Seven Fundamental Concepts in Spacetime Physics [2 ed.] 9783031497292, 9783031497308

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SpringerBriefs in Physics Vesselin Petkov

Seven Fundamental Concepts in Spacetime Physics Second Edition

SpringerBriefs in Physics Series Editors Egor Babaev, Department of Physics, Royal Institute of Technology, Stockholm, Sweden Malcolm Bremer, H. H. Wills Physics Laboratory, University of Bristol, Bristol, UK Xavier Calmet, Department of Physics and Astronomy, University of Sussex, Brighton, UK Francesca Di Lodovico, Department of Physics, Queen Mary University of London, London, UK Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig, Leipzig, Germany Maarten Hoogerland, University of Auckland, Auckland, New Zealand Eric Le Ru, School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn, Wellington, New Zealand Dario Narducci, University of Milano-Bicocca, Milan, Italy James Overduin, Towson University, Towson, MD, USA Vesselin Petkov, Montreal, QC, Canada Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany Charles H. T. Wang, Department of Physics, University of Aberdeen, Aberdeen, UK James D. Wells, Department of Physics, University of Michigan, Ann Arbor, MI, USA Andrew Whitaker, Department of Physics and Astronomy, Queen’s University Belfast, Belfast, UK

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Vesselin Petkov

Seven Fundamental Concepts in Spacetime Physics Second Edition

Vesselin Petkov Minkowski Institute Montreal, QC, Canada

ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefs in Physics ISBN 978-3-031-49729-2 ISBN 978-3-031-49730-8 (eBook) https://doi.org/10.1007/978-3-031-49730-8 1st edition: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To my wife Svetla and our son Vesselin (Jr) for their unconditional love and support

Preface to the Second Edition

There are two main reasons for the second edition of the book. First, despite positive reviews of the first edition and constructive discussions with colleagues, especially at the sixth spacetime conference (September 2022) and at the third Minkowski Meeting (September 2023), it became clear that elaborations on a number of complex and counter-intuitive explanations are necessary mostly in five chapters (Chaps. 2 and 3 needed only some small changes). Second, to reflect in the corresponding chapters relevant papers and books that appeared after the publication of the first edition, particularly the books Physics of Gravitational Waves: Sources and Detection Methods1 and Regular Black Holes: Towards a New Paradigm of Gravitational Collapse.2 The first three chapters (mostly Chap. 1) contain additional explanations and clarifications, prompted by reviews of and reactions to the first edition, which will prove helpful, I hope, for the proper understanding of the discussed concepts. To emphasize even further (i) the essence of Minkowski’s explanation of length contraction and (ii) its importance for the adequate understanding of the nature of spacetime, Fig. 1.9 and the corresponding explanation were added in Chap. 1. As Chap. 1 contains a quote from Hermann Weyl, where he suggests that it is the consciousness which creates our feeling that time flows, a second Appendix “Ancient Logical Arguments Against the Everyday View of Time” was added, where it is shown that the same idea had been discussed in ancient philosophy and later. The fourth chapter (on relativistic mass) now includes an authoritative statement of what the accepted definition of mass is. Such a statement turned out to be an essential part of any discussion of mass, because physicists, who reject the concept of relativistic mass, have been employing, almost explicitly, an inadequate everyday notion of mass—as quantity of matter—including in discussions after the publication of the first edition. The use of this inadequate notion of mass has been causing 1 A. Kenath, C. Sivaram, Physics of Gravitational Waves: Sources and Detection Methods (Springer 2023). 2 C. Bambi (ed.), Regular Black Holes: Towards a New Paradigm of Gravitational Collapse (Springer 2023).

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most objections to the concept of relativistic mass. That is why Max Born’s explicit warning about the danger of improper understanding of mass in relativity has been added to the chapter3 : In ordinary language the word mass denotes something like amount of substance or quantity of matter, these concepts themselves being defined no further... In physics, however, as we must very strongly emphasize, the word mass has no meaning other than... the measure of the resistance of a body to changes of velocity.

Also added to the chapter is Feynman’s brilliant concise explanation4 of both the role of relativistic mass in physics and its experimental confirmation. Reading this explanation makes one wonder whether the controversy over relativistic mass would have occurred if Feynman had been alive. The fifth chapter (on gravitation) contains a further emphasis on the full explanation of all gravitational phenomena (following Minkowski’s program of geometrizing physics) as nothing more than manifestations of the non-Euclidean geometry of spacetime without the need to assume that gravitation is a physical interaction, confirming Eddington’s 1921 observation that “gravitation as a separate agency becomes unnecessary.”5 Also added is an explanation of why only loop quantum gravity (LQG) makes a relevant effort to unify general relativity and quantum physics—LQG is rigorously based on the mathematical formalism of general relativity (gravitation = spacetime geometry) without trying to modify it in order to make it amenable to quantization. This is precisely what distinguishes LQG from the other approaches to create a theory of quantum gravity—LQG quantizes the (curved) spacetime itself, not the regarded as self-evident gravitational interaction. The sixth chapter (on gravitational waves) contains a more detailed elaboration (also prompted by comments on the first edition and latest publications on the subject) of the explanation of why no gravitational waves are emitted by inspiraling stars in binary systems before they collide and merge. It is specifically stressed that in general relativity it is impossible to have the following two statements both correct—(i) stars (modeled as point masses) in binary system are geodesic worldlines and (ii) inspiraling stars in binary systems emit gravitational waves before they collide. The reason is that in general relativity, a body, represented by a geodesic worldline, does not emit gravitational waves. Gravitational waves are emitted only when the stars’ worldlines are not geodesic (i.e., when the stars are subjected to absolute acceleration as Einstein demonstrated), that is, when they collide. When the stars are realistically regarded as extended bodies, weak gravitational waves are emitted before the collision, but they originate from strong tidal effects, not from the relative motion of the stars as often and misleadingly stated. 3

Max Born, Einstein’s Theory of Relativity (Dover Publications, New York 1965), p. 33. The Feynman Lectures on Physics. The New Millennium Edition. Vol. 1 (Basic Books, New York 2010), p. 15-9. See also Sect. 16-4 Relativistic mass. 5 A. S. Eddington, The Relativity of Time, Nature 106, 802–804 (17 February 1921), reprinted in: A. S. Eddington, The Theory of Relativity and its Influence on Scientific Thought: Selected Works on the Implications of Relativity (Minkowski Institute Press, Montreal 2015) pp. 27–30, p. 30. 4

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The seventh chapter (on black holes) is expanded with a more detailed discussion (again prompted by comments on the first edition) of the term “asymptotically,” which is sometimes used in the explanation of the formation of black holes for us (as distant observers). It is explained that the term leads to confusion, because it hides the real problem and, by the same argument (that black holes somehow form “asymptotically”), it implies that, if light asymptotically approaches the event horizon, it will reach it and eventually escape (which is not the accepted understanding). In any case, if black holes do form asymptotically, then, by absolutely the same logic, the event horizon will be as bright as a star. I would like to acknowledge many helpful discussions (mostly on issues covered in Chaps. 1, 4, 5 and 7) with colleagues who attended the sixth spacetime conference (12–15 September 2022, Albena, Bulgaria) and the third Minkowski meeting (11–14 September 2023, Albena, Bulgaria). I owe special thanks to Robert Geroch for his constructive criticism and for many insightful discussions on most of the concepts examined in the first edition of this small book. Finally, I would like to thank the Springer Executive Editor Dr. Angela Lahee for her exemplary professionalism. Montreal, Canada October 2023

Vesselin Petkov

Preface to the First Edition6

The main reason for writing about the seven selected fundamental concepts in spacetime physics—spacetime, inertial and accelerated motion in spacetime physics, origin and nature of inertia in spacetime physics, relativistic mass, gravitation, gravitational waves, and black holes—is that their present understanding does not appear to reflect adequately and fully the deep ideas and even the mathematical formalism of spacetime physics initially introduced by Hermann Minkowski for flat spacetime and developed by Albert Einstein for curved spacetime. The second reason for writing about these fundamental concepts by discussing the problems with them is to express not only my worry (I know I am not alone) that fundamental physics might not have been heading in the right direction especially in the 21st century. Whether or not it is explicitly admitted, it is a fact that in the last several decades there have been no major breakthroughs in fundamental physics as revolutionary as the theory of relativity (the discovery of the spacetime structure of the world) and quantum mechanics.7 This is particularly difficult to explain given the efforts of many brilliant physicists and the unprecedented advancements in applied physics and technology which made it possible to increase enormously the precision of experiments that can test new hypotheses in the search for even more fundamental physics. I think the most probable reason for the lack of major advancements in fundamental physics is that the art of doing physics, inherited from such giants of scientific thought as Galileo, Newton, Maxwell, Einstein, Minkowski and the founding fathers of quantum mechanics, does not appear to have been fully and creatively followed. It seems to me that three meta-theoretical issues might have been behind the inability even of some of the most talented physicists to make revolutionary contributions to 6

This Preface was updated in the second edition. From time to time, including at the last month’s (11–14 September 2023) third Minkowski meeting (held in the famous Black Sea resort Albena, near Varna in Bulgaria), some colleagues have been trying to question this (unquestionable, not only in my view) statement by giving different examples of theoretical advancements, which in all cases turn out to be nothing deeper than merely further developments of the quantum and the spacetime ideas.

