Fortschritte der Physik / Progress of Physics: Band 27, Heft 6 1979 [Reprint 2021 ed.] 9783112522707, 9783112522691

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FORTSCHRITTE DER PHYSIK H F . R A l ' S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

D FR IU I l Si II EN DEMOKRATISCHEN

REPUBLIK

VON F. K ASC II EL H N. V. LÖSCHE. K. R I T S C H L U M ) R. ROMPE

H E F T 6 • 1979 • B A N D 27

A K A D E M I E -

V E R L A G EVP i o , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an das Zeitungsvertriebsamt, an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST U N D WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 701 Leipzig, Postfach 160, oder an den AKADEMIE-VERLAG, DDR - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kasohluhn, Prof. Dr. Artur Lasche, Prof. D r . Rudolf RiUchl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36221 und 22 36229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznuxnmer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. "" Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik** erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 1 8 0 , - M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/27/6. © 1979 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618 i/

Fortschritte der Physik 27, 261-312 (1979)

Einstein-Maxwell Equations: Gauge Formulation and Solutions for Radiating Bodies MOSHE CARMELI*

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794 and MICHAEL K A Y E

Department

of Physics,

Ben Gurion University

of the Negev, Beer Seva 84120,

Israel

Abstract The subject of the SL(2, c) gauge theory of gravitation is reviewed. A detailed discussion is given on the differential geometry and the fibre bundle structure of such a theory. The coupling of Maxwell's field equations to those of gravitation is also given. The field equations obtained, which are shown to be equivalent to the coupled Einstein-Maxwell equations, are subsequently solved. The solutions sought after are radiating type ones of the kind of the Kerr metric, but with the mass of the body being variable and is a function of the retarded time. A generalization of the Kerr metric is presented and its energy-momentum tensor is analyzed in detail/The classification of the field obtained according to the Petrov scheme is also given.

National Conventions The following notation for indices will be used: Lower case Greek letters fi,v,Q,... run from 0 to 3 and stand for holonomic coordinate components. Lower case Greek letters enclosed in round brackets ( y>', where y>' = iexpiA°(x))j f . To preserve invariance of the differential equations governing the dynamics of the field it is necessary to counteract the variation of A0 with space-time coordinates by introducing a gauge field A/x, which is interpreted as the electromagnetic potential, and to replace S^tp by a "covariant deri-

Einstein-Maxwell Equations

269

vative" — iqA^) y, where q is the electromagnetic coupling constant. One then finds that, under the gauge transformation, the potential A^ transforms into AJ = AM + i - 1 ^ / ! 0 . Gauge transformations form a group, called the gauge group. The gauge group in this case is the one-dimensional unitary group U( 1), which is an Abelian group and hence Maxwell's theory is also known as an Abelian gauge theory [1—S\. 2.5. Non-Abelian Gauge Fields A similar situation arises in isotopic spin gauge theory where the gauge group is the nonAbelian group SU(2) [4]. Isotopic spin gauge suggests that all strong interaction processes, which do not involve electromagnetic interactions, be invariant under the isotopic spin transformation y> —> tp', where y>' = S^ip, with S being space-time dependent and is an element of the group SU(2). As in the case of electrodynamics, a gauge field Bp is then introduced to counteract the dependence of S on coordinates and all derivatives d^ip are replaced by (e>u — iB^) yi. Under the isotopic spin gauge transformation the potentials By, transform into B^ = S^B^S + iS-1 8US. The fields are then defined as the matrices F^, = — d^By -f [Bh, Bv], where [B^, B,~\ = B^BV — ByB^, a natural generalization of the Maxwell field tensor fh, = d^A^ — d^A, that occurs in the electromagnetic case associated with the group 17(1). Under a local 8U{2) transformation one then finds that Fuv transforms into F'uv = S^F^S. 2.6. Spinors and Space-Time Structure The spinor approach to space-time structure is based on the knowledge that two component spinors [56—66] are the most fundamental building blocks out of which all tensor and spinor fields of standard field theory can be constructed. Furthermore, if we consider the underlying manifold, which is to be the "stage" for the space-time theory, to be primarily a carrier of 2-component spinors then we can infer a number of things about its structure viz., its dimension and its signature. In the language of fibre bundles, the space-time manifold, M, is the base of a complex vector bundle, B, with structure group SL(2, C). (The definition of a complex vector bundle is given in the next section). The typical fibre consists of a pair, (C, C) of complex two dimensional vector spaces each equipped with a spinor metric, and related by the operation of complex conjugation c :C —>C. The elements ipA of C are acted upon by the structure group in the following manner y>'A = SabVb,

