM 2 d2 u dθ 2 + u = J which can be solved exactly. We leave this for the reader – Exercise 7 following this lecture leads to Kepler’s first law . Instead, we are going to perform a qualitative analysis of the solutions u(θ). Note that 2 J2 E ≥ −u + 2(GM )2 u , with the equality holding if and only if du dθ = 0. Figure 4.3 is the graph J2 2 of the function Veff (u) = −u+ 2(GM )2 u in the case when J = 0. (The function Veff is often called the effective potential.) Note that we are only focusing on u > 0, and that u → 0 is equivalent to r → ∞. 16 The minus sign here is tied to our sign convention that F = gradΦ as opposed to F = −gradΦ; it is the latter that is compatible with the interpretation of Φ as the potential energy.

4.3. Schwarzschild space-time Veff

−u +

J2 2 2(GM )2 u

E2 E1 u=0 r=∞

169 E1

Veff

E2

E2 u

E1 r=0

r − GM r

+

J2 2r 2

Figure 4.3. Keplerian orbits

Each energy level above (or on) Veff corresponds to an orbit. The lowest energy level can only be obtained when du dθ identically vanishes, and thus signifies an orbit where u (and, consequently, r) remains constant. This is a circular orbit. The orbits corresponding to slightly higher energy levels are bounded orbits (both u and r are bounded, and bounded away from zero) while the high energy orbits describe an escape to infinity (u → 0, i.e., r → ∞). The explicit solution for r(θ) reveals that the bounded orbits are ellipses, that the orbit corresponding to E = 0 is a parabola, and the orbits corresponding to E > 0 are hyperbolas. The elliptical orbits are periodic in the sense that r is a 2π-periodic function of θ. In particular, the so-called perihelion – the point on the orbit closest to the point mass M – is always the same.

4.3.4. Orbits in Schwarzschild space-time. To find orbits of small bodies (whose contribution to the gravitational field is negligible) in Schwarzschild space-time we need to find time-like geodesics parametrized by proper time τ . For the same reasons as in the Newtonian case it suffices to study the geodesics in the equatorial plane φ = π2 . In particular, it suffices to compute the time-like geodesics of the 3-dimensional rotationally symmetric static Lorentzian metric −c2 Adt2 + A−1 dr 2 + r 2 dθ 2 , and apply the obtained results to the function A = 1 + D r . Recall from (4.13) and (4.14) that the only non-vanishing (and for us here

170

4. General relativity

relevant) covariant derivatives are: ∇∂t ∂r = ∇∂r ∂t =

A 2A ∂t ,

∇∂θ ∂r = ∇∂r ∂θ = 1r ∂θ ,





A ∇∂t ∂t = c2 AA 2 ∂r , ∇∂r ∂r = − 2A ∂r , ∇∂θ ∂θ = −r∂r .

These expressions imply the following geodesic equations: ⎧  ⎪ ¨ A ˙ ⎪ ⎨t + A t r˙ = 0,   ˙2 ˙2 A 2 (4.18) r¨ + (c2 AA 2 t − 2A r˙ − r θ ) = 0, ⎪ ⎪ 2 ⎩θ¨ + θ˙ r˙ = 0, r

where ˙ denotes the derivative with respect to proper time τ . The first and the third equation can be expressed as     2˙ d d ˙ dτ At = 0 and dτ r θ = 0, ˙ The latter clearly leading to the conservation laws for At˙ and r 2 θ. corresponds to the angular momentum J; as we shall soon see the former corresponds to energy. From now on we set ˙ ˙ J = r 2 θ. E = At, Although we could go back to the geodesic equation starting with r¨, like we did in the Newtonian case earlier, it is easier to extract an equivalent equation from the fact that geodesics we are looking for are parametrized by proper time: ˙ 2 = −c2 . ˙ 2 + A−1 (r) ˙ 2 + r 2 (θ) −c2 A(t) We have − E Ac + 2 2

r˙ 2 A

+

J2 r2

= −c2 .

After an algebraic simplification we are brought to the conservation law   2 E 2 c2 1 2 A J2 = r ˙ + + c 2 2 2 2 r  2  and an effective potential A2 Jr2 + c2 . It can be shown that the collection of the three conservation laws  2  2 2 ˙ J = r 2 θ˙ and E c = 1 r˙ 2 + A J2 + c2 (4.19) E = At, 2 2 2 r is equivalent to the geodesic equations we found earlier in (4.18). (See Exercise 6 following this lecture.)

4.3. Schwarzschild space-time

171

In the case of Schwarzschild metrics we have A = 1 + effective potential 2 J2 DJ 2 c2 + Dc 2r + 2r 2 + 2r 3 .

D r

and the

The presence of the c2 term, being a constant itself, can be ignored17 . However, two of the remaining terms carry a great deal of resemblance to the terms in the Newtonian effective potential (4.17). In fact, by setting Dc2 2G 2 = −GM, i.e., D = − c2 M the terms become identical to those in (4.17)!! Overall, we have a conservation law for  2  J2 GM J 2 E˜ = c2 E 2 − 1 = 12 r˙ 2 − GM r + 2r 2 − c2 r 3 and the Schwarzschild effective potential: Veff = − GM r +

J2 2r 2

−

GM J 2 c2 r 3 .

Note that in the Newtonian limit (c → ∞, see Lecture 4.1) the expression for the Schwarzschild effective potential becomes the expression (4.17) for the Newtonian effective potential. It is this connection that allows us to think of the Schwarzschild metric (4.20)

g = −c2 (1 −

2 2GM c2 r )dt

+ (1 −

2GM −1 2 dr c2 r )

+ r 2 gS 2

as the relativistic model of an otherwise isolated massive object (e.g. a star) of mass M . Analysis of orbits in Schwarzschild space-time can be done by introducing the auxiliary variable u = GM r , just as in the Newtonian theory. We obtain  du 2 2 3 J2 J2 J2 − u + 2(GM E˜ = 2(GM )2 dθ )2 u − c2 (GM )2 u . The conservation law

dE dθ d2 u dθ 2

= 0 implies the differential equation 2  + u − c32 u2 = GM . J

The analysis of the solutions of this differential equation can be done qualitatively, using E˜ ≥ Veff where Veff (u) = −u +

2 J2 2(GM )2 u

−

3 J2 c2 (GM )2 u .

17 In hindsight, it is most fitting to put the c2 -term next to expression c2 + 12 r˙ 2 is akin to mc2 + 12 mv 2 .

1 2 2 r˙

because the

172

4. General relativity

We consider the case of non-zero fixed angular momentum J = 0, and determine “the fate of a particle” based on its energy-like conserved quantity E˜. Unlike the Newtonian scenario the general shape of the graph J of of Veff varies depending on the coefficients, i.e., on the ratio GM the angular momentum of the particle to the mass of the “star”. J2 12 Specifically, when (GM )2 < c2 the function Veff has no critical points, J2 12 (GM )2 = c2 it has exactly one 2 J 12 (GM )2 > c2 there are two critical

when

critical (saddle) point, and when

points (a local minimum and a local maximum). Critical points correspond to constant (equilibrium) solutions u(θ). Expressed in terms of r(θ) this is to say that critical points correspond to circular orbits. Veff = −u + E˜2

J2 2 2(GM )2 u

−

u

E˜1

J2 3 c2 (GM )2 u

Veff = − GM r +

E˜2

J2 2r 2

− r

GM J 2 c2 r 3

E˜2

E˜1 u=0 r=∞

E˜1 r=0 E˜2

Figure 4.4. Schwarzschildean orbits: the case of c2 J 2 < 12(GM )2

The case of c2 J 2 < 12(GM )2 . In this case there are no circular orbits. All orbits either escape to infinity (e.g. outgoing orbits with high energy) or crash18 into the “star” (e.g. orbits in low energy or ingoing orbits with high energy). This is different from the Newtonian case where any non-zero amount of angular momentum prevents the particle from crashing into the “star”. The bottom line is that in this relativistic case the angular momentum is simply too small. 18 Due to conservation of angular momentum, small r means high θ˙ which in turn means that the crash is probably more accurately described by the word “in-spiral”. Having said this, the geometric nature of Schwarzschild space-time for small r (i.e., ) deserves a more thorough investigation, and without it all comments about r < 2GM c2 crashes and in-spirals are on shaky ground. See Lecture 4.4 for further details.

4.3. Schwarzschild space-time Veff = − GM r +

J2 2r 2

−

173

GM J 2 c2 r 3

E˜max E˜2 E˜max rin

E˜2

E˜min rout

E˜1 E˜min r = 0 rin rout

E˜1

Figure 4.5. Schwarzschildean orbits: the case of c2 J 2 > 12(GM )2

The case of c2 J 2 > 12(GM )2 . In this case there are two circular orbits, one corresponding to the local minimum and one to the local maximum of Veff . The circular orbit corresponding to the local minimum is outer (r = rout ), and the circular orbit corresponding to the local maximum is inner (r = rin ). The outer circular orbit qualitatively resembles the Newtonian circular orbit in that nearby orbits are bounded; we investigate these nearby orbits below. The inner circular orbit only exists in a relativistic framework. It provides an inner bound for all bounded orbits. More precisely, the fate of the particle of relatively low energy (E˜ < 0 and E˜ < E˜max , where E˜max corresponds to the inner circular orbit) depends on whether the particle is on the inside or on the outside of the inner circular orbit. Particles inside r = rin are on a crash orbit, while the particles outside are on a bounded orbit. Note that due to the preservation of angular momentum, θ˙ is bounded away from zero which ultimately means that the particle is in one way of the other spiraling (within bounds). In contrast, particles in really high energy are on escape/crash orbits – all depending on whether they are ingoing or outgoing. The situation for intermediate levels of energy depends on whether the energy level E˜max is bigger or smaller than zero. A computation shows that

174

4. General relativity

E˜max < 0 when 12(GM )2 < c2 J 2 < 16(GM )2 , and that E˜max > 0 when c2 J 2 > 16(GM )2 . Veff = − GM r +

J2 2r 2

−

GM J 2 c2 r 3

E˜2 E˜1 E˜max

E˜2

E˜1 r

E˜min rout E˜max rin

rin rout

Figure 4.6. Schwarzschildean orbits: 12(GM )2 < c2 J 2 < 16(GM )2

Consider first the case of 12(GM )2 < c2 J 2 < 16(GM )2 and the intermediate energy level E˜max < E˜ < 0. In this case the particle is on a crash orbit, even if initially pointed outward. Basically, for how much angular momentum it has the particle has too much energy to stay “near” the stable orbit and yet not enough energy to escape. Veff = − GM r +

J2 2r 2

−

GM J 2 c2 r 3

E˜1 E˜3

E˜3 E˜max E˜2 E˜1

r

E˜max rin

E˜2

E˜min rout

rin rout Figure 4.7. Schwarzschildean orbits: the case of c2 J 2 > 16(GM )2

Now consider the case of c2 J 2 > 16(GM )2 and the corresponding intermediate energy level 0 < E˜ < E˜max . If at some point the particle is located outside the unstable circular inner orbit r = rin , then the particle is on an escape orbit. (This is very much like the

4.3. Schwarzschild space-time

175

hyperboloidal orbits in Newtonian gravity.) On the other hand, if the particle finds itself inside r = rin then it is on a crash orbit. This is true even if the particle is initially pointed outward. In both cases the particle has too much angular momentum for how much energy it has and is forced to turn around (if pointed “the wrong way”). 4.3.5. Perihelion advance. As promised earlier, we now investigate small perturbations of the stable outer circular orbits. Note that these orbits correspond to the smaller of the two solutions of 2 u − c32 u2 = ( GM J ) . The said solution will be labeled by u∗ . Actual GM 2 c2 2 values of u∗ depend on ( GM J ) but, since ( J ) < 12 , they are always 2 less than c6 . The orbits under investigation are governed by  GM 2 d2 u 3 2 , (4.21) dθ 2 + u − c2 u = J with u ≈ u∗ . This is not the kind of an equation whose solution we can hope to find in closed form. So, we retreat to linearization at u∗ . Let u = u∗ + v. If v ≈ 0 then 3u2 ≈ 3u∗ + 6u∗ v which in turn, upon inserting into (4.21) and simplifying, yields d2 v dθ 2

+ (1 −

6 c2 u∗ )v

≈ 0.