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the never ending efforts to achieve a more profound understanding of the physical world. 1. Failure to recognize the need to extract from the accepted physical theories, and particularly from the existing unambiguous experimental evidence, some foundational (true) knowledge that can serve as the basis on which future theories should be built. Such knowledge, comprising seeds of true information about features of the physical world, rigorously confirmed by experiment, can indeed form un unshakable foundation of fundamental physics because it cannot be challenged by future experiments since experiments do not contradict one another. The lack of such, explicitly accepted, foundational knowledge hampers the advancement of fundamental physics in different ways; for instance, sometimes it results in virtually wasting a lot of intellectual potential by developing theoretical models and even theories that contradict the already existing and accepted physical knowledge. For example, mostly prompted by the attempts the create a theory of quantum gravity, there have been proposals for alternative theories of gravitation containing the concept of gravitational force, whereas it is an experimental fact that such a force does not exist in Nature (see Chapter 5); another example is even the Hamiltonian formulation of general relativity which involves a preferred time, whereas there is no such thing in Nature. An integral and important role of established foundational knowledge is to identify research directions (like the two examples above) that should be excluded. Excluding research directions is nothing new in science (e.g., perpetuum-mobilerelated research) but should be employed more rigorously in order intellectual efforts, time and funds be focused on promising novel ideas, which do not contradict the accumulated foundational knowledge. 2. Inadequate scientific philosophy or perhaps even lack of scientific philosophy,8 i.e., lack of true understanding of the nature of physical theories, which is a necessary condition not only for making a scientific discovery but also for productive research in fundamental physics. For example, part of the art of doing physics is to identify what in the mathematical formalism of fundamental physics is, as some physicists say, “just a description” (like the Newtonian, Lagrangian and Hamiltonian formulations of classical mechanics) and what represents entities in the physical

8 The present situation is sometimes even worse—some physicists appear to think they do not need in their research any scientific philosophy (i.e., a system of guiding meta-theoretical principles such as the understanding of physical theories not as “just descriptions,” but as adequately representing features of the physical world). This is an unfortunate misconception—as explained in Chapter 1 an inadequate (not just lack of) scientific philosophy can prevent even great scientists such as Poincaré from making a discovery (Thibault Damour, ref. 8 in Chapter 1):

The sterility of Poincaré’s scientific philosophy: complete and utter “conventionality” … stopped him from taking seriously, and developing as a physicist, the space-time structure which he was the first to discover.

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world. If this is not explicitly and rigorously done, there is often no clear distinction between mathematics and physics in a physical theory; for example, (i) spacetime is often declared “just a description,” precluding any study of the implications of a real four-dimensional world for physics (in Chapter 1 we will see that the physics of a three-dimensional world is radically different from the physics of a four-dimensional world), (ii) the opposite situation—a pseudo-tensor of gravitational energy and momentum has been regarded as representing an entity in the physical world, whereas no such entity exists as a result of the non-existence of a gravitational force and a physical gravitational field (Chapter 5), (iii) claiming (not using the proper general relativistic formalism) that a physical model of a binary system, whose stars are regarded as point masses, demonstrates the emission of gravitational waves before the two stars collide, ignoring the fact that their worldlines are geodesic and no gravitational radiation is associated with geodesics (Chapter 6). 3. Ignoring arguments9 such as the mentioned above—no gravitational force, no gravitational energy and no preferred time in Nature and that the members of a binary system (whose members are modeled as point masses) do not emit gravitational waves before they collide, because their worldlines are geodesic. Ignoring conceptual analyses and arguments can have serious negative implications for the advancement of fundamental physics since they provide enormous help for interpreting physical theories. Excluding conceptual analyses (that had been an integral part of the research strategies of the most successful physicists behind the greatest discoveries in physics such as Galileo, Einstein and Minkowski) from the art of doing physics in the 21st century gives rise to confusion, misconceptions and misunderstanding of the accepted physical theories that can persist for decades. The seven fundamental concepts in spacetime physics have been selected because their present understanding involves such confusion, misconceptions and misunderstanding. They are discussed here by • closely following Galileo’s and Einstein’s research strategy of exploring the internal logic of physical ideas by analyzing real and thought experiments and Minkowski’s approach of exploring the internal logic of the mathematical formalism of physical theories10 • employing Minkowski’s program of regarding physics as spacetime geometry 9

It looks like ignoring arguments has become widespread practice in the 21st century. Another disturbing example is ignoring Minkowski’s arguments for the reality of spacetime by some philosophers and even by some physicists. In recent years several well-known physicists have been arguing effectively against Minkowski’s spacetime view of the world (which is firmly based on the experimental evidence as Minkowski himself demonstrated and as is shown in Chapter 1), claiming, without addressing Minkowski’s arguments, that flow of time and some kind of becoming are real features of the world. I am among the majority of physicists who think that this is not how science works; one can ignore arguments in politics and even in philosophy, but not in science, particularly in physics. In physics (and science in general), arguments are faced and addressed. 10 Galileo’s, Einstein’s and Minkowski’s way of doing physics, together with Minkowski’s program of geometrizing physics, is the essence of a research strategy that is being developed and employed at the Minkowski Institute in Montreal. This research strategy identifies, synthesizes and develops the successful methods behind the greatest discoveries in physics.

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• explicitly pursuing the most profound goal of Physics—understanding the physical world.11 This is done by employing Wheeler’s first moral principle12 —“Never make a calculation until you know the answer”—which is an enormously powerful guiding principle not only when learning and teaching physics, but most importantly when doing research, especially in fundamental physics. Early versions of it seems to have been behind the greatest discoveries in physics which demonstrates that carrying out conceptual physical analyses, i.e., physics without equations, is probably physics at its best (physics with equations comes second in terms of both chronological order and crucial contribution to obtaining new results and particularly to creating/discovering a new theory). For a very quick reference of what has been achieved by this approach, here are one-sentence summaries of the seven concepts13 : Spacetime Minkowski’s arguments, extracted from experimental physics (emphasized by Minkowski himself), demonstrate that the introduced by him absolute world (die Welt) does represent a real four-dimensional world, because relativistic experiments would be impossible if this were not the case. Inertial and Accelerated Motion in Spacetime Physics Employing Minkowski’s program for regarding physics as geometry of the real spacetime produces spectacular14 results—it explained the principle of inertia, the principle of relativity (including the constancy of the speed of light), why acceleration is absolute in both special and general relativity, that absolute acceleration is a manifestation of an absolute geometric feature (the curvature of the worldline of the accelerating body) and does not imply the existence of absolute space.

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Unfortunately, in recent decades this goal has not always been explicitly followed. The (in)famous “Shut up and calculate” reveals the desperation at the failure to understand the nature of quantum phenomena (particularly what the quantum object itself is, not its states), but it seems that that advice was appropriate to give at this time to all studying quantum mechanics provided that it is not intended to discourage them from asking deep and relevant questions about what exactly happens at the quantum level of the physical world. 12 E. F. Taylor, J. A. Wheeler, Spacetime Physics, 2nd ed. (W. H. Freeman and Company, New York 1992), p. 20. 13 For a quick reference, more extended summaries of the concepts are given in Summary of Chapters after the Preface. 14 The results are spectacular because Minkowski’s program provided unprecedented 100% explanations of most mechanical phenomena (as we will see in Chapter 3 the explanation of the origin of inertia is an almost self-evident conjecture, given the reality of worldtubes, but is still a conjecture). I suspect such a statement maybe irritating, most probably to philosophers, but at least I will be glad if they react and initiate a discussion where not only the meaning of explanation in physics can be discussed, but also the broader issue that Minkowski’s discovery of the spacetime structure of the world reveals what scientific realism actually is.