(2.24a)

where SAB is a 2 X 2 unimodular complex matrix. Similarly, for the elements fA' of C we have = SA'n>xpB', (2.24 b) where SA'& is the complex conjugateoiS A B . The prime on an index denotes the fact that it transforms according to the complex conjugate representation S, whereas, the prime on the spinor, yi', (or any other object), denotes a transformed quantity. The pair of spinors spaces (G, C) determines a real four dimensional vector space V by means of the "complexification" isomorphism. C (x) C

V, AB

(2.25)

consisting of Hermitian spinors of the form H \ These can be displayed as 2 X 2 Hermitian matrices with four real independent entries. V is the lowest dimensional real

270

M. Cabmbli and M. Kaye

vector space which can be built up from the typical fibre of B. Clearly then, if M were anything but four dimensional, it would not be possible to tie the fibers of B to it in a simple manner, that is, by identifying smoothly the real vector spaces V with the tangent spaces of M. The product maps SAB (xj SA'B', induced on M by the action of SL{2, C) on preserve the determinant of H A S (which corresponds to the square of the magnitude of a real four vector on M) and hence correspond to proper orthochronous Lorentz transformations, a well known fact. If follows that the Map (2.25) induces a local Minkowski frame at a given point of M and therefore the base space acquires a pseudo-Riemannian structure in addition to a spinor structure [59, 67—69], The usual reason offered for using spinors in general relativity is the possibility of exploiting the very powerful and simple tools of spinor calculus for the analysis of spacetime. This is certainly an excellent reason, however, the argument that we have put forward here is much stronger than this and shows that the use of spinors is in fact very fundamental and is closely tied up with the actual structure of a pseudo-Riemannian space-time manifold.

G,

2.7. The Geometry of the 8L{2, C) Gauge Theory In this section we shall develop the geometrical aspects of the SL(2, C) gauge theory using the language of fibre bandies. The approach used here is different from that originally used by Carmeli, but the final results are the same. We begin by constructing a fibre bundle known as a complex vector bundle and which will be denoted by B(M, G, n, G, ip) and which consists of the following objects: (i) the bundle space B, (ii) the base space M (called space-time), covered by a family of coordinate neighbourhoods \U], (iii) a two dimensional complex vector space G called the typical fibre (or simply fibre), (iv) a mapping n of B onto M called the projection (v) a Lie group 0 called the structure group of the bundle, which will be taken for the time being to be the group OL(2, C) of arbitrary 2 x 2 nonsingular complex matrices, and which acts effectively on C, (vi) a family W of diffeomorphisms {y>u} corresponding to the open cover'{i7| of M mapping each coordinate neighbourhood U x G onto nr^U). This is the property of local triviality which states that locally B is the topological product of an open set U of M and a fibre G. If p is a point in M then J!T1(p) = Cp is called the fibre over p. A cross section (or simply section) of the bundle B is a differentiable map & •. M —> B such that n o 0 — idM (where idM is the identity map in M). By a local cross section we mean a cross section of a sub-bundle ar^U) (U, C, G, f ) , where is the restriction of n to the domain of nr^U). The bundle B can be taken to be trivial, that is, B = M X C, in which case we can define the following two sections on B: lA{p) = { 1,0),

(2.26a)

nA(p) = (0, 1),

(2.26b) £aA

for all p £ M. This is equivalent to saying that we introduce a spinor basis = (lA,nA), at each point p of M such that = daA. Any other complex vector bundle B' isomorphic to B will be trivialized by a section, which we call a gauge, but every gauge leads to a different trivialization (a trivialization is an isomorphism of a fibre bundle onto a trivial fibre bundle [70]). In other words, two globally (coordinate independent) defined

Einstein-Maxwell Equations

271

spinor bases CaA a n ( i £'aA are related by a global gauge transformation = (S-V