This approximate equation is a simple harmonic oscillator equation; its solutions are periodic with period 2π  ≈ 2π + c62 u∗ π. 1 − c62 u∗ This means that the orbits we are investigating are not circular! In fact, they are not even closed! The locations where the minimum (or the maximum) values of r are achieved (or, more intuitively, where the particle in a bound orbit is closest/furthest from the “star”) themselves rotate. This phenomenon is called the perihelion advance. (Perihelion refers to the point on the orbit which is closest to the “star”.) This is something that one could check/observe within our Solar System. To be honest, perihelion advance exists in the Newtonian description of the Solar System as well: it is caused by gravitational interaction of a planet (say, Mercury) with other planets (say,

176

4. General relativity

Jupiter). Such “Newtonian” perihelion advance can and has been numerically estimated. In the case of Mercury Newtonian mechanics predicts the perihelion advance of about 532 seconds of an arc19 per century. Observations of the French astronomer Urbain Le Verrier, reported in 1859, show that the perihelion advance of Mercury is actually higher: somewhere about 575 seconds per century. The only Newtonian explanation of this anomaly would involve an existence of another planet, possibly closer to the Sun. This actually was the explanation suggested by Le Verrier. He named this hypothetical planet (Star Trek fans rejoice!) Vulcan and made a prediction of when and where it would be visible from the Earth. However, no planet was observed in accordance with Le Verrier’s predictions. In 1915 Albert Einstein presented “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity”20 which addressed the relativistic perihelion advance discussed in this lecture. Our analysis, when applied to Mercury, suggests a perihelion advance of about 40 seconds of an arc per century. A more careful analysis based on an approximation near an elliptical rather than a perfectly circular orbit puts the perihelion advance at about 43 seconds of an arc per century! This not only solved the anomaly of Mercury’s perihelion, but also served as early evidence of the validity of the theory of general relativity. 4.3.6. Exercises for Lecture 4.3. Computations in spherical symmetry. (1) Geometries of the standard unit sphere and the standard hyperbolic space are spherically symmetric. Express them in the form ds2 = B(r)dr 2 + r 2 gS n−1 used heavily in this lecture. Also discuss the domain of the new r-variable. (As usual, gS n−1 denotes the standard metric on the unit (n − 1)-dimensional sphere.) 19 This is a unit of angular measurement. There are 3, 600 seconds in one degree of an arc. 20 Erkl¨ arung der Perihelbewegung des Merkur aus der allgemeinen Relativit¨ atstheorie.

4.3. Schwarzschild space-time

177

(2) Verify the expressions (4.13) – (4.16). (3) Use symmetries to argue that up to permutations of indices Riemtrrt , Riemθφφθ , Riemtφφt = sin12 φ Riemtθθt and Riemrφφr = sin12 φ Riemtθθt are the only non-vanishing components of Riem of a static spherically symmetric Lorenztian metric. (4) Show that static spherically symmetric solutions of the vacuum Einstein equations with the cosmological constant21 Λ take the form of g = −(1 −

2GM c2 r

−

2 Λ 2 3 r )dt

+ (1 +

2GM c2 r

−

Λ 2 −1 2 dr 3r )

+ r 2 gS 2 .

(5) Compute the Riemann, Ricci and scalar curvature of ndimensional cylinders R × S n−1 with respect to the metric g = dr 2 + gS n−1 . Employ arguments based on symmetries whenever possible. Conserved quantities. (6) (a) Show that the differential equation r¨ + V  (r) = 0 is equivalent to the conservation law for 12 r˙ 2 + V (r). 2

J (b) Show that 12 r˙ 2 − GM r + 2r 2 is the conserved quantity J2 for the equation r¨ + GM r 2 − r 3 = 0.

(c) Show that the collection of the three conservation laws (4.19) is equivalent to the geodesic equations (4.18). Analysis of orbits. (7) Prove Kepler’s First Law: The orbit of every planet is an ellipse with the Sun at one of the focal points. 21 For Λ > 0 the space-time is called Schwarzschild-de Sitter space-time and for Λ < 0 the space-time is called Schwarzschild-anti de Sitter space-time, in honor of the Dutch physicist Willem de Sitter who was one of the first researchers to study space-times with non-zero cosmological constant.

178

4. General relativity (8) Go through all of the algebraic details and the special cases of our treatment of particle orbits in Schwarzschild spacetime. Make sure you address the details such as: why are c2 J 2 c2 J 2 (GM )2 = 12 and (GM )2 = 16 critical, what happens when c2 J 2 (GM )2

= 12 or

c2 J 2 (GM )2

= 16, what happens when J = 0, etc.

(9) Find photon orbits in Schwarzschild space-time. (This can be done in the exact same manner as what was done in the lecture except that the line element needs to be null along a photon orbit.) (10) Investigate particle orbits in a spherically symmetric vacuum with cosmological constant Λ < 0. Use the example of Schwarzschild - anti de Sitter space-time from Exercise 4.

4.4. Kruskal-Szekeres extension of Schwarzschild space-time 4.4.1. Introduction. One issue we blatantly ignored in Lecture 4.3 is that the expression (4.20) for the Schwarzschild metric is undefined at r = 2GM c2 . Though we never explicitly said so in Lecture 4.3 itself, we did make an implicit assumption throughout it: that r > 2GM c2 . Even though there is a sense in which the expression (4.20) defines 2 2 a Lorentzian metric when r < 2GM c2 , the signs on dt and dr are reversed from what we expect them to be. The work in Lecture 4.3 thus really only finds the expression for the metric in one coordinate patch. The catch is that this particular coordinate patch is only a part of a bigger manifold. The goal of this lecture is to describe the Schwarzschild space-time in its entirety. Along the way we offer the reader an introduction into the concepts of black holes and geodesic incompleteness. To avoid cumbersome expressions in this lecture we use the notation m = GM c2 . In effect m is the mass in units in which c = 1 and G = 1. Metaphorically speaking, if the choice of c = 1 makes it so that time is measured in meters then the choice of G = 1 and c = 1 makes it so that mass is also measured in meters.

4.4. Kruskal-Szekeres extension

179

4.4.2. Radial trajectories of light. The process of discovering “the rest” of the Schwarzschild space-time relies on a sequence of coordinate changes. The first of the coordinate changes achieves the repositioning of the trajectories of light so that they – much as trajectories of light in Minkowski space-time diagrams – trace out lightcones in coordinates. To initiate this process we explicitly compute radial trajectories of light, that is, equatorial (φ = π2 ; see Lecture 4.3) null geodesics with no angular momentum (J = r 2 θ˙ = 0). Since φ˙ = θ˙ = 0 in this situation, the condition that the geodesics are null leads to  2  −1 2  dt + 1 − 2m dr = 0. − 1 − 2m r r The latter can be rewritten as a separable differential equation dr dt

= ±(1 −

2m r ).

A direct computation shows that the general solution of this equation takes the form of    r −1 =C t ± r + 2m log 2m for a constant C. For different values of the constant on the righthand side we get trajectories depicted on the left portion of Figure 4.8. Due to the vertical asymptote at r = 2m the values   r −1 ρ = r + 2m log 2m range throughout the entire R as r ranges through (2m, ∞). t t

r

ρ

Figure 4.8. Radial trajectories of light in Schwarzschild space-time

180

4. General relativity

To “straighten” these trajectories we introduce   r  ρ = r + 2m log 2m − 1 or equivalently, (4.22)  r  eρ/(2m) = er/(2m) 2m −1 . With respect to this new coordinate the radial trajectories of light take the form t ± ρ = C; these trajectories are depicted on the right portion of Figure 4.8. Note that coordinates (t, ρ) now range throughout R2 . A direct computation shows that dρ =

1 1− 2m r

dr.

It follows that the Schwarzschild metric in the (ρ, t)-coordinates can be expressed as    −dt2 + dρ2 + r 2 gS 2 . ds2 = 1 − 2m r Attempts to eliminate r completely do not lead to an algebraically manageable situation. One alternative coming from (4.22) is (4.23)

ds2 =

 e−r/(2m) ρ/(2m)  −dt2 + dρ2 + r 2 gS 2 , ·e r/(2m)

which at this stage is worth noting only because at first sight it appears that the expression is defined all the way to r = 0. (We should be cautious about any such claim because (4.22) requires r > 2m.) 4.4.3. Conformal mappings of Euclidean R2 and Minkowski R1+1 . In terms of straightening out the null geodesics, any choice of coordinates (u, v) would do in place of (ρ, t) so long as −dv 2 + du2 and − dt2 + dρ2 are conformally equivalent to each other, that is: they are scalar multiples of each other. (The proportionality factor for two conformally equivalent metrics is called the conformal factor .) The subject of conformal mappings is a cornerstone of any Complex Variables (Analysis) course. For the benefit of readers who have not yet gone through a course in Complex Variables we examine conformal mappings on a very specific example. Consider the function

4.4. Kruskal-Szekeres extension w = exp(z), i.e., the function u + iv = exp(x + iy) given by

181 

u = ex cos(y), v = ex sin(y).

Due to periodicity in y it is prudent to restrict the domain to the horizontal strip −π < y < π only. The effect of this mapping on coordinate grid-lines is depicted in Figure 4.9. y

v

x

u

Figure 4.9. Example of a conformal mapping of a Euclidean plane

The image of grid-lines x = const are circles of radii ex , while the images of y = const are rays. Note that a hypothetical gridline x = −∞ is mapped to the origin. A direct computation shows (Exercise 1 below) that du2 + dv 2 = e2x (dx2 + dy 2 ). Similar theory applies to Minkowski R1+1 in place of Euclidean R2 . On a purely formal level the change amounts to the replacement of the imaginary i which satisfies i2 = −1 with an imaginary i which satisfies22 i2 = +1. In terms of the example we just looked at the change amounts to the replacement of sin and cos with sinh and cosh. To understand the induced changes consider the mapping given by  u = ex cosh(y), v = ex sinh(y). 22 Numbers of the form a + ib where i2 = +1 are sometimes called paracomplex numbers.

182

4. General relativity

Its effect on coordinate grid-lines is indicated in Figure 4.10. The image of grid-lines x = const are hyperbolae with vertices at u = ex , while the images of y = const are rays. Note that x = −∞ is mapped to the origin of the uv-plane, while y = ±∞ are mapped to v = ±u. Finally, a direct computation (Exercise 1 below) shows that du2 − dv 2 = e2x (dx2 − dy 2 ). y

v

y=∞ x = const.

x

u

y = const.

y = −∞ Figure 4.10. Example of a conformal mapping of R1+1

4.4.4. Kruskal-Szekeres extension. Inspired by the last example we rewrite (4.23) as  e−r/(2m) ρ/(2m)  ds2 = 16m2 · d (ρ/(4m))2 − d (t/(4m))2 + r 2 gS 2 ·e r/(2m) and observe that it is fruitful to use the mapping  u = eρ/(4m) cosh(t/(4m)), v = eρ/(4m) sinh(t/(4m)). This mapping inserts the ρt-plane onto the wedge u > |v|. In fact, we now have (4.24)

ds2 = 16m2 ·

e−r/2m · (−dv 2 + du2 ) + r 2 gS 2 , r/2m

where the connection between r and (u, v)-variables is  r  u2 − v 2 = eρ/2m = er/2m 2m −1 .

4.4. Kruskal-Szekeres extension

183

The function ζ → eζ (ζ − 1) increasingly maps (0, ∞) onto the interval (−1, ∞). This in particular means that for all (u, v) with u2 −v 2 > −1 there is a unique value of r such that  r  −1 . u2 − v 2 = er/2m 2m In other words, the subset u2 − v 2 > −1 of the uv-plane can be equipped with the Lorentzian metric (4.24). This new space-time is known as the Kruskal-Szekeres extension of Schwarzschild space-time, and is named in honor of Martin Kruskal and George Szekeres who discovered it independently in the 1960’s. The static Schwarzschild space-time which we originally introduced is only a portion – the wedge u > |v| – of the extended space-time.

v

t=∞ r = 2m

r=0 u r=0 t = −∞ r = 2m Figure 4.11. Kruskal-Szekeres diagram

The Kruskal-Szekeres extension is schematically represented in Figure 4.11. The hyperbolae in the diagram correspond to lines of constant r. The region(s) where u2 − v 2 < 0 correspond to 0 < r < 2m. The hyperbolae u2 − v 2 = −1, marked in bold, correspond to r = 0. Lines of constant t are represented by the rays passing through the origin. The lines u = ±v both correspond to t = ∞ and r = 2m. The only misleading feature of the diagram is that each point in it, due to spherical symmetry and suppressed θ and φ variables, actually represents an S 2 -worth of space-time events.

184

4. General relativity

Note that, due to evenness of all of our expressions in u and v, the static Schwarzschild space-time can also be identified with the wedge u < −|v|. In other words, there are two symmetric copies of the static Schwarzschild space-time within the Kruskal-Szekeres space-time. 4.4.5. The four zones of the Kruskal-Szekeres diagram. There are four zones of the Kruskal-Szekeres diagram that should be distinguished: Zone I where u > |v|, Zone II where v > |u|, Zone III where u < −|v| and Zone IV where v < −|u|. The zones are illustrated in the diagram in Figure 4.12. t=∞ r = 2m

v II

A

III

u A

C

D IV

B I

t = −∞ r = 2m Figure 4.12. The four zones of the Kruskal-Szekeres diagram

Zone I corresponds to static Schwarzschild space-time as was initially introduced. Suppose two observers in Zone I: observer A heading for Zone II and observer B who is “standing still” in Zone I (in the sense that its r-coordinate is constant). The thick dashed lines in Figure 4.12 are (light) signals that the observer A is sending to the observer B. Signals emitted prior to A reaching r = 2m are received by B. While for A the signals appear to be sent at regular “time” intervals, for B it appears the signals are getting less frequent. Once A goes beyond r = 2m its signals are no longer reaching B. In fact, no information can escape Zone II. It is even worse: observer B will never know that A reached Zone II.