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Origin and Nature of Inertia in Spacetime Physics Minkowski’s program for regarding physics as geometry of the real spacetime leads to the natural conjecture that inertia is another manifestation of the fourdimensionality of the world (like the kinematic relativistic effects)—the resistance a body offers to its acceleration originates from the four-dimensional stress which exists statically in the deformed (real) worldtube of the accelerating body. Relativistic Mass Like mass, which reflects the experimental fact that a body resists its acceleration (because mass is defined as the measure of that resistance), relativistic mass also reflects an experimental fact—the increasing resistance a body offers when accelerated to velocities close to that of light (and relativistic mass is similarly defined as the measure of that increasing resistance as the body’s velocity increases). Gravitation Employing Minkowski’s program for regarding physics as geometry of the real spacetime again produces spectacular results—(i) 100% explanation of Einstein’s principle of equivalence (including why the inertial force and mass are equivalent to the force of weight, traditionally regarded as gravitational, and to the passive gravitational mass, respectively), (ii) gravitational phenomena are fully explained as nothing more than mere manifestations of the non-Euclidean geometry of spacetime, which implies that gravitation is not a physical interaction. Gravitational Waves Gravitational waves, carrying away some gravitational energy of a binary system,15 are not emitted by the system before the collision of its members, which is best seen by initial models regarding the stars as point masses (because the stars’ worldlines are geodesic); when gravitational waves are emitted, during the collision of the stars, they do not carry gravitational energy, because what is released when the stars collide is not gravitational energy, but inertial energy (exactly like the inertial energy released when two bodies collide in flat spacetime). Black Holes It seems using double standards in physics can explain why in recent years physicists have started to talk about black holes as something established and accepted (by not interpreting “infinite time” as “never”), whereas, according to the Schwarzschild solution of the Einstein equation, black holes will never form for distant observers (like all of us) since they require infinite time for that (in the case of light, the “infinite time” needed for it to leave a black hole has been interpreted by the same physicists to mean “never”); black holes require finite time to form only for observers falling together with the collapsing star. 15

As we will see in Chapter 6 the system does not possess such energy. When the stars are realistically regarded as extended bodies, gravitational waves are emitted due to tidal effects in the stars.

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With the exception of most of Chapter 1, the other chapters reflect research done at the Minkowski Institute (Montreal), one of whose goals is to examine fundamental concepts that contain misconceptions, misunderstanding or cause confusion because such concepts hamper the advancement of fundamental physics. I would like to clarify two remarks I made in the text. I criticized some physicists for their over-confidence—(i) some particle physicists (p. 53) who were behind “what has probably been the most vigorous campaign ever waged against the concept of relativistic mass”16 and (ii) Feynman (p. 78) for being so sure that gravitational energy exists17 (even expressing irritation that other physicists do not see such an obvious “fact”). At the same time my style demonstrates recognizable over-confidence, but it is, I hope, clear that this style is intentional in my case: • to stress that strong arguments are presented that have serious implications for the proper understanding of key concepts in spacetime physics • to state (even provokingly): these are the arguments (especially Minkowski’s arguments) and not only “doing physics right” but even common sense require that they be faced and explicitly addressed. I would like to express special thanks to my wife Svetla for her patience and willingness to endure for years my efforts to achieve such explanations even of the most abstract issues that a non-expert can understand conceptually, and particularly for her suggestion specifically for this dense book—to include a summary of the chapters at the beginning of the book (after the Preface). Montreal, Canada March 2021

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M. Jammer, Concepts of Mass in Contemporary Physics and Philosophy (Princeton University Press, Princeton 2000) p. 51. 17 Feynman et al., Feynman Lectures on Gravitation (CRC Press, Boca Raton 2018), pp. 219–220.

Summary of Chapters

Chapter 1: Spacetime Hermann Minkowski successfully decoded the profound physical message hidden in the failed experiments (since Galileo’s time and the Michelson-Morley experiment) to detect absolute (uniform) motion—all experiments failed because there is no such thing in Nature as absolute space (with respect to which bodies move uniformly). As a mathematician, Minkowski had immediately realized that if observers in relative motion have different times (as Lorentz conjectured and Einstein postulated), they also have different spaces, which is impossible in a three-dimensional world and which implies that reality is a four-dimensional world with time as the fourth dimension (it seems almost certain that Minkowski arrived independently at what Einstein called special relativity and at the concept of spacetime, but Einstein and Poincaré published first while Minkowski had been developing the four-dimensional formalism of spacetime physics reported on 21 December 1907 and published in 1908 as a 59-page treatise; see Appendix B). As a worrying misconception seems to exist (even among physicists) that spacetime does not represent anything in the physical world and is nothing more than a four-dimensional mathematical space, it is explained in great detail in the chapter that Minkowski’s arguments, extracted from experimental physics (emphasized by Minkowski himself), demonstrate that the introduced by him absolute world (die Welt) does represent a real four-dimensional world, because relativistic experiments would be impossible if this were not the case. It should be also stressed that not only the experiments that confirmed the kinematic relativistic effects would be impossible if reality were a three-dimensional world, but all experiments that failed to discover absolute motion since Galileo (from Galileo’s own experiments and the MichelsonMorley experiment) would be also impossible—if reality were a three-dimensional world (which means that there would exist a single and therefore absolute space), all experiments would detect absolute motion, because absolute space would exist. As space is defined as a class of simultaneous events, if space were absolute (i.e., if

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reality were a three-dimensional world), both simultaneity and time would be also absolute in an obvious contradiction with the theory of relativity. Overcoming the common misconception that spacetime does not represent a real four-dimensional world is of crucial importance for the advancement of fundamental physics because the physics of a three-dimensional world is radically different from the physics of a four-dimensional world: • In a three-dimensional world, physics deals with motion and interactions of threedimensional bodies. • In a four-dimensional world, physics deals with relations between worldlines (worldtubes) and deformations in curved worldtubes since there is no dynamics in (the static) spacetime; it is even incorrect to say that a particle follows a timelike geodesic worldline because in spacetime the particle itself is a timelike geodesic worldline.

Chapter 2: Inertial and Accelerated Motion in Spacetime Physics Employing Minkowski’s program for regarding physics as geometry of the real spacetime produces spectacular results (because Minkowski’s program provided unprecedented 100% explanations)—it explained • Why accelerated motion is experimentally detectable, whereas uniform motion is not (and the experimentally indistinguishable rest and uniform motion), given that they both are motions in space. Minkowski first explained that there is no such thing in Nature as motions in space since particles are a forever given web of worldlines in spacetime: A straight worldline parallel to the t-axis corresponds to a stationary substantial point, a straight line inclined to the t-axis corresponds to a uniformly moving substantial point, a somewhat curved worldline corresponds to a non-uniformly moving substantial point.

and stressed that “Especially the concept of acceleration acquires a sharply prominent character.” So rest and uniform motion are indistinguishable since these “states” are represented by indistinguishable entities is spacetime—straight timelike worldlines. The experimentally distinguishable accelerated motion (experimentally detectable) and uniform motion (experimentally undetectable) are represented by distinguishable entities is spacetime—curved timelike worldline and straight timelike worldline, respectively (as shown in Chap. 3 a curved, i.e., deformed worldline should resist its deformation, which explains why accelerated motion is experimentally detectable). • The status of accelerated motion—acceleration is absolute in both special and general relativity because it is a manifestation of an absolute geometric feature (the curvature of the worldline of the accelerating body) and therefore does not imply the existence of absolute space.