(2.27)

where 8 is a global element of GL(2, C). Further structure can now be introduced by forming the tensor product bundle S = B (x) B and defining the global cross section 6 A B : M B ® B. Since dAB belongs to G (x) G it transforms under OL(2, G) according to

e'AB

= eCDscAsDB =

(det S) e[AB] + e (CD) scasdb,

(2.28)

where, det S stands for the determinant of the matrix SAB. In view of the remarks made in the previous section we now restrict the gauge group to SL(2, C), in which case M acquires a pseudo-Riemannian structure endowed with a metric of signature ( + , —, —, —). Furthermore, we choose € A B to be antisymmetric so that it is invariant under an SL(2, G) transformation. CAB has then all the properties of a metric spinor and can be used for raising and lowering spinor indices. In terms of the dyad basis lA, nA we have

eAB = lAnB — nAlB,

(2.29 a)

hence

eAB(p) = l

?

(2.29b)

for all p £ M. It should be noted that had we taken € A B to be symmetric and invariant under the structure group 0, then G would satisfy GT £ G — 6. Here € is the matrix B given by c : y>A

so that

•p = xpA',

(2.37)

c(Ca A) = Ca' A',

l A'(p) =

(2.38)

n A'(p)

_(1, 0) and = (0, 1) for all p £ We jiow define the tensor product with bundle B (x) B and the cross sections : M B ® B as the complex conjugate of Eq. (2.29). Finally, the dual conjugate bundle is defined by the composition of the maps d and c, that is, c o d : B -> B* such that c o d(y> A) = xp^AB = v* •

(2.39)

Hence, we have the following

hip) = ( 0,1), and

Ml») = (-1,0),

(2.40a) .

(2.40b)

ÍA'*(P) = J), (2.41) for all p 6 M. Having introduced the fundamental quantities and basic definitions used in spinor analysis and in the SL(2, C) theory, we now discuss the concept of local gauge transformations. By a local gauge transformation we mean a local transformation of the spinor dyad basis, i.e., a transformation of the type given by Eq. (2.27) with the matrix 8 a function of the coordinates. The effect of such a transformation is to cause the bundle B to be non-trivial and the section £aA (and their dual) to be non-global. There are, however, two relations which are invariant under such transformations and they are the normalization conditions (2.35) and the completeness relations (2.36). ' It is important to note that the gauge transformations are defined as rotations in an internal space in analogy with the approach of Y A N G and M I L L S \4\. 2.8. Gauge Potentials and Field Strengths The gauge potentials of our theory are interpreted in terms of a connection in the complex vector bundle B. Let y>(x) be a spinor field on B and $a{x) a spinor basis. Denote by (£x), the result of parallel transport of from the point P(x) to the point

Einstein-Maxwell Equations

273

Q(x + dx). The connection F is the result of the comparison of ip(x + dx) — $(x) = (Prp(x + dx),

+ dx) with i|>(«)>

E^x + dx) dxf) = Pip,

(2.42)

where (,) denotes the value of the first entry on the second as a linear functional on C. E = {E^} is a (in general, non-holonomic) basis of TV{M) the tangent space of M at P. Let E* = {£>) be the basis of TP*(M), the contangent space of M at P, then V = E^V^ and t|>(a;) == y>A(x) (a;). Hence, with respect to the coordinate neighbourhood U, vy =

(2.43)

The spinor connection coefficients are simply the components of VEJSA with respect to the basis C,B, i.e., VE£A = [VM£AY SB = R*JSB, (2.44) the F B A are the sixteen complex components of the spinor connection V. From Eq. (2.44) we arrive at the following two equations, r * A = (5 s , VE^A) >

and

y$A = r»AE>

(2.45)

® g*.

(2.46)

We shall now find the local coordinate representation of the covariant derivative. First, note that Vip = F ( r ^ ) = (dyS) now putting dipA — E^d^,

+

(2.47)

and using Eq. (2.43) and Eq. (2.46) we find that = ^

+

(2-48)

The transformation properties of the P^B can be read off directly from Eq. (2.45). Under a coordinate transformation they transform like the components of a covariant vector and under a local gauge transformation, 5 " = -$b(S-i)b\

S'a = S A b S b ,

(2.49)

they have the following inhomogenous transformation law I"/a = (S-V / y W +

W .