4.4. Kruskal-Szekeres extension

185

Zone II is called a black hole; the term is motivated by the fact that not even light can leave this zone. Note that the term black hole does not refer to a material astrophysical body! Instead, it refers to a region of a space-time whose geometry imposes a very peculiar behavior of time-like and null geodesics and curves. The 3-dimensional submanifold of the space-time described by r = 2m, the boundary of the black hole region, is called the horizon. Once the observer A crosses the horizon they start receiving signals from Zone III. Zone III is, as mentioned earlier, simply another copy of Zone I. However, no (light) signals sent by the observer C in Zone III will ever reach Zone I. (Likewise, no signal from Zone I will ever reach Zone III.) Yet, by crossing the horizon, observer A finds out about the existence of Zone III. For better or for worse, observer A is not able to share that information with their friend in Zone II. Finally, there is Zone IV. Observer D who found itself in Zone IV has no choice but to send signals out of it and ultimately leave it. In some ways Zone IV is a complete opposite of the black hole region, and is named a white hole.

4.4.6. Inextendibility of Kruskal-Szekeres metric. It is interesting to observe that r can still be used as a coordinate in the black hole (and the white hole) region, even though Figure 4.11 suggests that ∂r is time-like. In fact, inspired by the algebraic patterns that brought us to (4.24) in the first place one may want to examine the black hole region by means of the following reparametrization: 

  r sinh(t/(4m)), u = er/4m 1 − 2m   r r/4m v=e 1 − 2m cosh(t/(4m)).

The exact computation of the metric (4.24) in (u, v)-coordinates features a negative sign due to reversed roles of cosh and sinh, and a r r − 1 with 1 − 2m . Overall, the negative sign due to replacement of 2m signs cancel each other and one arrives at the same old expression for the Schwarzschild metric: ds2 = −(1 −

2 2m r )dt

+ (1 −

2m −1 2 dr r )

+ r 2 gS 2 .

186

4. General relativity

When 0 < r < 2m the ± signs of the first two metric components are opposite from what they are when r > 2m. As suspected from Figure 4.11 it is r that serves as a time coordinate in the black hole region. The computations of the Levi-Civita connection and the Riemann curvature tensor from Lecture 4.3 do not take into account the ± sign −1 . Thus, all the expressions of the functions A = 1 − 2m r and B = A we obtained in Lecture 4.3 still apply in the black hole region! Up to permutations of indices the following are the only non-vanishing components of Riem, even in the black hole region: Riemtrrt =

A 2 ,

Riemtφφt = 

Riemrφφr = − rA 2A =

rAA 2

=

1 Riemtθθt , sin2 φ

1 Riemtθθt , sin2 φ

Riemθφφθ = r 2 sin2 φ(1 − A).

(Compare with (4.15) and Exercise 3 from Lecture 4.3.) It is natural to wonder if the Kruskal-Szekeres extension can be extended further, past r = 0. To this end inspect the behavior of the sectional curvature 

Sec(Span{∂t , ∂r }) = − A2 =

2m r3

as r → 0. It becomes infinite! This is in sharp contrast to what happens as r → 2m, when it is finite. The point here is that the behavior of curvature – exactly because it is coordinate independent – can serve as an insight into (in)extendibility. If a scalar quantity based on curvature becomes infinite, then an issue at hand is a genuine geometric issue and not a problem of our choice of coordinate patches. The investigation of the scalar quantity which in our case proves that the Kruskal-Szekeres extension is indeed maximal is left for the reader; see Exercise 5 following this lecture. 4.4.7. Geodesic incompleteness. There is another computation from Lecture 4.3 which we can recycle in the black hole region – that of time-like geodesics. Recall the conserved quantities (4.19) associated with unit time-like geodesics. Since in the black hole region we have A = 1 − 2m r < 0 it follows that  2   dr 2 2 J = E − A + 1 ≥ 2m 2 dτ r r − 1,

4.4. Kruskal-Szekeres extension

187

with equality if and only if E = At˙ = 0 and J = r 2 θ˙ = 0. In particular, proper time τ satisfies the differential inequality  2m −1/2 dτ . dr ≤ r −1 The equality holds only when the geodesic on the Kruskal-Szekeres diagram of the black hole region follows a radial path t = const. The estimate on dτ dr provides an upper bound on the proper time along a time-like geodesic entering a black hole region. Integration using substitution w2 = 2m r − 1 yields  r=2m  u=∞  2m −1/2 4m τ≤ − 1 dr = dw = mπ. r (1 + w2 )2 r=0 u=0 This is an absolutely fascinating result! Since time-like geodesics maximize proper time, mπ is the longest time of existence of any time-like curve which enters the black hole region. In plain words, whoever falls into the black hole only has mπ to live! On the flip side, the “birth” from the white hole cannot possibly take longer than mπ. The behavior we see here is that of geodesic incompleteness, i.e., existence of inextendible time-like geodesics. Geodesic incompleteness is the subject of much-celebrated Singularity Theorems of Stephen Hawking and Sir Roger Penrose from the 1960’s. In addition to black holes another source of geodesic incompleteness is the phenomenon of big bang. The classic example on which to explain the concept of big bang is that of FLRW23 space-times, widely used mathematical toy models for studying the universe at large. These space-times take the form g = −dt2 + a(t)2 gE , where gE = dx2 + dy 2 + dz 2 and solve the Einstein equations for the perfect fluid T = −(ρ + P)dt2 + Pg. Readers who have completed Exercise 10 from Lecture 4.2 probably recall that  2 3 aa˙ = 8πρ and 3 aa¨ = −4π(ρ + 3P) 23 Named in honor of Alexander Friedmann, Georges Lemaˆıtre, Howard P. Robertson and Arthur Geoffrey Walker who worked independently on various aspects of these space-times throughout the 1920’s and 1930’s.

188

4. General relativity

are the necessary and the sufficient conditions for g = −dt2 + a(t)2 gE to be a solution of the Einstein equations. Under any kind of realistic assumptions on ρ and P we have a ¨ < 0, meaning that a is concave down. A positive concave down function must have a root! In other words, at least one of the following two scenarios must occur: a had a root in the past or a will have a root in the future. The situation where a had a root in the past is the classic example of what is called the big bang; its counterpart in the future is often called the big crunch. Explicit computational details regarding FLRW space-times are contained in Exercises 6 and 7 below. The big bang situation is tied to a being an increasing function, at least initially. The Hubble24 Law, the observation that astrophysical objects are moving away from us at a rate proportional to their distance to us, is evidence that a is currently increasing. The proportionality factor H = aa˙ , which admittedly does change with time, is called the Hubble parameter and can be estimated from astrophysical observations. To the extent to which the FLRW model of the universe is to be believed, there must have been a big bang in the past when all the universe was condensed in a single point! Predictions of general relativity are not to be taken too literally at small scales because at such scales one has to take into account effects of other physical theories (e.g. quantum theories). However, a feature of past-directed time-like geodesics here is worthy of notice: they are inextendible! In fact, the (proper) time t of their existence cannot be more than the current value of a · (a) ˙ −1 = H −1 ! The reason behind this is that the amount of time that a function which increases at least at the rate of a˙ (which indeed is the case due to a ¨ < 0) takes to reach the value of a is at most a · (a) ˙ −1 . Based on current values of the Hubble parameter we can compute that the time of existence of past-directed time-like geodesics is at most 2 · 1010 years. One could say that our universe is at most 2 · 1010 years old!

24 The law is named in honor of the American astronomer Edwin Powell Hubble. However, the discovery of the law seems to go back to the work of Georges Lemaˆıtre in the late 1920’s.

4.4. Kruskal-Szekeres extension

189

The claim of many of the Singularity Theorems is that such an incompleteness of past-directed time-like geodesics is a common feature amongst space-times and is neither a side effect of the simplicity nor the symmetries of the FLRW space-times. 4.4.8. Prototype black hole initial data. Developing theorems which predict singularity or black hole formation based on initial data alone is one of the many tasks of those who study initial value formulation in general relativity. The v = 0 slice of the Kruskal-Szekeres extension provides a good background on which to learn about the subject. We now study initial data for the Kruskal-Szekeres extension arising from the v = 0 slice. The values of the function r on the v = 0 slice, looking from right to left on the Kruskal-Szekeres diagram, range from r = ∞ down to r = 2m and then symmetrically back up to r = ∞. As each point on the diagram represents a (two-dimensional) sphere, it follows that the geometry of the slice is of the hour-glass type; see Figure 4.13. r = ∞,  = ∞, Zone I r = 2m,  = m/2 r = ∞,  = 0, Zone III Figure 4.13. Schwarzschild initial data

The 2-dimensional sphere located at r = 2m is a minimal surface in the sense that small perturbations of it have larger surface area. (See Exercise 10 for further details.) Given the nature of r = 2m within the larger space-time, this minimal surface is often also referred to as a horizon. The configuration in Figure 4.13 is sometimes called a wormhole. It is our obligation to point out that this nomenclature is somewhat misleading. There is no way for a signal from an observer in Zone I to ever reach Zone III (and vice versa). No observer could use

190

4. General relativity

the wormhole to “travel” between “parallel universes”. In fact, the wormhole gives rise to the black hole! We now examine the metric g and the second fundamental form K of the v = 0 slice in more detail. First of all, it follows from the symmetry v → −v of the space-time that K = 0. (For reasons that are apparent now, initial data with K = 0 are said to be timesymmetric.) Each of the two “halves” of the slice are parametrizable by r and standard spherical variables. The metric g in each coordinate patch takes the form of −1 2  dr + r 2 gS 2 . g = 1 − 2m r Note that we have g ≈ gE as r → ∞. It follows that the metric has two asymptotically Euclidean ends. Asymptotically Euclidean behavior is something we expect when addressing an isolated gravitational system because we expect gravitational effects to be negligible far away from the system. The exact definition of the phrase asymptotically Euclidean varies slightly in the literature, but the common feature of all is more precise information about the decay of g − gE far into an asymptotic end. For us here asymptotically Euclidean metric means a set of coordinates {xi } on each open non-compact end in which the components of g − gE = g − i (dxi )2 are on the order of O( r1 ), their first derivatives are on the order of O( r12 ), their second derivatives are on the order of O( r13 ), etc. On any given asymptotically Euclidean end there may be many different choices of such coordinates {xi }. We refer to any such choice of coordinates as coordinates in which the metric is manifestly asymptotically Euclidean. It is valuable to observe that the metric g is conformally flat, i.e., that (after a change of coordinates) it is conformally equivalent to the Euclidean metric gE . To find an appropriate conformal factor, which we henceforth denote by ξSchw , and an appropriate change of variables25 r ↔  one needs to solve ⎧   −1 2 ⎨ √ dr 4 2 = d/, 1 − 2m dr = ξ d , Schw r r(r−2m) i.e.,  2 4 2 ⎩ r = ξSchw  . ξSchw = r/. 25 Variable used here has nothing to do with the variables and ρ used earlier in these lecture notes; it is just a temporary auxiliary variable.

4.4. Kruskal-Szekeres extension

191

Upon integration we obtain r =+m+

m2 4

and ξSchw = 1 +

Consequently, the metric g takes the form of  −1 2  ds2 = 1 − 2m dr + r 2 ds2S 2 = 1 + r

m 2 .

m 2

4 gE .

The relationship between r and  is not one-to-one. Viewed as a function of ρ the function r has the following features: • It decreases for 0 <  < m/2. Specifically, we have r → +∞ when  → 0+ and the local minimum of r = 2m for ρ = m/2. • It increases for m/2 <  < ∞. Specifically, we have r → +∞ when  → +∞. • It is symmetric in the sense that r() = r(m2 /4). Exercise 8 below asks the reader to manually verify that the inversion  ↔ m2 /4 indeed serves as a symmetry of the metric g. The latter is completely in correspondence with the symmetry between Zone I and Zone III of the Kruskal-Szekeres metric. In particular, we get to think of  as a spatial radial coordinate which simultaneously encompasses the two halves of the initial data. For  > m/2 we get one of the two halves of the hour-glass, while for  < m/2 we get the other. These two halves are connected at  = m/2 (i.e., r = 2m), the minimal sphere which yields the horizon and the black hole in its evolution. Precise statements of singularity/black hole formation results in general are well beyond the scope of these lecture notes. Their overall sense is that the existence of minimal surfaces within initial data (or generalizations of these surfaces when K = 0) is an indicator of singularity formation. 4.4.9. More general remarks on conformally flat initial data. To anyone familiar with Newtonian gravity the expression for the conformal factor ξSchw appears to be remarkably similar to the expression

192

4. General relativity

for the gravitational potential of the point mass! We end the lecture with an examination of this similarity. 4 Schwarzschild initial data (g, K) = (ξSchw gE , 0) belong to the genre of conformally flat, asymptotically Euclidean, time-symmetric initial data. More generally such data take the form of