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• The principle of inertia—a body preserves its state of rest or uniform motion (by inertia), because in reality, i.e., in spacetime (where there is no motion), it is a straight timelike worldline, which is perceived by an observer as being at rest (when the body’s and the observer’s worldlines are parallel) or moving uniformly on its own (without a mover) when the body’s worldline is inclined to the worldline of the observer. • The principle of relativity (including the constancy of the velocity of light)— physical phenomena look the same for all inertial observers (i.e., in all inertial reference frames), because the observers describe the four-dimensional reality (spacetime) in the same way—in three-dimensional language (imposed on us by the way our senses present the external world to us) in terms of their own times and three-dimensional spaces; the velocity of light is the same for all inertial observers because each of them measures it in their own spaces using their own times.

Chapter 3: Origin and Nature of Inertia in Spacetime Physics For centuries inertia has been an outstanding puzzle, which has been leading to a lot of misconceptions. An example is the very existence of inertial forces—whether they are fictitious or real. By employing Minkowski’s program for regarding physics as geometry of the real spacetime, it is explained when inertial forces are fictitious and when real. What further reveals the depth and power of Minkowski’s program is that it leads to the natural conjecture18 that inertia is another manifestation of the four-dimensionality of the world (like the kinematic relativistic effects)—the resistance a body offers to its acceleration originates from the four-dimensional stress which exists statically in the deformed (real) worldtube of the accelerating body (like the static three-dimensional stress existing in a deformed three-dimensional rod). The static resistance existing in a deformed worldtube can be traced down to the self-forces acting on accelerated elementary constituents of the body which have contributions from electromagnetic, weak and strong interactions. This conjecture also explains that the force of weight (traditionally regarded as gravitational force) is in spacetime physics an inertial force (making it additionally clear why there is no gravitational force in both general relativity and Nature)—the worldtube of a body at rest on the Earth’s surface is deformed and resists its deformation (i.e., as the falling body’s worldtube is geodesic the body moves by inertia (non-resistantly) and when it hits the ground the body resists its being prevented 18

This conjecture is natural, because if the reality of spacetime (and worldtubes) is regarded as experimentally confirmed (as I think, following Minkowski, it is overwhelmingly experimentally confirmed), then a real worldtube should indeed resist its deformation. The conjecture fully explains (i) why only an accelerated body resists its acceleration—because its worldtube is deformed, and (ii) why a body in uniform motion does not resist its motion—because its worldtube is not deformed; it is a straight worldtube.

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from moving by inertia and an inertial force acts on the ground)—that resistance manifests itself as the force of weight of the body. This conjecture also fully explains the equivalence of inertial mass and (passive) gravitational mass: they are the same thing—inertial mass: an accelerating body resists its acceleration (i.e., its worldtube resists its deformation) and the measure of this resistance is the body’s inertial mass; the worldtube of a body on the Earth’s surface is deformed and resists its deformation, i.e., the body is prevented from moving by inertia (prevented from falling) and the measure of that resistance is the body’s inertial mass (in curved spacetime), misleadingly called in the past (passive) gravitational mass. Minkowski’s program of geometrizing physics also led to an essential clarification that is crucial for the proper understanding of gravitational phenomena in spacetime physics: what so far has been called kinetic energy of a body is in reality inertial energy because the true physical meaning of the origin of that energy is the body’s inertia—more precisely, the work done by inertial forces.

Chapter 4: Relativistic Mass During the last three decades, physicists have witnessed (or rather endured), as Max Jammer put it “what has probably been the most vigorous campaign ever waged against the concept of relativistic mass.”19 The over-confidence of some critics of the concept of relativistic mass (mostly particle physicists) went as far as to use even expressions like “the virus of relativistic mass” (see the relevant reference in Chap. 4). However, looking at the issue open-mindedly and, more importantly, rigorously, particularly at the relevant experimental evidence, clearly shows that relativistic mass is an experimental fact. Like mass, which reflects the experimental fact that a body resists its acceleration (because mass is defined as the measure of that resistance), relativistic mass also reflects an experimental fact—the increasing resistance a body offers when accelerated to velocities close to that of light (and relativistic mass is similarly defined as the measure of that increasing resistance the body offers as its velocity increases). As some colleagues have been firmly rejecting the concept of relativistic mass, the two most relevant objections are addressed in Chap. 4 and it is shown that those theoretical objections have proper explanations and that theoretical objections have no chance of rejecting the experimental fact that mass increases with velocity (since a body’s resistance to its acceleration, bringing its velocity close to that of light, √ increases)—(i) the double argument of Taylor and Wheeler and (ii) that γ = 1/ 1 − β 2 should not be “attached” to the mass, because it comes from the 4-velocity.

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M. Jammer, Concepts of Mass in Contemporary Physics and Philosophy (Princeton University Press, Princeton 2000) p. 51.

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Chapter 5: Gravitation Einstein’s general relativity, which identified gravitational phenomena with the nonEuclidean geometry of spacetime, has been considered to be the greatest intellectual achievement in fundamental physics along with Minkowski’s discovery of the spacetime structure of the world, because without it general relativity would not have been possible. However, the present interpretation of general relativity, coming from Einstein himself, resembles an incomplete version of that greatest intellectual achievement because it regards gravitation as a physical interaction and the pseudotensor of the gravitational energy and momentum as representing real quantity in the physical world. It seems until 1948, when Einstein wrote “I do not agree with the idea that the general theory of relativity is geometrizing Physics or the gravitational field,” he had been having doubts that spacetime was real—and indeed, how it could be stated seriously that gravitation is a manifestation of the curvature of something that does not exist; but as of 1952, when Einstein added the fifth appendix “Relativity and the Problem of Space” to the fifteenth edition of his book Relativity: The Special and General Theory,20 he might have been changing his views (see Chap. 5). A rigorous analysis of the mathematical formalism of general relativity unambiguously demonstrates that no gravitational interaction (and no gravitational energy and momentum) is causing the gravitational phenomena since they are fully explained as effects of the curvature of spacetime. What I think is unprecedented in physics is that three obvious and independent arguments against the concept of gravitational energy (each of which, taken alone, is sufficient to rule it out) have been merely ignored21 : • the mathematical formalism of general relativity stubbornly refuses to yield a proper tensorial expression for gravitational energy and momentum, which is a clear indication that foreign concept in general relativity does not represent a real physical quantity; • there is no gravitational field as a physical field whose energy is gravitational; at best, gravitation can be regarded as a geometrical field, which, however, does not possess any energy; • the experimental fact that there is no gravitational force in Nature also demonstrates that there is no gravitational energy for the obvious reason—gravitational energy should be defined as the work done by gravitational forces, which do not exist. Most helpful for the proper understanding of the profound physical meaning of general relativity is to employ Minkowski’s program for regarding physics as 20

A. Einstein, Relativity: The Special and General Theory, new publication in the collection of five works by Einstein: A. Einstein, Relativity, edited by V. Petkov (Minkowski Institute Press, Montreal 2018). 21 This is an example of what I think is a serious problem in the twenty-first century fundamental physics that might be partially responsible for the lack of breakthroughs as revolutionary as spacetime physics and quantum physics—ignoring what some physicists might call conceptual arguments (trying to justify their ignoring).

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geometry of the real spacetime. Such an analysis again produces spectacular results— (i) 100% explanation of Einstein’s principle of equivalence (including why the inertial force22 and mass are equivalent to the force of weight, traditionally regarded as gravitational, and to the passive gravitational mass, respectively), (ii) gravitational phenomena are fully explained as nothing more than mere manifestations of the nonEuclidean geometry of spacetime, which implies that gravitation is not a physical interaction.

Chapter 6: Gravitational Waves Until recently the widespread understanding of gravitational waves contained two misconceptions—(i) that they are emitted by binary systems while the stars inspiral before they collide,23 (ii) that they carry gravitational energy. Fortunately, the first misconception was essentially corrected when gravitational waves were detected as seen from the official LIGO GW170104 Press Release24 “LIGO Detects Gravitational Waves for Third Time” (1 June 2017): As was the case with the first two detections, the waves were generated when two black holes collided to form a larger black hole.