(2.50)

In order that spinor indices may be raised, lowered and summed through the operation of covariant differentiation, we demand that (iAB and is also zero. The gauge potentials in the SL(2, C) gauge theory are taken to be the dyad components of the spinor connection. This means that the gauge potentials are nothing more than the Newman-Penbose spin coefficients [6]. We conclude this discussion of the gauge potentials by rewritting equations (2.44) and (2.50) in the form originally given by Carmeli. Equation (2.44) written in component notation becomes: = WoB-

(2.54)

Introducing the dyad components of the spinor connection defined by £aACbB> Eq. (2.54) can be written as W

and Eq. (2.50) as

=

(BM)ab

=

rMB (2.55)

(B„ V = ( £ - V ( B J S SJ> - ( £ - V W

•

(2-56)

The field strengths of a gauge theory are what as known in the language of differential geometry as curvature tensors. The existence of a curvature tensor is equivalent to the nonexistence of a parallel vector field (spinor field, in the present case) in the differentiable manifold under consideration. This can be expressed alternatively as the nonintegrability of the equivalence transport, for the basis spinors Ca as defined by Eq. (2.42) and implies the existence of nonvanishing field strengths (spinor curvature tensor) FM,AB given by f„ab

- d,rrJB + r,Acr„cB - j

= w

w

,

(2.57)

which have dyad components F h J = S,B,a» - e ^ B J + B ^ B , * - B^B, C K

(2.58)

The transformation properties of the gauge fields under a gauge transformation can be deduced from Eq. (2.56), and it is found that the fields transform homogeneously, (2.59) Equations (2.55), (2.56), (2.58) and (2.59) are the defining equations of the SL(2, C) gauge theory of gravitation. It is important to emphasize the fact that we have not assumed from the outset that our space-time is curved and we certainly have not any connection between the 8L(2, C) gauge theory and Einstein's theory of gravitation. However, at this stage of the development we point out that the existence of the gauge fields F^ does infact imply that space-time is curved, since it can easily be shown that F,J

= j iZVew^',

(2.60)

where is the Riemann tensor and aaab- = aaAB,CaAC b,B'• Finally we note that the gauge fields obey the Bianchi identities M W + VyFaiiAB + 17i>FyaÀB = 0,

(2.61a)

Einstein-Maxwell Equations

275

which can be derived directly from Eq. (2.57) or from Eq. (2.60) written with spinor indices together with the well known Bianchi identities obeyed by the components of the Riemann tensor. The dyad components of Eq. (2.61 a) are given by V>Fiy + VJP* + VpFya = [Ba, FPr] + [By, Fafi] + [Bf, Fra],

(2.61 b)

and should be considered as a matrix equation in the dyad indices. 2.9. Free Field Equations In analogy with the Yang-Mills approach we take a Maxwellian type Lagrangian density LF = — j i~gTr

F^F»',

(2.62)

where the trace is carried out on the undisplayed dyad indices and g = det oIMtl-ofb'. The first order form of this Lagrangian density is l

f = —j

Tr

j

+

- d,B, + [B„ £ v ])j,

(2.63)

and variation with respect to the matrix elements of B^ leads to the following field equations d,\i~g F>»\ - [Br, F»ea + %xgmi}} = hJ",

where, J14, a 2 x 2 matrix, is the current representing the source of the gravitational field and is given by J" = - [B„ raP}}. (2.73) The Lagrangian density giving rise to the field equations (2.72) was recently proposed by C A B M E L I [ 1 9 ] and is given by L=

—j T r J ^ ' i ^

F„ +_8rBM - 8,By + [B„

.

(2.74)

From the gauge theoretical point of view the field variables are the potentials Bp and the fields F^,. The field equations are Eqs. (2.58) and (2.72). However, these equations are not sufficient if we require our gauge solutions to be solutions also of Einstein's equations. We still need a set of differential equations for the variables bbc'c'.

(2.89)

Now y>aied is completely symmetric and 4>abe'd' is symmetric in its first and last pair of indices, hence -2(7tv*av and (Bfl)ttb. Let us consider first the Lagrangian density for the gravitational field L0 = - 2 ( ( B „ W

-

+ (BJb< (B„)c° - (B,)b< {BJ°\.