(g, K) = (ξ 4 gE , 0) with |ξ − 1| = O( r1 ), |∂ξ| = O( r12 ), ... as r → ∞. To see that the conformal factor ξ in such a set-up generalizes the notion of the gravitational potential consider the constraint equations (4.11). With the absence of the second fundamental form K there really is only one constraint equation and it places restrictions on the scalar curvature of g. In the example of relativistic dust—i.e., perfect fluid with P = 0—this constraint equation reads26 as Scal g = 16πρ, where ρ denotes the mass-energy density. This equation already resembles a Poisson equation: it suggests that we are solving for a metric g which is a response to the presence of matter ρ. However, there is a certain amount of circular reasoning here because ρ is density! Informally speaking, it makes little sense to prescribe matter in kg units of m 3 in order to find the metric which informs us about the meaning of meters in the first place. It would be far more appropriate to, metaphorically speaking, prescribe matter in units of kg. To accommodate this perspective we multiply the constraint equation with dvol and seek the metric g for which Scal g dvolg is prescribed. A computation which is best left as an exercise for the reader (see Exercise 3 following this lecture for guidelines) shows that for conformally flat metrics g = ξ 4 gE we have Scal g dvolg = −8ξΔgE ξdvolgE . Prescribing matter in the “kilogram-format” now means a specification of ωdvolgE for some function ω. Note that ω and ρ do not have the same physical interpretation: ω measures the number of “kilograms” per unit of Euclidean volume and thus ω = ξ 6 ρ. The constraint equation reads as −8ξΔgE ξdvolgE = 16πωdvolgE , i.e., ξΔgE ξdvolgE = −2πω. 26

When G and c are restored the equation becomes Scal g =

16πG ρ. c2

4.4. Kruskal-Szekeres extension

193

The latter clearly reveals the interpretation of the conformal factor ξ as a generalization of the Newtonian gravitational potential. For the purposes of our lecture the asymptotic boundary value problem  (4.25) ξΔgE ξdvolgE = −2πω, ξ ∞ = 1 is referred to as the relativistic Poisson problem27 . The following theorem can be seen as a relativistic counterpart to Newtonian Theorems 4.1 and 4.2. Theorem 4.3. Suppose that ω is a non-negative, compactly supported and smooth function on R3 . There exists a unique positive solution ξ of the relativistic Poisson problem (4.25). Furthermore, the function ξ satisfies the asymptotic conditions |ξ − 1| = O( r1 ), |∂ξ| = O( r12 ), ... as r → ∞. Techniques needed to establish this theorem are PDE-based and representative of the field of mathematics called geometric analysis. These techniques are central to many areas of geometry and general relativity. For this reason we chose Theorem 4.3 and its further consequences as a playground on which to explain basic ideas of geometric analysis, the concluding subject of the course. 4.4.10. Exercises for Lecture 4.4. Conformal changes. (1) (a) Consider the mapping u = ex cos(y), v = ex sin(y) from the horizontal strip −π < y < π of the xy-plane to (the punctured) uv-plane. Show that   du2 + dv 2 = e2x dx2 + dy 2 . (b) Consider the mapping u = ex cosh(y), v = ex sinh(y) 27

G With G and c restored the equation states ξΔgE ξdvolgE = −4π 2c 2 ω.

194

4. General relativity from the xy-plane to wedge u > |v| of the uv-plane. Show that   −dv 2 + du2 = e2x −dy 2 + dx2 . (2) Consider the inversion mapping x → y =

x |x|2

from the punctured R2 to itself. (a) Describe the images of circles centered at the origin under this mapping. Examine the behavior of the images as the radius of the circle approaches zero, and as the radius of the circle approaches ∞; (b) Show that, with the exception of the coordinate axes, the Cartesian coordinate grid-lines map to circles passing through the origin and with the diameter positioned orthogonally to the grid-line. (c) Sketch the effect of the inversion mapping on the Cartesian coordinate grid-lines. (d) Show that the inversion mapping is a conformal mapping of the Euclidean R2 . Specifically, show that |dy|2 =

2 1 |x|4 |dx| .

(e) Describe what happens if instead you consider inversion as a mapping on punctured Rn . (f) Repeat the prompts above with the expression for inversion changed into x x → y = R2 |x| 2.

(3) Consider two conformally equivalent Riemannian metrics g ˜ Riem,  and g˜ = e2f g on an n-dimensional manifold. Let Γ, # Scal denote the Christoffel symbols and the Riemann curvature of g˜. (a) Show that ˜ kij = Γkij + Lijk (∂f ) Γ

4.4. Kruskal-Szekeres extension

195

where Lijk denotes some linear polynomial of the first partial derivatives of f with coefficients which are expressions in components of g and g −1 . (b) Suppose that at a point P the coordinates {xi } are inertial with respect to g. Show that at P we have l

 ijk = Riemijkl + Lijk l (∂ 2 f ) + Qijkl (∂f ). Riem Here Lijk l denotes some linear polynomial in second 2

f l coordinate derivatives ∂x∂α ∂x β and where Qijk denotes some quadratic polynomial in first coordinate deriva∂f tives ∂x α , both with constant coefficients which are independent of g at P .

(c) Argue that   # = e−2f Scal + AΔg f + B|gradf |2 Scal g for some constants A and B which are independent of g and potentially dependent on the dimension n. (d) Use the fact that the standard metric on S n is conformally equivalent28 to the Euclidean metric to show that A = −2(n − 1) and B = −(n − 1)(n − 2). (e) Suppose that n ≥ 3 and set e2f = ξ 4/(n−2) . Show that # = ξ −4/(n−2) Scal − Scal

4(n−1) n−2

ξ −(n+2)/(n−2) Δξ.

 2 m gE on punctured R3 . (4) Consider the metric g = 1 + 2|x| (a) Compute the scalar curvature of g. (Use Exercise 3.) (b) The geometry of g has two asymptotic ends: one when |x| → ∞ and one when |x| → 0. Investigate the behavior of the scalar curvature on each asymptotic end. 28

Recall the stereographic projection!

196

4. General relativity

Geometry of the black hole region. (5) (a) Let Riem denote the Riemann curvature tensor of a pseudo-Riemannian metric g. Show that the scalar quantity  gia gjb gkc gld Riemijkl Riemabcd R= ijkl,abcd

is independent of choice of coordinates. (b) Let Riem denote the Riemann curvature tensor of the Kruskal-Szekeres extension. Show that R = Cr −6 for some non-zero constant C. Geodesic incompleteness. (6) Consider the relativistic perfect fluid T = (ρ + P)dt2 + Pg where g = −dt2 + a(t)2 gE for some non-negative function a(t) and where ρ(t) and P(t) are spatially constant. (The reader would benefit from reviewing Exercises 4 and 10 from Lecture 4.2.) (a) Show that the system of equations consisting of the Einstein equations and the conservation laws div T = 0 is equivalent to the system   2 3 aa˙ = 8πρ, ρ˙ + 3(ρ + P) aa˙ = 0. (b) Suppose the equation of state P = λρ for some constant λ ≥ 0. Show that ρa3+3λ = Const. (c) Show that a(t) satisfies the differential equation of the form C (a) ˙ 2 − 1+3λ = 0 a for some non-negative constant C. (d) Find the explicit solutions of the system of equations consisting of the Einstein field equations and the perfect fluid conservation laws.

4.4. Kruskal-Szekeres extension

197

(e) What can you say about the (proper) time of existence of such a perfect fluid “universe”? Distinguish the cases when a˙ is found to be positive, and the case when a˙ is found to be negative. (7) Consider the setting of Exercise 6 with g = −dt2 + a(t)2 gS 3 where gS 3 denotes the standard metric on unit S 3 . (a) Show that   2 3 aa˙ = 8πρ − a12 , ρ˙ + 3(ρ + P) aa˙ = 0. (b) Suppose in addition the equation of state P = λρ for some constant λ ≥ 0. Show that a(t) satisfies the differential equation of the form C +1=0 a1+3λ for some non-negative constant C. (a) ˙ 2−

(c) Show that a is concave down and bounded from above. (d) What can you say about the (proper) time of existence of this perfect fluid “universe”? Geometry of Schwarzschild initial data. 4  m gE on punctured R3 . (8) Consider the metric g = 1 + 2|x| (a) Show that the inversion mapping x→

m2 4|x|2

·x

is a symmetry of g. Describe this symmetry pictorially, in terms of Figure 4.13. (b) The metric g has two asymptotically Euclidean ends: one when |x| → ∞ and one when |x| → 0. For each of the ends find coordinates in which the metric g is manifestly asymptotically Euclidean.

198

4. General relativity (9) Describe the evolution of initial data 4  m gE , K = 0 g = 1 + 2|x| with respect to time function v of (4.24). Specifically: (a) Sketch the overall shape of each geometry g(v); (b) Compute locations and sizes of any horizons and/or minimal surfaces of g(v). (10) Let f : S 2 → (0, ∞) be a smooth function and consider the surface Σf given by r = f , i.e.,

x = f (θ, φ) cos θ sin φ, y = f (θ, φ) sin θ sin φ, z = f (θ, φ) cos φ where standard domains of θ and φ variables are assumed. (a) Show that the surface area of Σf computed with respect 4  m gE is given by to g = 1 + 2|x|

4   m f f 2 + |gradf |2 dvolS 2 , Area(Σf ) = 1+ 2f S2 where dvolS 2 is the area element of standard unit S 2 . (b) Suppose now that fε is a perturbation of f . Compute d d2 Area(Σfε ) and Area(Σfε ). dε dε2 (c) Show that Area(Σfε ) reaches its local minimum at the constant function f = m/2.

Chapter 5

Introduction to geometric analysis

5.1. The (relativistic) Poisson problem 5.1.1. Introduction. In this lecture we provide an analysis of the Poisson equation promised in Lecture 4.2, and address the corresponding relativistic problem we introduced in Lecture 4.4. Rather than just stating and applying theorems from analysis or the theory of partial differential equations (PDEs), we choose to examine strategies and overarching themes of solving (non-linear) PDEs. The reason for this is perhaps best articulated by Jerry L. Kazdan in [5]. In many ways, partial differential equations is a subject whose essence is more a body of techniques rather than a body of theorems. One of the easiest ways to learn these techniques is to see how they can be applied to simple examples. To avoid the situation where the burden of extracting “the body of techniques” from “the body of theorems” entirely falls on the reader we present the proofs below in a somewhat unconventional way, organizing steps in the proofs according to the corresponding PDE concept or technique as opposed to the logically most efficient route to 199

200

5. Geometric analysis

the final conclusion. Also for the benefit of the reader we begin the narrative with a review1 of background information from analysis. 5.1.2. Background in analysis. Recall the multi-index notation we introduced in Lecture 4.2, according to which α = (α1 , ..., αn ) denotes an n-tuple of non-negative integers while ∂xα =

∂ |α| for |α| = α1 + ... + αn . (∂x1 )α1 ...(∂xn )αn

When the variable x is clear from context (or not important) we drop x from the subscript and simply write ∂ α . In addition, for a vector h we let hα be a shorthand for (h1 )α1 ...(hn )αn . Recall that Taylor’s Theorem in the multivariable context states that for a smooth (that is, infinitely differentiable) scalar function f of several Euclidean variables x and a small displacement h the following expansion applies:   cα ∂ α f x hα + E, f (x + h) = 0≤|α| 0 there is the “proximity” δ > 0 such that all values f (y) with y within the δ-proximity of x are themselves within ε from f (x): |y − x| < δ −→ |f (y) − f (x)| < ε. 5.1.3. Proofs of Theorems 4.1 and 4.2. The proofs of the two theorems feature four clusters of topics and techniques characteristic of PDEs. 5.1.3.1. Addressing regularity of a function/solution. The phrase regularity of a function commonly refers to the number of continuous derivatives of a function or even its integrability status. One of the common features of proofs behind many PDE results is that we first get ahold of a “solution” any which way, i.e., in low regularity. We then improve our knowledge of the regularity later. Note that we are in that exact situation with the expression for the gravitational potential! A physically intuitive reasoning brought us to the expression  ρ(x) dvolx , Φ(y) = G |y − x| and expressing the integral in spherical coordinates assures us that the integral converges and that Φ(y) is defined. However, the regularity of Φ(y) as a function of y is much less clear. It is at this stage that we establish that Φ is differentiable and compute its derivatives. We claim that derivatives of Φ satisfy  α  α ∂ ρ(x) ∂ ρ(y + x) dvolx = G dvolx . (5.1) ∂ α Φ(y) = G |y − x| |x| Consider a small displacement vector h and fix an integer l ≥ 1. Since ρ is itself smooth, it, according to Taylor’s Theorem, expands as follows:   cα ∂ α ρy+x hα + E. (5.2) ρ(y + h + x) = 0≤|α| 0 and consider a subdivision of (a fixed box containing) the support of ω into little boxes with diagonal no bigger than ε/(3C1 ). The overall number of such boxes depends on ε (and it may well be very large!) but it is finite. Thus, by increasing M finitely many times if necessary we can arrange it to be such that for all m ≥ M and for all centers y∗ of all the subdivision boxes we have |ξ2m (y∗ ) − ξ− (y∗ )| < ε/3 and

|ξ2m+1 (y∗ ) − ξ+ (y∗ )| < ε/3.