Gravitational waves that are emitted by the stars of a binary system before they collide are generated by tidal effects in the stars themselves (not due to the relative motion of the stars as often and misleadingly stated); the forces, holding together the stars’ constituents (at fixed distances), cause deformations in the constituents’ worldlines (since the worldlines are not congruent meaning that the distances between the constituents change); as a result gravitational waves are emitted. The strongest emission of gravitational waves occurs when the stars merge; during the collision of the stars, the worldlines of their constituents are deformed much greater than due to tidal effects; that is why the emitted gravitational waves are stronger. However, those waves do not carry gravitational energy, because what is released when the 22

As explained in Chap. 3, the origin and nature of inertial forces is a natural (but still) conjecture. However, this does not affect the full explanation of the principle of equivalence, which postulates the equivalence (not the origin and nature) of inertial and gravitational forces. 23 There is a double misconception here—the first, that a physical model, regarding the stars as point masses, predicts the emission of gravitational waves before the stars collide, which is incorrect because before the merger the worldlines of the point stars are geodesic (when the stars are realistically modeled as extended body gravitational waves are indeed emitted before the merger, but the waves are generated by tidal effects in the stars themselves) and the second is that the gravitational waves carry away orbital gravitational energy (as pointed out in Chap. 5, the proper interpretation of general relativity demonstrates that there is no such energy in the theory itself; in addition, the experimental fact that there is no gravitational force in Nature—falling bodies do not resist their fall—demonstrates that there is no gravitational energy not only in general relativity, but in Nature as well). 24 But the paper that reported the first detection of gravitational waves still contains this misconception (see Chap. 6).

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stars collide is not gravitational energy, but inertial energy (exactly like the inertial energy released when two bodies collide in flat spacetime).

Chapter 7: Black Holes The Schwarzschild solution of the Einstein equation, which has been widely understood to predict the existence of black holes, gives rise to two problems that have not been properly addressed so far: • it contains a singularity, and many physicists have doubts that the physical world can contain such a feature; • according to the Schwarzschild solution black holes will never form for distant observers (like all of us) since they require infinite time for that; black holes require finite time to form only for observers falling together with the collapsing star. The reason that in recent years physicists have started to talk about black holes as something established and accepted appears to be a result of using of double standards in physics: • it is assumed that light will never leave a black hole because it needs infinite time to do so; in this case “infinite time” is interpreted as “never;” • black holes are almost unanimously assumed to exist, despite that they also require infinite time to form for distant observers (like us); so “infinite time” in this case inexplicably does not mean “never.”

Contents

1 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Inertial and Accelerated Motion in Spacetime Physics . . . . . . . . . . . . . . 21 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Origin and Nature of Inertia in Spacetime Physics . . . . . . . . . . . . . . . . . . 31 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 Relativistic Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Appendix A: Ancient Logical Arguments Against the Everyday View of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Appendix B: Minkowski Developed his Own Ideas of Spacetime Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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I see it, but I don’t believe it. Georg Cantor [1] Beyond the divisions of time and space which are imposed on our experience, there lies a higher reality, changeless, and independent of observer. P. L. Galison [2] Things are never quite the way they seem. Stan Ridgway, Lyrics “Camouflage”

Major confusions and misconceptions are involved in the understanding of what the concept of spacetime represents. Unlike the adoption of the developed by Hermann Minkowski four-dimensional (spacetime) formalism, Minkowski’s arguments for the reality of spacetime1 have been effectively ignored. Even worse—some physicists (including a number of relativists) claim that spacetime and Minkowski’s four-dimensional formalism are “just a description” and that the question of the reality of spacetime belongs to philosophy. Most relativists appear to understand well both (i) that, as the dimensionality of the (macroscopic) world is a fundamental feature of the world (arguably as fundamental as its very existence), it is not just a description, and (ii) that it is physics (not philosophy) which can determine the dimensionality of the world. Nevertheless, as the question of the nature of spacetime naturally plays a central role in the foundations of spacetime physics, it is necessary to state explicitly and in great detail

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More precisely: Minkowski’s arguments that the concept of spacetime (what Minkowski called die Welt) represents a real absolute (observer-independent) four-dimensional world. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8_1

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what Minkowski himself advocated, i.e., what the concept of spacetime actually represents. In this chapter Minkowski’s arguments—general and specific (based on the nature of length contraction)—are stated, elaborated and extended by demonstrating that none of the experiments, which confirmed the kinematic relativistic effects, would be possible if spacetime were not real [3, 5]. There are three main reasons of why physicists should deal explicitly with the question of whether the concept of spacetime is indeed “just a description” (like the Newtonian, Lagrangian and Hamiltonian formulations of classical mechanics) or represents a real four-dimensional world with time as the fourth dimension as Minkowski insisted. 1. Determining the status of spacetime is of utmost importance for physics itself. That is why, to ensure that the foundations of spacetime physics are firmly established, physicists should explicitly answer the question (on the basis of the relativistic experimental evidence): Is the world three-dimensional (then spacetime is nothing more than a mathematical space) or four-dimensional (then spacetime adequately represents the external world). It seems even this foundational question can be a source of confusion and misconceptions to such an extent that one can hear the objection “It is an incorrect question to ask what the dimensionality of the world is.” In discussions with colleagues I realized that some of them regarded the issue of dimensionality as indeed “just a description.” I think such a position cannot survive a closer scrutiny. A three-dimensional world and a four-dimensional world are experimentally distinguishable. Consider a rod. Everyone, including physicists, regard it as a three-dimensional object. Minkowski, however, believed that it was a fourdimensional worldtube (the three-dimensional rod existing en bloc at all moments of its proper time, i.e., at all moments of its history). So we have a clear alternative regarding the dimensionality of the rod—the rod, which we perceive as a threedimensional object, is in reality either three-dimensional (the everyday concept of a rod) or four-dimensional (the rod’s worldtube). One cannot say “there is no alternative, because the rod’s dimensionality is a matter of description” exactly like one cannot say “there is no alternative regarding the dimensionality of a square (a two-dimensional object) and a cube (a threedimensional object), because their dimensionality is a matter of description.” I cannot imagine one would argue that it is an incorrect question to ask what the dimensionality of a square and of a cube are, because their dimensionality is a matter of description. The three-dimensional rod and the four-dimensional rod (the rod’s worldtube) are like a square and a cube—two objects of different dimensionality. Minkowski’s explanation of length contraction of a rod (as we will see later) demonstrates that this relativistic effect is possible if and only if the rod’s worldtube is a real four-dimensional object. That is why the experimental confirmation of length contraction, being an experimental confirmation of the reality of the rod’s worldtube, constitutes an experimental distinction between a three-dimensional object (the rod we perceive) and a four-dimensional object (the rod’s worldtube) and, therefore, an experimental distinction between a three-dimensional world and a four-dimensional world.

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Answering the question of the dimensionality of the world (i.e., of what the concept of spacetime represents) is of crucial importance for the advancement of fundamental physics because the physics of a three-dimensional world is radically different from the physics of a four-dimensional world: • In a three-dimensional world physics deals with motion and interactions of threedimensional bodies. • In a four-dimensional world physics deals with relations between worldlines (or worldtubes in the case of extended bodies) and deformations of curved (nongeodesic) worldtubes because there is no dynamics in spacetime; it is incorrect even to say that a particle follows a timelike geodesic worldline because in spacetime the particle itself is a timelike geodesic worldline. In a three-dimensional world there are particles moving with constant velocity or with acceleration, whereas in a four-dimensional world (spacetime) there are • straight and curved (non-geodesic) worldlines (in flat spacetime) • geodesic and deformed (non-geodesic) worldlines (in curved spacetime). In a three-dimensional world the nature and origin of inertia and inertial forces have been a mystery, whereas in a four-dimensional world, as we will see in Chaps. 2 and 3, Minkowski’s explanation of inertial motion (represented by straight worldlines) and accelerated motion (represented by curved, or deformed, worldlines) naturally leads to the conjecture that inertia (the resistance a particle offers to its acceleration) is a manifestation of the resistance arising in the deformed worldline (rather worldtube) of the accelerated particle (in a real spacetime worldtubes are real fourdimensional objects which should resist when deformed). Therefore it becomes clear why accelerated motion is experimentally detectable through the resistance that a particle offers to its acceleration (that resistance comes from the deformed worldtube of the accelerated particle), whereas inertial motion (motion with constant velocity) cannot be discovered experimentally because it is represented by geodesic worldlines (which are not deformed and therefore there is no resistance that can be detected). When experts in spacetime physics allow the status of spacetime to stay undetermined, the price of such an inaction turns out to be significant—in addition to the examples above, gravitational phenomena also remain not fully understood as manifestations of the non-Euclidean geometry of spacetime. As we will see in Chaps. 5 and 6 in a real spacetime Einstein’s general relativity has a natural explanation (without smuggling in it the concept of gravitational energy and momentum, which the theory itself refuses to accept by not yielding a proper tensorial expression)—gravitation is completely geometrized since it is fully explained as manifestation of the curvature of spacetime without the ad hoc assumption that it is a physical interaction. Then it is virtually obvious that (i) gravitational waves are emitted by a body only when its worldtube is deformed (i.e., when the body is absolutely accelerating as originally predicted by Einstein, Chap. 6), (ii) gravitational waves do not carry