(2.95)

Varying the action formed from (2.95) with respect to the a'ef leads to the following Euler-Lagragian equations 8 l 8L„ \ 8La 23*

280

M . CAEMELI a n d M . K A Y E

t h a t is - 2 a{-e^et'h'aKc^]bc'(Fflt)ba

+

o « ' a ^ t f W (F et)ba]

2a

= 0.

(2.97)

Expanding out the field equations (2,97) and substituting in for (F b i ' M ') p i from (2.88) we find the following set of equations 2el24> tgr r + X e t t e r r } = 0 ,

(2.98)

where 2 = i?/4. If we add to this equation the variation of the matter Lagrangian density with respect to a e e f we obtain 2egf'i,' +

^SegSfh' =

„.

aegh'

(2.99)

xTef'gh>,

where 8Lm

are the dyad components of the energy momentum tensor of the matter T e f gh - = o f y X a'gh'Tp,. Equations (2.99) are, of course, simply Einstein's gravitational field equations written in dyad notation. This is clear, since 4>egfh' a r e the dyad components of the trace free part of the Ricci tensor, so t h a t Eqs. (2.99) are equivalent to ^

- j

Rg^

-

(i?/4) gMV =

kT,v,

i.e. Rfit

=

"2~

py .

We now consider the variation of the Lagrangian density with respect to (B/i)ab and assume t h a t the matter Lagrangian is independent of this variable. The Euler Lagrange equations t h a t result from varying (2.95) with respect to (B/i)ab are _8_ [

8L0

8L

a ? [ H B l j J \ ~

t h a t is, 1»"')

d^a^-a*

-

es'(SP"'alpq-)\

d n W i d W o w ) '

'+ •

er{h°\dP*'aipg.)

+ =

a

(B° ')h

+

s

(B„ 'r

8

h c

+

a'^'id/a^)} ^ } (Bds')dll

ere -

( # * ' ) /

(" is the dual of /'", V

and

(3.1b)

=

Y

e^hß,

(3.2)

is the electric four vector current. We shall also define, hi", the complex dual of /' h>" = i*p",

(3.3 a)

which also has the following representation A"' = afab-a'cd'(fu:b'ai""t'faß.

(3.3 b)

Einstein-Maxwell Equations

285

Half the self-dual part (up to a factor of -i) of the Maxwell tensor is defined by fiv =

V).

,(3.4)

so that Maxwell's equations (3.1) can be written =

(3.5)

The dyad components of f^, ffi and )pt are fab'cdT — fn*G l'ab' a'cd i,

f Mab'cd'

/j>V*V

=

=

J

(fab'cd'

=

jab'

(3.6)

+

(3.7)

fcVad'),

•

(3.8)

Using the antisymmetry of the Maxwell field tensor equation (3.6) can be decomposed as follows fab'cd;

=

aeSt>b'd' +

ac eb'd' >

E

(3.9)

where ac is symmetric in its indices. Introducing the compact notation =

4>a+c=

(3.10)

ac>

we can write the three complex independent components of f M ab ' C i- a s follows f M0b-ci'

=

4>n,

n ^ a

+

(3.11)

c.

The fields f ^ are related to the electromagnetic potentials A^ through the relation A* =

which has dyad components

=

fab'cd'

- M "

ffVoV(P^K

-

(3-12) M»)-

(3.13)

By using equation (2.75 a) we can rewrite equation (3.13) in terms of partial derivatives and the 8L(2, C) gauge potentials, B^, the result being: fab'cd'

=

(Cci'Aab' +

-

+ [(-Baft')/ ¿pd'

8ab-Aei')

[Acq.{B^b-a)"'i-

-

(-Bcd')aP

~

¿pb']

(3,14)

Aaq.{B^.cYb.].

In the above equation Aab- = a^^-A^. Combining equations (3.9) and (3.14) we get the following expressions for ^>ab, ab =

\

fac'b c'

+

[(Bae'V

+

[A

b q

=

€

-J

Apf. .{B\.

a

y

e r

((8

(Bb/,)aP -

o e

'A

b /

'—

8brAae.)

A.] .

(3.15)

286

M . CABMELI a n d M . K A Y E

The dyad components of Maxwell's equations (3.5) can be written, with the aid of Eq. (2.75a), as follows: -

((£"