Each y within (a fixed box containing) the support of ω belongs to one of the subdivision boxes and so we have |ξ2m (y)−ξ− (y)| ≤|ξ2m (y) − ξ2m (y∗ )| + |ξ2m (y∗ ) − ξ− (y∗ )| + |ξ− (y∗ ) − ξ− (y)| ε ε + ε/3 + C1 · =ε 0 satisfying (4.25), their difference would satisfy ω (5.12) Δ(ξ1 − ξ2 ) = 2π · (ξ1 − ξ2 ). ξ1 ξ2 If we had ξ1 −ξ2 = 0 somewhere, then – without loss of generality – the function ξ1 − ξ2 would reach a positive internal maximum. However,

212

5. Geometric analysis

since 2π ξ1ωξ2 ≥ 0 the Strong Maximum Principle implies that ξ1 − ξ2 is a constant. This is a contradiction due to ξ1 − ξ2 → 0 as |x| → ∞. 5.1.5. Establishing the convergence of the limit sequence and addressing the regularity of the limit function(s). One of the delicate steps in the proof of Theorem 4.3 involved the rate of convergence of sequences {ξ2m (y)} and {ξ2m+1 (y)}. Ultimately, we were able to prove that after a certain “moment” M all values of ξ2m (y) are close to the corresponding values of ξ− (y), at least in cases when y is limited to compact sets K. In such situations we say that convergence ξ2m (y) → ξ− (y) is uniform in y ∈ K. Uniform convergence of functions means convergence as a whole and not just of the individual output values: ∀ε > 0, ∃M ∈ N, ∀m ≥ M, max |ξ2m (y) − ξ− (y)| < ε. y∈K

In the field of analysis one introduces the concept of a metric space: a set equipped with a distance function. In these lecture notes we have dealt with two genres of metric spaces. One was (subsets of) Rn where the distance between two elements x and y is given by |x − y|. The other genre was complete Riemannian manifolds where the distance between two points is determined by the length-minimizing geodesic. There is no reason why the set in question could not be a set of functions, and there is no reason why a distance between two functions could not be something that addresses not the individual outputs of two functions but a proximity of the two functions as a whole. When working with PDEs it is often essential to set the stage right by choosing a good function (metric) space to work with. This entails the choice of the domain, level of regularity and asymptotic conditions. For our purposes here4 it is sufficient to work within C l (K): the vector space of all scalar functions over a compact domain K which have l continuous (covariant) derivatives. The C l (K)distance between two functions u and v is defined to be u − vC l (K) where uC l (K) := max{|u| + |∇u|g + ... + |∇l u|g }. K

4

The reader should be warned that this is uncharacteristically simple.

5.1. Poisson equation

213

The latter expression is most commonly referred to as the C l (K)norm of u. The statement about uniform convergence we made earlier is nothing but the statement that the sequence of functions {ξ2m } converge to the ξ− in C 0 (K)-sense of the word. Through bootstrapping we ultimately showed that convergence of {ξ2m } to ξ− is in fact C l (K) convergence, for every l and every K. An alternative way of formulating the same is to say that {∂ α ξ2m } converges to ∂ α ξ− for all |α| ≤ l uniformly over all compact sets. See Exercise 4 following this lecture for further practice of this terminology and these concepts. In general, the phrase uniform refers to the ability to choose a quantity independently of a parameter. For example, we could say that the sequence {ξm (y)} is uniformly bounded from below by 1 and is uniformly bounded from above by the value C = max ξ1 : 1 ≤ ξm (y) ≤ C for all m, y. Uniform boundedness from above and below over a domain K is equivalent to C 0 (K)-boundedness, ξm C 0 (K) ≤ C. The statement (5.8) is the statement about equicontinuity of the collection of functions {ξm }, a concept we now explain. The definition of continuity requires an existence of δ for each ε; consult Section 5.1.2 if necessary. This value of δ is not actually required to be uniform in the domain variable x. In circumstances when δ can be chosen independently of x we say that a function is uniformly continuous. In circumstances when we have a collection of functions (such as a sequence {ξm } from our proof) and when in addition the value of δ can be chosen uniformly across this collection of functions we say that the collection is equicontinuous. The ε/3-argument following (5.8) is one of the major steps in the proof of the following theorem5 . Theorem 5.2 (Arzel´ a Ascoli Theorem). If a collection of functions on a compact domain is uniformly bounded and equicontinuous then it has a uniformly convergent subsequence. Compactness of the domain is essential for the result. A careful reader has probably noticed that we emphasized the finite number 5 The theorem is a result of combined efforts of Italian mathematicians Cesare Arzel´ a and Giulio Ascoli in the 1880’s.

214

5. Geometric analysis

of the subdivision boxes in our proof. Without compactness such a claim would be impossible. Our strategy towards ensuring equicontinuity was to make use of the Mean Value Theorem; see transition from (5.7) to (5.8). In general, uniform boundedness of the first derivatives of a collection of functions implies subsequential convergence. As indicated in our proof this strategy can even be further bootstrapped, leading to what is often called Rellich6 Lemma. Theorem 5.3 (Rellich Lemma). If for some l ≥ 0 a collection of functions {fα } on a compact domain K is uniformly bounded in C l+1 (K) then there is a sequence {fαn } which converges in C l (K). The crux of so many proofs of PDE results is in providing bounds for a family of functions and its derivatives. In our case here we are blessed to have such an explicit Green’s Representation Formula. Had this not been the situation we would have probably had to employ some far less elementary results, such as Elliptic Regularity Estimates. Finally, note that the Arzel´a Ascoli Theorem and the Rellich Lemma only guarantee subsequential convergence. In our case we were able to use monotonicity of the sequence to ensure the convergence of the entire sequence. For further ideas the reader is referred to Lecture 5.2 below; the lecture is meant to serves as a playground for “the body of techniques” developed in this lecture. 5.1.6. Exercises for Lecture 5.1. Practice with analysis details. (1) Prove the formula (5.3). (2) Prove Theorem 5.1 in the more general case. (3) (a) Consider the functions fn (x) := n2 xe−nx of a single real variable x. 6 The result is commonly attributed to Franz Rellich and is known under several different names such as the Rellich Lemma or the Rellich-Kondrakov Compactness Theorem.

5.1. Poisson equation

215

(i) Show that for all 0 ≤ x ≤ 1 the sequence of numbers fn (x) converges to zero as n → ∞.  x=1 (ii) Compute x=0 fn (x) dx. Is it true that  x=1  x=1 lim fn (x) dx = lim fn (x) dx ?

n→∞

x=0 n→∞

x=0

(b) Suppose the sequence of functions fn (x) defined on a compact domain K converges toward the function f (x) in C 0 (K)-sense: ∀ε > 0, ∃N ∈ N, ∀n ≥ N, ∀x ∈ K, |fn (x) − f (x)| < ε.   Assuming that both K fn (x) dvolx and K f (x) dvolx converge, show that    lim fn (x) dvolx = lim fn (x) dvolx = f (x) dvolx .

n→∞

K

K n→∞

K

(4) Suppose {un } is a sequence of smooth functions over R3 . Suppose that for all compact domains K and all multiindices α the sequence of functions {∂ α un } converges in C 0 (K)-sense. (a) Show that the function u(x) defined by u(x) = lim un (x) n→∞

is smooth and that for all compact domains K and all multi-indices α the sequence of functions {∂ α un } converges to ∂ α u. (b) Show that the sequence {un } converges to u in C l (K)sense, for all l and all K. Linear theory/Green’s Representation Formula. (5) (a) Suppose x ranges through a fixed compact domain in R3 . Use Taylor’s Theorem to show that there exist constants C and R such that for all y with |y| ≥ R we have 1 y, x

C 1 = + + E, with |E| ≤ . 3 |y − x| |y| |y| |y|3

216

5. Geometric analysis (b) Let ρ be a non-negative, smooth and compactly supported function in R3 . Furthermore, let  M = ρ(x) dvolx . Show that

 1 ρ(x) M dvolx = +O |y − x| |y| |y|2 as |y| → ∞. Furthermore, show that 



M 1 ρ(x) dvolx = grady grady . +O |y − x| |y| |y|3 (c) Show, under the same assumptions, that there exists a unique p ∈ R3 such that

 1 M ρ(x) dvolx = +O |y − x| |y − p| |y − p|3 as |y| → ∞. What is the expression for p? (6) How does the statement of Green’s Representation Formula from Theorem 4.2 differ between R3 and Rn when n = 3? Prove your claim! (Hint: what are spherically symmetric solutions f (r) of the equation Δf = 0 on Rn ?) (7) Suppose that ρ is a compactly supported and smooth function on R3 . Prove that there is a unique solution to the asymptotic boundary value problem Δu − u = −4πρ,

lim u(y) = 0

|y|→∞

and that it is given by  e−|y−x| u(y) = ρ(x) dvolx . |y − x| Non-linear problems. (8) (a) Let f (x) be a smooth function of x ∈ [a, b]. Show that the equation u (x) = f (x) has a unique smooth solution within the set of functions satisfying the boundary conditions u(a) = u(b) = 0.

5.1. Poisson equation

217

(b) Show that if u(x) and v(x) are two smooth functions satisfying the boundary conditions u(a) = u(b) = v(a) = v(b) = 0 and u ≤ v  then u ≥ v. (c) Suppose that F (u) is a non-increasing smooth function and that for some smooth functions u− ≤ u+ satisfying the boundary conditions u− (a) = u− (b) = u+ (a) = u+ (b) = 0 one has u− ≥ F (u− ) and u+ ≤ F (u+ ). Show that the smooth sequence defined by u0 = u− , un = F (un−1 ), un (a) = un (b) = 0 is monotone and bounded: u− ≤ u0 ≤ u1 ≤ ... ≤ u+ . (d) Show that the sequence of derivatives {un } is bounded in C 0 [a, b]. (e) Apply the Arzel´a Ascoli Theorem/Rellich Lemma to get a convergent subsequence {unk } in C 0 [a, b]. (f) Use monotonicity to prove {un } converges in C 0 [a, b]. (g) Use the Mean Value Theorem to show that the sequence {un } converges to a function u in C 2 [a, b]. (h) Show that u is a smooth solution of the boundary value problem u = F (u), u(a) = u(b) = 0. (9) Assume that F (u) is a strictly increasing smooth function of the scalar variable u. Show that if the equation Δu = F (u) has one (smooth) solution u, then the solution is unique. (10) The following problem builds on Exercise 7 above. Suppose that ρ is a compactly supported and smooth function on R3 for which 4π max |ρ|  1.

218

5. Geometric analysis Prove that there is a unique solution to the asymptotic boundary value problem Δu − u = −4πρe−u ,

lim u(y) = 0.

|y|→∞

5.2. On the concept of mass 5.2.1. Introduction. We began our discussion of gravity in Lecture 4.2 by describing the gravitational attraction between two (Newtonian) point masses, and by introducing the gravitational potential M Φ(x) = G |x|

for the (Newtonian) gravitational point-source. It is interesting to note that Δ Φ = 0, which to a naive mind might suggest absence of matter. The crux, of course, is that ΔΦ only vanishes away from the origin. To fully understand point masses one has to take into account the situation at the origin as well. The flux of grad Φ across spheres centered at the origin is easily computed to be −4πGM , making the rate of flux ΔΦ = div(grad Φ) at the origin infinite. In other words, ΔΦ in a point-source situation is zero everywhere except at the location of the point-source where it is infinite. Such a situation is commonly described in terms of the Dirac7 delta function. One of the goals of this lecture is to provide ways of making the concept of the Dirac delta function, as well as its role in Newtonian theory of gravity, more rigorous. We then use methods of geometric analysis to present reasons why the Dirac delta function is at least somewhat inadequate for modeling relativistic point-sources. This inadequacy is just one of very many reasons why the concept of mass in general relativity is so difficult. We conclude the lecture and the course with a brief survey of (the difficulties with) the concept of mass in general relativity. 7 Even though its appearance in mathematical literature goes back to at least Augustin Louis Cauchy and the 1820’s, the Dirac delta function is named in honor of the English (quantum) physicist Paul Adrien Maurice Dirac whose books from the first half of the 20th century introduced much of the notation currently in use in theoretical physics.

5.2. On the concept of mass

219

5.2.2. Why gravitational point-sources? Point-sources are always an idealization. There are at least two genres of situations where having such an idealization is beneficial. One situation arises when we are studying a compact object from a great distance, making the length scale of the object itself miniscule. Any deviations, inhomogeneities or lack of symmetry within a small object can then be assumed to be negligible, and one may hope to employ a mathematically simple (if “unrealistic”) toy model with a great deal of symmetry in which the entire object is concentrated at a single point. For the purposes of these lecture notes let us refer to such a standpoint as the macroscopic standpoint. Another situation when having a pointsource idealization is helpful is microscopic and was already described when we talked about continuum in Lecture 4.1. “Realistic” objects are made out of a great number of smaller constituents whose cumulative behavior manifests itself in the behavior of the object as a whole. However, keeping track of attributes of each individual constituent is, due to their great number, at least impractical if not impossible. Instead, we introduce averaged attributes which, in an infinitesimal form, are attached to idealized point-source constituents. A classic example of this line of reasoning is the employment of the concept of mass density, i.e., mass per unit volume, ρ(x). The idea here is to pretend that the idealized point-source at location x has mass is then the “sum” of the ρ(x) dvolx . The total mass of the object  masses of all the point-sources: M = ρ(x) dvolx .

5.2.3. The macroscopic approach: Newtonian gravity. Let us take the Poisson equation as a starting point of Newtonian gravity. M From such a standpoint we can see that Φ(x) = G |x| serves as the macroscopic point-source model in at least two different ways. One is captured in Subsection 5.1.3.2 and Exercise 5 from Lecture 5.1: the gravitational potential for any compactly supported dust cloud can be approximated as  Φ(x) = G assuming M =

M ρ(z) dvolz = G +O |x − z| |x| 

ρ(x) dvolx .