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gravitational energy, and (iii) the actual open question in gravitational physics seems to be how matter curves spacetime, not how to quantize the apparent gravitational interaction [9]. In a real spacetime the issue of whether or not there is a contradiction between quantum physics and spacetime physics can be adequately addressed—now there exists a wide-spread misconception that the probabilistic nature of quantum mechanics contradicts the deterministic nature of spacetime physics. The source of that misconception is the uncritical assumption that spacetime physics is deterministic, which is just wrong: spacetime physics states that the world is four-dimensional, i.e., given at once (en bloc), which equally accommodates deterministic and probabilistic phenomena. For example, the deterministic behaviour of a classical particle is represented by a worldline, whereas the probabilistic behaviour of an electron can be represented by a more complex (probabilistic) spacetime structure [4, 5, Chap. 10]—e.g., the electron can be represented by the “disintegrated” worldline of the classical electron, whose constituents are probabilistically scattered all over the spacetime region where the electron’s wavefunction is different from zero.2 In other words, the probabilistic distribution of the electron’s constituents is “forever given” in spacetime.3 Perhaps the greatest advancement achieved by spacetime physics is the replacement of the postulates of three-dimensional physics with .100% explanations in a real spacetime—e.g., replacing the postulates of relativity and the constancy of the velocity of light with the explanations given by Minkowski himself (later in this chapter); also, as we will see in Chap. 5, the principle (postulate) of equivalence is fully explained by employing Minkowski’s program of geometrizing physics. 2. Determining the status of spacetime is also of utmost importance not only to experts in spacetime physics but to all, because it crucially affects our view of what exists. Indeed, what could be more important for our view of the world than to understand whether reality is an evolving in time three-dimensional world4 or it is a four-dimensional world (spacetime),5 which contains en bloc the entire histories in their proper times of all particles and bodies in the Universe (like a film strip contains 2 In the ordinary three-dimensional language this probabilistic spacetime structure, which brings the idea of atomism to its logical completion (discreteness not only in space, but in time as well— four-atomism or atomism in spacetime), manifest itself as appearance and disappearance of the constituents of the electron at the Compton frequency. It is unfortunate that this novel idea [4] remained unnoticed despite that it deals with a fundamental continuity at the heart of quantum mechanics—the almost explicit assumption that quantum objects exist continuously in time. 3 Had Minkowski lived longer he might have been thrilled to describe such a probabilistic spacetime structure by the mystical expression “predetermined probabilistic phenomena.” 4 This is the common worldview called presentism according to which only the constantly changing present exists (exists now), whereas the past and the future do not exist in any way (obviously, they could not exist by definition if “exist” is intuitively understood/defined as “exist now.”). 5 Sometimes such a world is called a block universe. One cannot talk about past, present and future in this four-dimensional world (spacetime), because all events of spacetime exist equally. Here, obviously, “exist” does not mean the intuitive everyday (unscientific) “exist now;” it means what Minkowski meant when stating that what exits is the four-dimensional world [10, p. 62]: “only a world in itself will exist.”

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en bloc the entire story of a movie from the old times when film strips were used), which means that there is no becoming6 (i.e., no flow of time) in such a world. The drastically counter-intuitive nature of a four-dimensional world makes it enormously difficult not only to non-experts but even to physicists to consider seriously the possibility that the concept of spacetime may represent a real four-dimensional world. Moreover, it seems that there exists a contradiction between two facts. First, the relativistic experimental evidence overwhelmingly supports Minkowski’s discovery (deduced from the failed experiments to detect absolute motion as we will see below) of the spacetime structure of the world—that reality is a static four-dimensional world containing en bloc the entire history of the perceived by us three-dimensional world. Second, it is an inter-subjective fact that we are aware of ourselves and the world only at one single moment of time—the present moment (the moment now)—which constantly changes. In 1919 Hermann Weyl was the first (and essentially the only one so far) who tried to reconcile these two seemingly irreconcilable facts—the reality of spacetime and our awareness of the world (and ourselves) only at the moment now. Weyl reached the conclusion that it is our consciousness (somehow “traveling” in the four-dimensional world along our worldlines) which creates our feeling that time flows [14]: The great advance in our knowledge .. . . consists in recognising that the scene of action of reality is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavours to crystallise out of direct experience. It is a four-dimensional continuum, which is neither “time” nor “space.” Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it, as history, that is, as a process that is going forward in time and takes place in space.

Unfortunately, Weyl’s reconciliation of the above facts has not been rigorously examined so far. The apparent contradiction that the consciousness7 “travels” in the “frozen” four-dimensional world (spacetime) is not an excuse because Weyl had surely been aware of it and nevertheless “went public” with his proposed resolution. I guess Weyl might have concluded that the apparent contradiction—the consciousness’ “motion” in spacetime—was not as serious as the apparent contradiction between the facts he tried to reconcile. It turned out that dealing with the status of spacetime does not always appear to be based on the experimental evidence (and even on common sense) most probably because of the counter-intuitive features of a real four-dimensional world—not only 6 It is logically obvious that in spacetime a “local becoming” or “local becomings” inevitably lead either to the untenable relativization of existence or event solipsism [5, p. 124]. If existence is regarded as absolute, as it should be, “becoming in spacetime” is a clear contradiction in terms, because, by definition, all events of spacetime have the same existential status, whereas becoming, by definition, presupposes that only some events exist. 7 The idea that it is the consciousness (or the mind or the soul) that creates our feeling of time flow (i.e., of three times—past, present and future) can be traced back to ancient times; see Appendix A “Ancient Logical Arguments Against the Everyday View of Time.”

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is there no becoming, but also there is no free will8 in such a world. And that seems to be too much for a lot of scientists, philosophers and especially for non-experts. I guess colleagues who work on the nature of spacetime have encountered ample examples of illogical resistance to the very possibility that reality might be a fourdimensional world (with no flow of time and no free will) not only by ordinary people, whose sometimes irrational and stubborn reaction to such a possibility—effectively: this cannot be because it cannot be—is neither science nor common sense, but also by researchers who publish on this subject. It is not only amazing, but rather disturbing that such researchers do not appear capable or willing even to entertain that possibility (let alone examine the overwhelming experimental evidence proving it)—that the world might be indeed what Minkowski described (and if it turns out that the world is four-dimensional, would they continue to deny what exists?). In my classes on relativity and on foundations and philosophy of spacetime I always tried to comfort the students by assuring them that we are all entitled to our views, but nevertheless I had to remind them that Nature does not care at all about our personal opinions. 3. Determining the status of spacetime is of meta-theoretical importance for physics itself. Wide-spread realization that not only Minkowski’s arguments for the reality of spacetime, but also the experimental confirmation of the kinematic relativistic effects provide overwhelming support for the understanding that the concept of spacetime represents a real four-dimensional world, will convincingly demonstrate the need of explicit foundational knowledge that will form the unshakable basis on which future physical theories will be built. Although meta-theoretical issues in fundamental physics are often (misleadingly) regarded as not proper physics, the history of physics unambiguously shows that behind the greatest discoveries in physics there has been an adequate understanding of the nature of physical theories—that they represent features of the external physical world. By examining which concepts of our physical theories represent entities and features of the physical world we are able to accumulate seeds of foundational (true) knowledge about the world (extracted from the experimental evidence) that will never be challenged by future experiments because experiments do not contradict one another. Not only is such growing foundational knowledge a necessary condition for the advancement of fundamental physics since it provides the firm (certain) basis 8 Despite that free will is impossible in a four-dimensional world (even in a three-dimensional world it may or may not exist) one can still read such titles “Why free will is beyond physics” (the January 2021 issue of Physics World; https://physicsworld.com/a/why-free-will-is-beyondphysics/). Such an assertion does not seem to be physics (and science) at its best. In spacetime our bodies are forever given four-dimensional worldtubes (best visualized by a film strip) and in no way what is happening in our brains can change the forever given web of our bodies’ worldtubes (as we will see below the relativistic experimental evidence would be impossible if the worldtubes of physical bodies did not exist as four-dimensional objects). So the actual situation in science is just the opposite—“free will is not beyond physics;” that is why any discussion of free will that does not address Minkowski’s arguments based on experimental physics (and the relativistic experimental evidence after Minkowski’s death) for the reality of spacetime is nothing more than an insignificant unscientific chat.