1 |x|2

when |x| → ∞,

220

5. Geometric analysis

Another macroscopic approach involves the change of length scale. Formula-wise, such a change can be achieved as follows: Start with a fixed, smooth, non-negative, compactly supported ρ(x) and introduce the sequence (5.13)

ρn (x) := n3 ρ(nx).

The idea here is that n enumerates different length scales and that we are ultimately considering n → ∞. The inclusion of the multiplicative n3 -term reflects the fact that ρ is a density – i.e., amount per unit volume – and that changing the sense of length scale also changes what we mean by unit volume. By Theorem 4.2 there exist unique solutions Φn to  ΔΦn = −4πGρn , Φn ∞ = 0. These solutions are given by  3  n ρ(nz) ρ(y) (5.14) Φn (x) = G dvolz = G dvoly . |x − z| |x − y/n| Let us examine the behavior of the sequence {Φn } as n → ∞. Theorem 5.4. Let Φn be the sequence of functions introduced in (5.14). If M = ρ(x) dvolx then Φn (x) → Φ(x) = G

M |x|

for all x = 0. Furthermore, the stated convergence is uniform with  all derivatives over all sets of the form {x  |x| ≥ a > 0}. Proof. For the purposes of this proof the reader should fix a > 0 and assume that y ranges through the support of ρ, which is a fixed, closed and bounded set. For each x = 0 we have 1 1 → as n → ∞ |x − y/n| |x| 1 by virtue of continuity of |x| . In fact, the stated convergence is uniform in y and we are allowed to come with the limit inside of the integral:   ρ(y) 1 M dvoly = G ρ(y) dvoly = G . Φn (x) → G |x| |x| |x|

5.2. On the concept of mass

221

M In particular, Φn (x) converges to Φ(x) = G |x| point-wise.   The convergence here is uniform over {x |x| ≥ a > 0}. Indeed, after increasing n so that all x and x − y/n are guaranteed to range within the (slightly bigger) set {z  |z| ≥ a/2 > 0}, the Mean Value Theorem implies an estimate of the form    1 1    |x − y/n| − |x|  ≤ C/n.

Here the size of the constant C depends on the diameter of the support 1 of ρ and the maximum of the first derivatives of z → |z| over |z| ≥ a/2. Integration of the stated inequality proves |Φn (x) − Φ(x)| ≤ GCM/n

 as well as the uniform convergence of Φn over {x  |x| ≥ a > 0}. The representation formula (5.14) can be differentiated under the integral sign as many times as desired, provided n is large enough so that |x|  |y|/n. The argument presented above, when performed on

1 1 α α ∂x − ∂x |x − y/n| |x| proves that the convergence towards Φ is uniform with all the derivatives.  5.2.4. Dirac delta function. The sequence of functions ρn we used in Section 5.2.3 does not converge over R3 —at least not in any straightforward sense of the word. On one hand, the fact that ρ is of compact support implies that for every x = 0 we have ρn (x) = 0 if n is large enough. Yet, the maxima of ρn are unbounded (they are growing at the rate of n3 ). Naively, at least, the limit of ρn as n → ∞ is exactly the situation we described in the introduction to this lecture: the limit is zero everywhere except at x = 0 where it is somehow infinite. For the purposes of these lecture notes Dirac delta function on R3 means (the hypothetical limit of) a sequence of the form (5.13) where ρ is a smooth, non-negative, compactly supported function with  ρ(y) dvoly = 1.

222

5. Geometric analysis

The word “function” here is at least somewhat inappropriate. The situation is not that (in principle, at least) there is some expression or a pattern into which we could input numerical values. Yet, “function” is a traditional word to use in this subject and so that is the word we use in these lecture notes as well. The traditional notation for the Dirac delta function is δ. Although we avoid doing so, it is also common to see δ(x). Note that in our definition the Dirac delta function is always centered at the origin. By replacing ρn (x) with ρn (x − p) we can recenter the delta function at any p. Notationwise, the reader could expect to see δp or δ(x − p) for the recentered function. We should mention that there are other perspectives on the Dirac delta function. Probably the most common one is rooted in so-called linear distribution theory. The upshot of that particular perspective, as it applies to us here, is captured by the following theorem. Theorem 5.5. Suppose u is smooth and compactly supported. We then have  u(p)δp . u(x) = p

A point of clarification is in order. Since δp is a shorthand for (the hypothetical limit of) the sequence of functions ρn (x − p), Theorem 5.5 is the statement that  u(x) = lim u(p)ρn (x − p) dvolp . n→∞

Proof. Using the very definition of the sequence {ρn } and the substitution y = n(x − p) we obtain   u(p)ρn (x − p) dvolp = u(x − yn )ρ(y) dvoly . Since u is continuous at x we have u(x − ny ) → u(x) as n → ∞. Given the compactness of the support of ρ the convergence u(x − yn )ρ(y) → u(x)ρ(y) as n → ∞

5.2. On the concept of mass

223

is uniform in y. It now follows that the limit as n → ∞ of the integral in question can be taken under the integral sign:   lim u(p)ρn (x − p) dvolp = u(x)ρ(y) dvoly . n→∞  The fact that ρ(y) dvoly = 1 now completes our proof.  Introductory paragraphs to this lecture toy with the idea that the M function Φ(x) = G |x| solves ΔΦ = −4πGM δ. The exact sense in which this indeed is the case has already been presented in Theorem 5.4. The reader should be informed though that in linear distribution theory the equation ΔΦ = −4πGM δ is understood in terms of weak derivatives of the function Φ itself. Exercise 4 below is designed to give the reader some insight into this alternative approach. Though the distribution theory approach works beautifully for linear equations and linear theories such as Newtonian gravity, it takes a considerable amount of technical skill to make it helpful for non-linear equations and general relativity. This particular circumstance was the main motivating factor for the approach to the Dirac delta function we chose. 5.2.5. The microscopic approach: Newtonian gravity. There is a huge aesthetic appeal in the expression  u(p)δp u(x) = p

of Theorem 5.5: it conveys the fundamental microscopic premise that a continuum of matter (in this case u(x)) is in effect made of pointsources (in this case u(p)δp ) in  a straightforward (if calculus-based) additive manner (in this case p u(p)δp ). According to Theorem 5.4 ρ(p) solves the linear Poisson problem for the pointthe function G |x−p| source ρ(p)δp . From this particular standpoint Green’s Representation Formula (Theorem 4.2) is simply a form of a superposition principle for the linear Poisson equation.

Admittedly, we already appealed to this particular superposition principle in Lecture 4.2 when we introduced the gravitational potential for the dust cloud and when we introduced the Poisson equation

224

5. Geometric analysis

in the first place! Yet the opening words of Section 5.2.3 were “Let us take the Poisson equation as a starting point of Newtonian gravity....”! It thus may appear that we are in a circular chicken-and-egg situation. More than anything the situation we have just found ourselves in is a reflection of the fact that in Newtonian gravity the macroscopic and microscopic approaches to point-sources are not only both possible but that they are compatible: the two are using the same Newtonian M . As we discuss below this particupoint-source model Φ(x) = G |x| lar set of circumstances is not something we can take for granted in general relativity. 5.2.6. The ADM mass and the macroscopic approach in general relativity. The universally accepted macroscopic approach in general relativity relies on the following observation: there indeed is a mathematically simple toy model with a great deal of symmetry which is a vacuum everywhere except at one point. What we are referring to here is the Schwarzschild body discussed in Lecture 4.3! While the phenomena discussed in Lecture 4.4 may want us to think twice about the phrase “everywhere except at one point”, we can at least be comforted by the fact that ξSchw = 1+ 2cG2 M r of Schwarzschild initial data 4  4 gE = 1 + 2cG2 M gE ξSchw r satisfies the relativistic Poisson equation ξΔξ = 0 on punctured R3 . The mass of a general asymptotically Euclidean Riemannian manifold ξ 4 gE is understood precisely in terms of comparison and/or ap4  4 proximation with some ξSchw gE = 1 + 2cG2 M gE when r → ∞. Inr formally, the idea is much in the spirit of what was done in the case of Newtonian gravity in Subsection 5.1.3.2 and Exercise 5 of Lecture 5.1. The idea is to look at the asymptotic expansion of ξ as r → ∞: 1 ξ = 1 + 2cG2 M r + O r2 , and define the mass of ξ 4 gE to be the constant M of this expansion. More precisely, the mass is defined by means of the flux of grad ξ across the coordinate spheres at infinity:  1 lim grad ξ, ∂r dvolSR2 . (5.15) − 4πG/(2c2 ) R→∞ |x|=R

5.2. On the concept of mass

225

It is remarkable that the last expression is extendible to any asymptotically Euclidean metric g: (5.16)

  ∂gij 1 ∂gjj mADM = lim − ν i dvolSR2 . 32πG/(2c2 ) R→∞ |x|=R ij ∂xj ∂xi Here {xi } are coordinates for an infinite end in which g is manifestly asymptotically Euclidean (see Section 4.4.8 in Lecture 4.4) and xi are the components of the standard outward pointing vecν i = |x| tor field ∂r . Discussion of where the formula (5.16) comes from and why it is independent of the choice of coordinates {xi } in which g is manifestly asymptotically Euclidean is very much beyond the scope of these lecture notes. An interested reader can find a very thorough presentation of the subject in [1]. The concept of mass defined in (5.16) is referred to as the ADM mass in honor of Richard Arnowitt, Stanley Deser and Charles W. Misner who first developed the concept in the early 1960’s. The reader should notice that the ADM mass is always computed 4 gE has relative to an asymptotic end. Schwarzschild initial data ξSchw two asymptotic ends: one where r → ∞ and the other where r → 0. The two ends are symmetric due to inversion symmetry (see Lecture 4.4), and thus the two ADM masses are the same and are equal to M . The reader is strongly encouraged to work through some examples of ADM mass computations. Exercise 5 below provides an excellent starting point. 5.2.7. Complications with the macroscopic approach in general relativity. In the case of Newtonian gravity there is an alternative macroscopic approach based on the Dirac delta function; see Theorem 5.4. We now perform the same kind of analysis in the relativistic context. In place of (5.14) we are now interested in asymptotic boundary problems   3  n ρ(nx) , ξn ∞ = 1. (5.17) ξn Δx ξn = −4π GM 2c2 Based on the understanding of the Newtonian situation, and the understanding that a Schwarzschild body is a relativistic counterpart to

226

5. Geometric analysis

the Newtonian point-source, we expect that ξn (x) → 1 +

G 2c2

·

M |x| .

As we shall see, the situation with ξn and the resulting geometries (ξn )4 gE is far from this simple. In fact, a surprise is waiting for us at the origin. To study the functions ξn near the origin we make use of coordinates y = nx; they in essence zoom in at the origin. In these coordinates our equation becomes ξn (y/n)Δy ξn (y/n) = −4π GM 2c2 nρ(y). The intention now is to absorb the factor of n into the left-hand side. This motivates us to define a new function Ξn (y) :=

√1 ξn (y/n). n

Note that this function satisfies

  Ξn Δy Ξn = −4π GM 2c2 ρ(y), Ξn ∞ =

√1 . n

It follows from Subsection 5.1.3.2 that the function Ξn (y) − the following asymptotic behavior:      α  ∂ (Ξn (y) − √1n ) = O |y|−1−|α| as |y| → ∞.

√1 n

has

Our next goal is to study the convergence of the sequence {Ξn }. Theorem 5.6. The sequence of functions {Ξn } converges to a smooth positive function ΞF on R3 . Furthermore, this convergence is uniform with all derivatives over all compact subsets of R3 . Proof. The overall plan is to obtain the convergence through applications of the Rellich Lemma. To be able to do so we need C l -bounds on the sequence {Ξn }. We obtain such bounds through the following observation: the functions Ξn satisfy Δ(Ξ2n ) = 2Ξn ΔΞn + 2|gradΞn |2 ≥ −8π GM 2c2 ρ  1 along with Ξ2n ∞ = n . In fact, because of the asymptotic conditions satisfied by Ξn we know that    α 2  ∂ (Ξn (y) − 1 ) = O |y|−1−|α| as |y| → ∞. n

5.2. On the concept of mass

227

It follows that the alternative version of Green’s Representation Formula (Theorem 5.1) applies to the function Ξ2n − n1 . Consequently, we obtain  ρ(z) GM 1 dvolz . Ξ2n (y) ≤ + 2 n c |y − z| The integral on the right-hand side of the stated inequality is, as seen in Lecture 5.1, a (non-negative) continuous function of y which approaches 0 at infinity. As such it reaches its maximum value in the interior on R3 . In particular, there is a positive constant A such that Ξ2n ≤

1 n

+A≤1+A

and consequently Ξn (y) ≤ 1 + A for all n and all y. Interestingly, such boundedness from above also proves that the sequence {Ξn } over the support of ρ is bounded from below by a positive value. To see this, note that as a continuous function over the support of ρ the function  GM ρ(z) y → 2 dvolz c |y − z| reaches it minimum value. We denote it by a and note that it is a strictly positive number. We now have  1 GM 1 a ρ(z) · dvolz > , Ξn (y) = √ + 2 c Ξn (z) |y − z| 1+A n provided that y is within the support of ρ. Next, our goal is to prove boundedness of the derivatives of Ξn . To do so we use the fact that we may differentiate Green’s Representation Formula under the integral sign once:

 ρ(z) 1 GM 1 ∂Ξn (y) = √ + 2 ∂ dvolz . c Ξn (z) |y − z| n It thus follows that |∂Ξn (y)| ≤ 1 +

GM 1 + A · c2 a



ρ(z) dvolz . |y − z|2

Boundedness of the last integral was something we established at several places in Lecture 5.1, most notably in the proof of (5.7). Note that the last estimate applies to all y. The same bootstrapping procedure used in Section 5.1.4.3 of Lecture 5.1 now shows that |∂ α Ξn (y)|

228

5. Geometric analysis

are bounded for each α. In other words, we now know that the sequence {Ξn } is bounded in C l (K) for each l and each compact K. The next step in the proof is to obtain a function over R3 which serves as a subsequential limit of {Ξn }. The Rellich Lemma only applies over compact domains, an obsticle we now circumnavigate through and employment of what is known as a diagonal argument. Specifically, consider a sequence {Ki } of compact subsets of R3 with $ K1 ⊆ K2 ⊆ K3 ⊆ ... such that Ki = R 3 i

and consider the table ⎡

Ξ1,1 ⎢Ξ2,1 ⎣ .. .