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on which any new theory must be build, but it also augments our understanding of the external world. The lack of proper understanding of the nature of physical theories may dramatically hamper the advancement especially of fundamental physics. Perhaps Poincaré’s failure to discover the spacetime structure of the world, caused by his unfortunate view on the role of mathematics in physics, is the most cruel example in the history of physics of how an inadequate view of the nature of physical theories can prevent even great scientists from making a discovery. Poincaré first published his observation that the Lorentz transformations could be viewed as rotations in a four-dimensional mathematical space with time as the fourth dimension [6] but failed to recognize that this mathematical space represented a real four-dimensional world (which Minkowski did) because he believed that physical theories are only convenient descriptions of the world and therefore it is really a matter of convenience which theory we would use.9 As Damour [8] stressed it, it was .. . .

the sterility of Poincaré’s scientific philosophy: complete and utter “conventionality” which stopped him from taking seriously, and developing as a physicist, the space-time structure which he was the first to discover.

.. . .

Why Damour’s assertion that Poincaré “was the first to discover” the spacetime structure should be more fairly understood as “was the first to publish” becomes clear from a careful investigation which reveals that what really happened in 1905 might have been truly tragic (see Appendix B). —– Before seeing how Minkowski discovered the spacetime structure of the world, let us stress the fact that he had tried to make it as clear as possible that he was announcing not a four-dimensional mathematical space, but a real four-dimensional world whose existence10 was pronounced by the ultimate judge in physics—the 9 This view implies that our physical theories do not adequately represent the physical world, which may explain why Poincaré did not believe that the four-dimensional mathematical space with time as the fourth dimension (which he discovered independently of Minkowski and published first) represented a real four-dimensional world. 10 Sometimes some philosophers object that one cannot say that spacetime exists because “exists” means “exists now”. I had the chance to address such an objection by reiterating the obvious to all who work on the nature and ontology of spacetime—that interpreting “exists” as “exists now” reflects the unscientific (everyday) use of the term “existence.” After Minkowski revolutionized our views of space and time 115 years ago, the mundane term “exist” should not be confused with (unfortunately) the same term “exist” that has been used since Minkowski when talking about spacetime and four-dimensional objects [10, p. 62]:

You see why I said at the beginning that space and time will recede completely to become mere shadows and only a world in itself will exist. That “powerful” (to non-experts) objection was raised after my talk at a special panel “Realism and determinism in Physics,” dedicated to the retirement of Mario Bunge, at the 2010 Meeting of the Canadian Society for the History and Philosophy of Science on 28–29 May 2010, Concordia University, Montreal.

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experiment. Minkowski called this four-dimensional world die Welt (the world); we now call it spacetime. He excitedly began his revolutionary lecture “Space and Time” on September 21, 1908 [10, p. 57]: The views of space and time which I want to present to you arose from the domain of experimental physics, and therein lies their strength. Their tendency is radical. From now onwards space by itself and time by itself shall completely fade into mere shadows and only a specific union of the two will still stand independently on its own.

Although it is self-evident, let me nevertheless specifically stress it, because some physicists regard spacetime as nothing more than a mathematical space—a mathematician could not be so excited if he was introducing just another mathematical space. That Minkowski had been indeed announcing not a four-dimensional mathematical space but a real four-dimensional world is perhaps best demonstrated in the draft version of the 1908 lecture, where Minkowski initially wrote [2] that the essence of the new views on space and time is mightily revolutionary, to such an extent that when they are completely accepted, as I expect they will be, it will be disdained to still speak about the ways in which we have tried to understand space and time.

Although Minkowski’s excitement from the realization that he had discovered “a higher reality, changeless, and independent of observer” that lies “beyond the divisions of time and space which are imposed on our experience” [2] had been toned down in the final version of the 1908 lecture, a careful reading of the lecture’s text (published in 1909) leaves no doubt that he was discussing and providing the mathematical description of a real four-dimensional world. Let us now see how Minkowski arrived at the idea of the spacetime structure of the world by decoding the profound physical message hidden in the unsuccessful experimental attempts to detect absolute motion (captured in Galileo’s principle of relativity and the Michelson-Morley experiment). He revealed that hidden message by doing a double analysis of those experiments and of the internal logic of the mathematical formalism of classical mechanics. First, Minkowski arrived independently11 from Einstein at the equivalence of the time .t of a stationary observer and a second time .t ' , which Lorentz introduced (as “an auxiliary mathematical quantity”12 to explain the Michelson-Morley experiment) calling it the local time of a moving observer (whose .x ' axis is along the .x axis of the 11

See Appendix B and also Max Born’s recollection of what Minkowski told him: He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other were pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendour.

12

Lorentz admited his failure to recognize the profound physical meaning of his idea to introduce a second time in physics (and that a second time is impossible in a three-dimensional world as Minkowski showed). In 1916 in a note added to the second edition of his The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat he wrote [12]:

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stationary observer). He figured out, on the basis of the Lorentz transformations, that t and .t ' should be treated equally, which implies that there exist more then one time in the physical world. Then, as a mathematician, Minkowski immediately realized that not only do observers in relative motion have different times, but they also have different spaces [10, p. 62]:

.

One can call .t ' time, but then must necessarily, in connection with this, define space by the manifold of three parameters.x ' ,. y,.z in which the laws of physics would then have exactly the same expressions by means of .x ' , . y, .z, .t ' as by means of .x, . y, .z, .t. Hereafter we would then have in the world no more the space, but an infinite number of spaces analogously as there is an infinite number of planes in three-dimensional space. Three-dimensional geometry becomes a chapter in four-dimensional physics. You see why I said at the beginning that space and time will recede completely to become mere shadows and only a world in itself will exist.

Minkowski specifically pointed out that [10, p. 65] Neither Einstein nor Lorentz disputed the concept of space.. . . To go beyond the concept of space in such a way is an instance of what can only be imputed to the audacity of mathematical culture.

So Minkowski’s mathematical analysis led him to a four-dimensional mathematical structure (his die Welt). When he assumed that this structure represented a real four-dimensional world it became immediately evident why it is impossible to detect experimentally absolute motion—in this four-dimensional world there is no absolute space (such a space can exist only in a three-dimensional world) with respect to which bodies move; observers in relative motion have different spaces and times (when the four-dimensional world is described in terms of the ordinary three-dimensional language) and every time they perform experiments to detect their motion with respect to the assumed absolute space they do it in their own spaces by using their own times and always determine that they are at rest with respect to their own spaces. Then Minkowski realized that the physical message, which all failed experiments have been trying to convey to us for centuries, is indeed of profound depth—those experiments all failed because absolute space does not exist since the physical world is four-dimensional. The way Minkowski realized that the physical world must be four-dimensional in order that absolute motion does not exist, which provided a straightforward explanation of why all experiments to detect it failed, makes it clear why he insisted at the beginning of his 1908 lecture that the strength of the new views of space and time comes from the fact that they “arose from the domain of experimental physics.” Due to its importance for the foundations of spacetime physics, let me state it again: Minkowski’s discovery of the spacetime structure of the world explained the physical meaning of the principle of relativity (the impossibility to discover absolute uniform motion and absolute rest), which Einstein merely postulated—all physical The chief cause of my failure was my clinging to the idea that the variable .t only can be considered as the true time, and that my local time .t ' must be regarded as no more than an auxiliary mathematical quantity.