Ξ2,1 Ξ2,2 .. .

Ξ3,1 Ξ3,2 .. .

⎤ ... . . .⎥ ⎦

in which rows are convergent subsequences of {Ξn } created as follows: Let {Ξn,1 } be a subsequence of {Ξn } (whose existence is guaranteed by the Rellich Lemma) which converges in C 1 (K1 ). Next, consider the sequence {Ξn,1 } and the fact that it is bounded in C 3 (K2 ). By the Rellich Lemma one can extract a subsequence {Ξn,2 } of {Ξn,1 } which converges in C 2 (K2 ). Note that because the sequence {Ξn,2 } is a subsequence of {Ξn,1 } the limit function of {Ξn,2 } in C 2 (K2 ) is identically the same over K1 as the limit function of {Ξn,1 } in C 1 (K1 ). In other words, the limit function of {Ξn,2 } over K2 extends the limit function of {Ξn,1 } over K1 . Loosely speaking, each row in the table is created by extracting a convergent subsequence from a previous row. The limit function of each row is an extension to a bigger (compact) set of the limit function from a previous row. The consideration of these limit functions yields ΞF which is now defined over all of R3 . The fact that ΞF is positive (for y in the support of ρ) is a consequence of ΞF (y) ≥

a 1+A .

Next, consider the diagonal subsequence of functions {Ξn,n }. By construction {Ξn,n } is a subsequence of each {Ξn,l } and it thus converges to ΞF in each C l (Kl ). In other words, the diagonal subsequence

5.2. On the concept of mass

229

{Ξn,n } converges uniformly with all derivatives over all compact subsets of R3 . The limit function ΞF is smooth and satisfies  1 GM ρ(z) · dvolz . ΞF (y) = 2 c ΞF (z) |y − z| Consequently, the function Ξ = ΞF solves the problem   (5.18) ΞΔΞ = −4π GM 2c2 ρ, Ξ ∞ = 0. Note that the uniqueness of solutions of (5.18) follows from the Strong Maximum Principle exactly as in Theorem 4.3. It remains to show that the whole sequence {Ξn } converges to ΞF with all derivatives on all compact sets K. To do so we suppose the opposite, that there are some l, K, ε0 > 0 and a subsequence Ξnk with (5.19)

Ξnk − ΞF C l (K) ≥ ε0

for all k. By applying what we have already proven to sequences {ρnk } (as opposed to the full sequence {ρn }) we obtain a subsequence of Ξnk which converges with all derivatives to a solution of (5.18). However, ΞF is the unique solution of (5.18). The latter contradicts (5.19) and  proves that Ξn → ΞF with all derivatives over all compact K. Here is an effectively immediate consequence of Theorem 5.6. Note that the result already constitutes a departure from the expected situation of recovering the Schwarzschild body in the limit. Theorem 5.7. Consider the sequence of functions {ξn } from (5.17). These functions converge to the constant function 1 over R3  {0}. The convergence is uniform with all derivatives over all sets of the  form {x  |x| ≥ a > 0}. ρn Proof. Since Δξn = −4π GM 2c2 · ξn the Green’s Representation Formula shows that  GM n3 ρ(nz) dvolz ξn (x) =1 + 2 2c ξn (z)|x − z|  GM ρ(y) dvoly =1 + 2 2c ξn (y/n)|x − y/n|  ρ(y) GM 1 √ dvoly =1 + 2 2c Ξn (y)|x − y/n| n

230

5. Geometric analysis

As consequences of Theorem 5.6 and continuity of we have convergences Ξn (y) → ΞF (y) and

1 |z|

at each z = 0

1 1 → . |x − y/n| |x|

These convergences are uniform in y, provided y only ranges over the support of ρ. In fact, by using the Mean Value Theorem the second of the two convergences can be shown to be uniform in x, provided we  restrict x to range over sets of the form {x  |x| ≥ a > 0}. (Compare the latter with the corresponding step in the proof of Theorem 5.4.) Overall, we have that   ρ(y) ρ(y) 1 dvoly → · dvoly and Ξn (y)|x − y/n| |x| ΞF (y)  ρ(y) 1 √ dvoly → 0 as n → ∞. Ξn (y)|x − y/n| n Just as in the proof of Theorem 5.4 the stated convergences are uni  form with all the derivatives over all sets of the form {x |x| ≥ a > 0}. This completes our proof.  Before we move on to a discussion of geometries ξn4 gE it is worthwhile to digress and observe the following. Theorem 5.8. Consider the function ΞF addressed in Theorem 5.6. The metric gS 3 = ξF4 gE defines a smooth metric on S 3 . Proof. Recall from Lecture 1.1 that the sphere S 3 can be covered by two coordinate charts arising from stereographic projections: one from the North Pole and the other from the South Pole. The transition map between the two charts is given by the inversion y x , i.e., y ↔ x = . x↔y= |x|2 |y|2 (Compare also with Exercise 5 following Lecture 1.1.) To prove that the metric Ξ4F gE = ΞF (x)4 |dx|2 extends to a smooth metric on S 3 it suffices to express it in y-coordinates and then verify that the found expression extends smoothly to y = 0. A computation shows that |dx|2 = |y|1 4 |dy|2 ; see also Exercise 2 following Lecture 4.4. This observation motivates us to consider the

5.2. On the concept of mass metric

⎛

231

⎞4 1 ρ(z)   dvolz ⎠ · |dy|2 4  y  |y| ΞF (z)  |y|2 − z ⎞4 ⎛  GM ρ(z) 1  dvolz ⎠ |dy|2 . =⎝ 2 · 2c ΞF (z)  y − |y| · z |y|

⎝ GM 2c2



The magnitude term is easily found to be 1 1   = . y  1 − 2 y, z

+ |y|2 |z|2  |y| − |y| · z The latter can be expanded into power series in −2 y, z + |y|2 |z|2 , provided that |y| is substantially smaller than the size of the support of ρ. Such power series expansion leads to Taylor series expansion of  1 ρ(z)  dvolz ·  y ΞF (z)  − |y| · z |y| in y centered at y = 0. It now follows that the stated integral, as well as the metric under consideration, are both smooth at y = 0.  When combined, Theorems 5.7 and 5.8 convey the limit behavior of geometries ξn4 gE described in Figure 5.1.

ρ=

GM 2c2 δ

gE gS 3 = Ξ4F gE . 4g Figure 5.1. Limiting behavior of the sequence of geometries ξn E

Instead of discovering a Schwarzschild body in the limit we have discovered a copy of Euclidean space with a smooth bubble attached

232

5. Geometric analysis

to it! The limiting ADM mass is zero and not M which the right-hand side −4π GM 2c2 δp of (5.17) would suggest. The reader can explicitly verify that the ADM mass of ξn4 gE is equal to  √1 ρ/Ξn dvol, n which by Theorem 5.6 converges to   √1 dvol → 0 · ρ/ΞF dvol = 0. ρ/Ξ n n (See Exercise 6 for guidelines.) Unlike the situation with the Poisson equation and Newtonian gravity, the two macroscopic approaches to point-sources based on the equation (5.20)

 ξΔξ = −4π 2cG2 ρ, ξ ∞ = 1

do not lead to the same point-source idealization! A reason for this situation lies in the fact that ρ in (5.20) refers to mass-energy density. When talking about the total mass-energy of a system of objects one not only has to include the raw sum of the masses but also all the (gravitational) interaction energies between them, e.g., (5.21)

M2 M1 c2 + M2 c2 + G |pM11−p 2|

where p1 and p2 denote the locations of M1 and M2 . When using the Dirac delta function approach the inclusion of the n3 -term in ρn (x) = n3 ρ(nx) reflects an assumption about how mass-energy density scales relative to how length (and hence volume) scales. However, 1 in the expression for mass-energy shows that the presence of |p1 −p 2| 3 n -scaling is not appropriate. If Newtonian mass density required n3 -scaling due to the presence of volume in the denominator of the schematic expression ρ = Mass/Volume, mass-energy density requires n4 -scaling due to the metaphorical denominator (|p1 − p2 | · Volume) in its expression. In Exercise 7 the reader is invited to investigate consequences of such a scaling. For yet further information on this alternative approach to point-sources the reader is referred to [2].

5.2. On the concept of mass

233

5.2.8. Complications with the microscopic approach in general relativity. Even with the scaling situation addressed the concept of ADM mass does not align with the straightforward integration of mass-energy density ρ. Indeed, the ADM mass of ξ 4 gE which corresponds to mass-energy density ρ via (5.20) turns out to be    −1 ρ dvol < ρ dvol. mADM = Δξ dvol = 4πG/(2c2 ) ξ (See Exercise 6 below.) Being, by definition, a quantity observable by “an observer at infinity”, ADM mass could be seen as an effective mass-energy of an object. In view of (5.21) the decomposition −4π 2cG2 ρ = Δξ + (ξ − 1)Δξ is to be seen as a decomposition of ρ into the effective and the interaction component. Very informally speaking, effective ADM mass is what is left from ρ when we disregard the interaction energy keeping the cloud of dust together in the first place! Even though it appears that Δξ (up to a constant) serves as an effective mass-energy density it should be noted that it is not a density: its values are not an average of a physically meaningful quantity within a small (infinitesimal) neighborhood. To understand this (perhaps subtle) point, suppose that ρ1 is one of several dust clouds constituting a bigger dust cloud ρ. Since a portion of ρ needs to be allocated to the interaction between ρ1 and the (far away) remainder of ρ, the effective mass-energy density Δξ1 has to be strictly bigger than the effective mass-energy density Δξ! In particular, the effective mass-energy density at locations of ρ1 also depends on remaining and possibly far away components of ρ. Furthermore, the relativistic equations are by their very nature non-linear (as is the case with (5.20)) and no superposition principle along the lines of    ρ(p) δp → M = ρ(p)δp → Φ(x) = G ρ(p) dvolp ρ(x) = p p |x − p| p of Newtonian gravity can even be anticipated. It is for all these reasons that the microscopic point of view, according to which the cumulative behavior of constituents of an object reveals the behavior of the object as a whole, is considerably more difficult to implement in general relativity than in Newtonian gravity.

234

5. Geometric analysis

Due to an overwhelming presence of horizons and alternative asymptotic ends it is certainly not clear that the Schwarzschild body should play the role of a point-source for the microscopic purposes, and it is not clear that the macroscopic and microscopic approaches in general relativity can even be compatible. To this day (May of 2020) there is no universally accepted way of computing the so-called quasi-local mass: mass-energy of a proper subset of an asymptotically Euclidean manifold. In fact, the problem of defining a suitable notion of quasilocal mass is still one of the most investigated questions in general relativity. 5.2.9. Exercises for Lecture 5.2. Dirac Delta function. (1) The description of the Dirac delta function provided in the lecture is specific to dimension 3. Alter this description and design a sequence of smooth, compactly supported functions {ρn (x)} over Rk whose hypothetical limit as n → ∞ is the Dirac delta function on Rk . How does the change you propose impact the statement of Theorem 5.5? (2) Let μ(z) be a smooth and compactly supported function of a real variable. (a) Design a sequence of smooth and compactly supported functions {ρn (x)} on R3 which in the limit as n → ∞ describe an infinitely thin wire of (linear) mass density8 μ(z), located along the z-axis. (b) Let u(x) be an arbitrary compactly supported smooth function on R3 . Show that   u(x)ρn (x) dvolx = u(0, 0, z)μ(z) dz. lim

n→∞

x∈R3

z∈R3

(3) Generalize Exercise 2 and design sequences of smooth, compactly supported functions {ρn (x)} on R3 which in the limit as n → ∞ describe: 8

The units here could, for example, be kg/m.

5.2. On the concept of mass

235

(a) An infinitely thin sheet located in the xy-plane whose mass-density is μ(x, y) kg/m2 . As usual, assume that μ is smooth and compactly supported. (b) An infinitely thin spherical shell of radius 1 and centered at the origin. Assume mass-density is described by some smooth function μ defined on the sphere. In both cases address the value of  u(x)ρn (x) dvolx lim n→∞

x∈R3

for compactly supported smooth functions u(x) on R3 . (4) Throughout this problem let H denote the function9 of real variable x defined as follows:  −1 if x < 0, H(x) = 1 if x ≥ 0. Throughout this problem let f and u be smooth functions of a real variable with u compactly supported.   (a) Show that u (x)f (x) dx = − u(x)f  (x) dx.   (b) Show that u (x)|x| dx = − u(x)H(x) dx. (c) A function h(x) is said to be a weak derivative of the function g(x) if   u (x)g(x) dx = − u(x)h(x) dx holds for all u. (i) Show that for a smooth function f (x) its derivative f  (x) also serves as its weak derivative. (ii) Is the function |x| differentiable? If so, what is its derivative? Is it weakly differentiable? If so, what is its weak derivative? 9 This function is called the Heaviside function in honor of the English physicist Oliver Heaviside. It was Heaviside who in the 1880’s reformulated Maxwell’s equations of electromagnetism into the language of vector calculus.

236

5. Geometric analysis (d) It is common to see a claim that H  = 2δ, where δ denotes the Dirac delta function on the real line. (Refer to Exercise 1 if needed.) Provide a rigorous explanation of this claim, based on the concept of weak derivative.

Mass in general relativity. (5) Consider the metric

4 M1 M2 G G gE g = 1+ 2 + 2c |x − p1 | 2c2 |x − p2 | on R3  {p1 , p2 }. (a) Compute the ADM mass of the asymptotically Euclidean end |x| → ∞. (b) Use the change of coordinates

2 GM1 x − p1 y= 2 2c |x − p1 |2 to show that g also has an infinite asymptotically Euclidean end x → p1 . Compute the ADM mass of the infinite asymptotically Euclidean end x → p1 . (c) Show that g also has an asymptotically Euclidean end as x → p2 , and compute its ADM mass. (d) Find a relationship between the three ADM masses. What is the (self-)interaction energy of this two-body system? (6) (a) Consider the metric ξ 4 gE where ξ solves (5.20) for some compactly supported, non-negative, smooth ρ. Show that 

1 G ρ/ξ dvol +O , ξ(x) = 1 + 2 2c |x| |x|2 

1 x G +O ρ/ξ dvol gradξ(x) = − 2 2c |x|2 |x|3 as |x| → ∞.

5.2. On the concept of mass

237

(b) Continue with the same assumptions. Show that the ADM mass of ξ 4 gE is   −1 ρ dvol. mADM = Δξ dvol = 4πG/(2c2 ) ξ (c) Now consider the metrics ξn4 gE where ξn solves (5.17). Show that the ADM mass of ξn4 gE is  ρ 1 mADM = √ dvol. Ξn n (7) Consider the metrics ξn4 gE where ξn solves the asymptotic boundary problem   4  n ρ(nx) , ξn ∞ = 1 ξn Δx ξn = −4π GM 2c2 instead. Show that the sequence of ADM masses of ξn4 gE converges to  ρ mADM = dvol, ΞF where ΞF is the unique positive solution of the problem   ΞF ΔΞF = −4π GM 2c2 ρ, ΞF ∞ = 0. (8) Prove that definitions (5.15) and (5.16) agree on metrics of the form ξ 4 gE . (Assume that ξ satisfies the asymptotic conditions ∂ α (ξ(x) − 1) = O(|x|−1−|α| ) as |x| → ∞.) Other problems in geometric analysis. (9) The equation (5.20) stems from the constraint equation Scal (ξ 4 gE ) dvolξ4 gE = 32π 2cG2 ω dvolgE . As such, (5.20) is related to (but not the same as) what is known as a prescribed scalar curvature problem: the problem of finding an asymptotically Euclidean metric whose scalar curvature is given: (5.22)

Scal (ξ 4 gE ) = f. For convenience assume that f is smooth, non-negative and compactly supported.

238

5. Geometric analysis (a) Show that the prescribed scalar curvature equation is equivalent to  ΔgE ξ = − 18 f ξ 5 , ξ ∞ = 1. (b) Suppose that f is very small, max f  1. Show that there exists a unique positive solution ξ of (5.22) satisfying the standard asymptotic boundary conditions ∂ α (ξ(x) − 1) = O(|x|−1−|α| ) as |x| → ∞. (c) Suppose that f is large enough so that  f (z) 1 dvolz ≥ 1 32π |x − z| for all x in the support of f . Show that (5.22) has no solutions. (Hint: show by induction that over the n support of f the solutions ξ satisfy ξ ≥ 25 for all n.) (10) Ricci flow is a subfield of geometric analysis which has received a lot of attention in the last 40 years. The main idea here is to consider an evolution g(t) of Riemannian geometry g(0) whose driving term is related to Ricci curvature of g(t). Let gS 3 denote the standard metric on unit S 3 . (a) Show that g(t) = (1 − 4t)gS 3 solves the Ricci flow d g(t) = −2Riccig(t) , g(0) = gS 3 . dt (b) Describe the evolution of gS 3 under Ricci flow pictorially.

Bibliography

[1] Robert Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), no. 5, 661–693, DOI 10.1002/cpa.3160390505. MR849427 [2] Noah Benjamin and Iva Stavrov Allen, The effects of self-interaction on constructing relativistic point particles, Gen. Relativity Gravitation 50 (2018), no. 1, Paper No. 4, 35, DOI 10.1007/s10714-017-2324-6. MR3736108 [3] Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 2nd ed., Universitext, Springer-Verlag, Berlin, 1990. MR1083149 [4] J¨ urgen Jost, Partial differential equations, Graduate Texts in Mathematics, vol. 214, Springer-Verlag, New York, 2002. Translated and revised from the 1998 German original by the author. MR1919991 [5] Jerry L. Kazdan, Applications of partial differential equations to problems in geometry, https://www.math.upenn.edu/˜ kazdan/. [6] John M. Lee, Introduction to topological manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 202, Springer, New York, 2011. MR2766102 [7] John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR2954043 [8] John M. Lee, Introduction to Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer, Cham, 2018. Second edition of [ MR1468735]. MR3887684 [9] Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR719023 [10] Bernard F. Schutz, A first course in general relativity, Cambridge University Press, 1986. [11] Jeffrey R. Weeks, The shape of space, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 249, Marcel Dekker, Inc., New York, 2002. MR1875835 [12] N. M. J. Woodhouse, General relativity, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2007. MR2268691

239

Index

Cl -norm, 213 -spaces, 212 4momentum, 135 velocity, 132 Arzel´ a Ascoli Theorem, 213 asymptotically Euclidean metric, 190 atlas, 5, 17 Bianchi identity first, 84 second, 85 big bang, 187 black hole, 185 boost, 130 causality, 132 center of mass, 204 chart, 17 Christoffel symbols, 34 clock, 132 commutator, 47 conformal equivalence, 180 factor, 180 flatness, 190 connection, 52 coefficients, 52

Levi-Civita, 54 torsion-free, 54 constraint equations, 157 continuity equation, 134 coordinatization, 5 cosmological constant, 154 covariant differentiation, 54, 76 tensor field, 62 covector field, 60, 61 curvature tensor, 81 extrinsic, 121, 155 operator, 80, 81 Ricci, 109 scalar, 109 sectional, 100 deviation vector field, 93 Dirac delta function, 221 directional derivative, 45 distance, 22, 23 divergence, 67, 152 theorem, 70 dual, 65 effective potential, 168, 171 Einstein equation, 151, 154 tensor, 153

241

242 energy functional, 31 energy-momentum, 135 equicontinuity, 213 Euler’s equation, 134 evolution equations, 157 exponential map, 90 FLRW space-times, 187 function space, 212 Gauss (Divergence) Theorem, 68 -Bonnet Theorem, 116, 119 Lemma, 93 Theorem, 115 Theorema Egregium, 120 geodesic, 34 completeness, 35 equations, 34 incompleteness, 187 normal coordinates, 90 polar coordinates, 92 time-like, 150 gradient vector field, 67 gravitational field, 144, 145 potential, 144 great circle, 23 Green’s Representation Formula, 146, 204 Hessian, 78 horizon, 185 Hubble parameter, 188 hyperbolic geometry, 40 space, 37 trigonometric functions, 39 index contraction, 65 notation, 63 inertial coordinates, 91 observer, 124 initial data, 158 value formulation, 155 integration by parts, 71

Index interval, 125 inversion, 10, 194 isometry, 36, 37, 55 Jacobi equation, 96 operator, 151 vector field, 96 Kepler’s laws, 168 Kruskal-Szekeres extension, 183 Laplacian, 71 lapse and shift, 161 length contraction, 130 length functional, 30 Lie bracket, 47 light-cone, 124 line element, 3 line element, 5, 20 Euclidean line element, 4 spherical line element, 8, 10 Lorentz manifold, 149 signature, 127 transformations, 130 lowering an index, 65 manifold, 16 compact, 26 complete, 24 path connected, 23 mass, 135, 224 -energy density, 232 ADM, 225 conservation, 134 density, 133, 219 effective, 233 mass-energy, 135 quasi-local, 234 maximum principle, 71 minimal surface, 189 Minkowski inner-product, 125 space-time, 126 momentum, 132 4-momentum, 135 conservation, 134 density, 133

Index Newtonian limit, 131 non-degenerate inner-product, 126 null geodesic, 150 vector, 126 parallel, 54 transport, 51, 55 parametrization, 5 perfect fluid, 137, 152 perihelion advance, 175 point-source, 218 Poisson equation, 146 relativistic, 193 pressure, 134 projective spaces, 10 pseudo-orthonormal basis, 127 pseudo-Riemannian manifold, 149 raising an index, 66 regularity, 201, 203 relativistic Poisson problem, 193 relativity general, 148 principles, 123 special, 123 Rellich Lemma, 214 rest spaces, 129 Riemannian geometry, 26 Riemannian metric, 20 Schwarzschild -anti de Sitter space-time, 177 -de Sitter space-time, 177 metric, 166, 171 space-time, 166 second fundamental form, 121, 155 signature, 127 simultaneity, 128 spaces, 129 space-like curve, 131 vector, 126 space-time, 124 static space-time, 163 stereographic projection, 8 stress-energy tensor, 136, 152 support of a function, 145

243 tangent vector, 18 field, 2, 4, 19 space, 18 tensor, 61 (k, l)-tensor, 62 components, 63 field, 62 product, 61 tensorial, 63 time, 128 dilation, 130 proper, 132, 150 time-cone, 127 time-like curve, 131 geodesic, 150 sphere, 128 vector, 126 trace, 65 transition mappings, 10, 17 twin paradox, 141 uniform, 213 convergence, 212 vector null, 126 space-like, 126 time-like, 126 velocity 4-velocity, 132 spatial, 129, 142 volume element, 24 weak derivative, 235

Published Titles in This Subseries 93 Iva Stavrov, Curvature of Space and Time, with an Introduction to Geometric Analysis, 2020 66 Thomas Garrity, Richard Belshoff, Lynette Boos, Ryan Brown, Carl Lienert, David Murphy, Junalyn Navarra-Madsen, Pedro Poitevin, Shawn Robinson, Brian Snyder, and Caryn Werner, Algebraic Geometry, 2013 63 Mar´ıa Cristina Pereyra and Lesley A. Ward, Harmonic Analysis, 2012 ´ 58 Alvaro Lozano-Robledo, Elliptic Curves, Modular Forms, and Their L-functions, 2011 51 Richard S. Palais and Robert A. Palais, Differential Equations, Mechanics, and Computation, 2009 49 Francis Bonahon, Low-Dimensional Geometry, 2009 33 32 7 3

Rekha R. Thomas, Lectures in Geometric Combinatorics, 2006 Sheldon Katz, Enumerative Geometry and String Theory, 2006 Judy L. Walker, Codes and Curves, 2000 Roger Knobel, An Introduction to the Mathematical Theory of Waves, 2000

2 Gregory F. Lawler and Lester N. Coyle, Lectures on Contemporary Probability, 1999

This book introduces advanced undergraduates to Riemannian geometry and mathematical general relativity. The overall strategy of the book is to explain the concept of curvature via the Jacobi equation which, through discussion of tidal forces, further helps motivate the Einstein field equations. After addressing concepts in geometry such as metrics, covariant differentiation, tensor calculus and curvature, the book explains the mathematical framework for both special and general relativity. Relativistic concepts discussed include (initial value formulation of) the Einstein equations, stress-energy tensor, Schwarzschild space-time, ADM mass and geodesic incompleteness. The concluding chapters of the book introduce the reader to geometric analysis: original results of the author and her undergraduate student collaborators illustrate how methods of analysis and differential equations are used in addressing questions from geometry and relativity. The book is mostly self-contained and the reader is only expected to have a solid foundation in multivariable and vector calculus and linear algebra. The material in this book was first developed for the 2013 summer program in geometric analysis at the Park City Math Institute, and was recently modified and expanded to reflect the author’s experience of teaching mathematical general relativity to advanced undergraduates at Lewis & Clark College.

For additional information and updates on this book, visit www.ams.org/bookpages/stml-93

STML/93

www.ias.edu