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phenomena look in the same way to observers in uniform relative motion (so they cannot tell who is moving as the experimental evidence proved) because they have different times and different spaces. Each observer performs experiments in his own space and time and for this reason the physical phenomena look in the same way to all observers, e.g., the speed of light is the same for them since each observer measures it in his own space by using his own time. The explanatory power of the new ideas of space and time explains Minkowski’s dissatisfaction with the principle of relativity, which postulates, but does not explain, the non-existence of absolute motion [10, p. 65]: I think the word relativity postulate used for the requirement of invariance under the group is very feeble. Since the meaning of the postulate is that through the phenomena only the four-dimensional world in space and time is given, but the projection in space and in time can still be made with a certain freedom, I want to give this affirmation rather the name the postulate of the absolute world.

.G c

Minkowski’s general argument for the reality of spacetime can now be explicitly formulated—the experiments, which failed to discover absolute motion (from Galileo’s to Minkowski’s time, including the Michelson-Morley experiment), would be impossible (i.e., they would discover absolute motion) if spacetime were not real. To see why, assume that the introduced by Minkowski four-dimensional world (spacetime) is nothing more than “an abstract four-dimensional mathematical continuum” [13] and that reality is what we perceive—an evolving in time threedimensional world. Then there would exist a single space (since a three-dimensional world allows the existence of one space), which as such would be absolute (the same for all observers). As a space constitutes a class of simultaneous events (the space points at a given moment), a single (absolute) space implies absolute simultaneity and therefore absolute time as well. Hence, a three-dimensional world allows only absolute space and absolute time in contradiction with the experimental evidence that uniform motion with respect to the absolute space cannot be discovered as encapsulated in the principle of relativity: as Minkowski explained it, the reason for the failure of all experiments to detect absolute motion was the non-existence of an absolute (i.e., a single) space; if such a space existed (i.e., if the world were three-dimensional), uniform motion relative to that space would be detected. I hope it is now clear why Minkowski’s general argument, taken even alone, effectively proves the reality of spacetime. Minkowski did not state explicitly that absolute motion would be discovered if the world were not four-dimensional, most probably because it looked completely obvious to him and therefore unnecessary to be included in his dense 1908 lecture. As Minkowski’s general argument demonstrates that many spaces (i.e., relative simultaneity) are possible only in a four-dimensional world, it also effectively proves that length contraction and time dilation are possible only in a four-dimensional world because these two relativistic effects are specific manifestations of the fact that observers in relative motion have different spaces (different sets of simultaneous events, i.e., relative simultaneity). To see that this is really the case, let us examine Minkowski’s explanation of length contraction as presented graphically in Fig. 1.1.

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Fig. 1.1 The transparency which Minkowski used at his lecture in Cologne on September 21, 1908. It shows Fig. 1 in his paper “Space and Time.” Source (with permission): Cover of The Mathematical Intelligencer, Volume 31, Number 2 (2009)

Minkowski considered two bodies in uniform relative motion represented by their red and green worldtubes13 as shown on the right half of Fig. 1.1. He demonstrated that, like the failed experiments to detect absolute motion are manifestations of the reality of spacetime, length contraction is another such manifestation. Consider the body represented by the vertical red worldtube. The three-dimensional cross-section . P P, resulting from the intersection of the body’s worldtube and the space (represented by the red horizontal line in Fig. 1.1) of an observer at rest with respect to the body, is the body’s proper length. The threedimensional cross-section . P ' P ' , resulting from the intersection of the red body’s worldtube and the space (represented by the inclined green line) of an observer at rest with respect to the green body (represented by the inclined green worldtube), is the relativistically contracted length of the red body measured by that observer (the cross-section . P ' P ' only appears longer than . P P because a fact of the pseudoEuclidean geometry of spacetime is represented on the Euclidean surface of the page). Note that while measuring the same body, the two observers measure two different three-dimensional bodies represented by the cross-sections . P P and . P ' P ' in Fig. 1.1 (this relativistic situation will not be truly paradoxical only if what is meant by “the same body” is the body’s worldtube). To see that length contraction is impossible in a three-dimensional world,14 assume, just for the sake of the argument, that the worldtube of the red body did not exist as a four-dimensional object and were nothing more than an abstract geo13

Minkowski used the term strip, not worldtube. Minkowski did not state this in his 1908 lecture most probably for the same reason (as in the case of his general argument)—it is exceeding obvious especially to a mathematician.

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metrical construction. Then, what would exist would be a single three-dimensional body, represented by the cross-section . P P, the red and green body would share the same three-dimensional space (in Fig. 1.1 the red and green line would coincide) and both observers, associated with the red and green bodies, respectively, would measure the same three-dimensional body of the same length because nothing else would exist. Therefore, not only would length contraction be impossible, but relativity of simultaneity would be also impossible since a spatially extended three-dimensional object is defined in terms of simultaneity—all parts of a body taken simultaneously at a given moment—and as both observers in relative motion would measure the same three-dimensional body (represented by the cross-section . P P) they would share the same class of simultaneous events in contradiction with relativity. Let us now summarize Minkowski’s specific argument for the reality of spacetime: the relativistic length contraction of a body is a manifestation of the reality of the body’s worldtube and therefore another manifestation of the reality of spacetime— (i) the red and the green bodies must have different three-dimensional spaces (i.e., relative simultaneity, which is impossible in a three-dimensional world) in order to intersect the red body’s worldtube at two different places (in the cross-sections . P P and . P ' P ' ), and (ii) the red body’s worldtube must be real in order to be intersected at two different places. Another relativistic effect—time dilation—is also such a manifestation of relativity of simultaneity (the fact that observers in relative motion have different threedimensional spaces); that is, a manifestation of the reality of spacetime. In other words, like length contraction, time dilation is also impossible in a three-dimensional world: in order that time dilation be possible (i) two observers in relative motion, measuring this relativistic effect, must have different three-dimensional spaces, and (ii) the worldtubes of the clocks involved in the measurement must be real fourdimensional objects [5, Chap. 5]. As length contraction and especially time dilation have been repeatedly confirmed by experiment15 none of these experiments would be possible if spacetime were not real. As a lot is at stake with proving the reality of spacetime, it might be tempting to resist Minkowski’s explanation of length contraction (which is the accepted natural explanation) and to try to think of this effect as a deformation phenomenon in line with the initial attempts of Lorentz and FitzGerald to explain the negative result of the Michelson-Morley experiment. Such a temptation is a sure recipe for a total failure to understand spacetime physics, because the Lorentz transformations demonstrate that the distance between two (space-like-separated) points relativistically contracts no matter whether these points mark the end points of a meter stick or are simply two points in space (where no deformation is involved). Also, the essence of length 15

Length contraction was tested experimentally, along with time dilation, by the muon experiment in the muon reference frame [15]: In the muon’s reference frame, we reconcile the theoretical and experimental results by use of the length contraction effect, and the experiment serves as a verification of this effect.

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Fig. 1.2 An ordinary meter stick which we believe we see; in fact, what we see is a bit more complex because we see the past light cone

contraction is not so much the contraction, but the fact that this relativistic effect is a manifestation of the reality of the worldtube of the contracting body and the possibility that the observers in relative motion, involved in the measurement of the length of the body, should have different three-dimensional spaces, which is impossible in a three-dimensional world (it is the different intersections between the body’s worldtube and the two spaces that are actually measured). To see this even clearer, let us examine a thought experiment visualizing Minkowski’s explanation of length contraction,16 which makes is exceedingly clear that what is measured is two different three-dimensional cross-sections (one of which is shorter) of the worldtube of the meter stick resulting from the intersections of the worldtube and two three-dimensional spaces of two observers in relative motion.17 Two ordinary meter sticks are involved in the thought experiment. In order to make this experiment as convincing as possible let see what is a meter stick in a threedimensional and in a four-dimensional world. If reality were a three-dimensional world, the meter stick would be what we perceive—a typical three-dimensional object (shown in Fig. 1.2)—and take for granted that it is what really exists. In reality, however, since we see the past light cone, what we perceive does not even constitute a three-dimensional object (whose parts exist simultaneously at a given moment), because what we see is a mosaic of fragments of the same meter stick at different moments of its history (lying on the past light cone). According to Minkowski, however, the three-dimensional meter stick exists equally at all moments of its history in a four-dimensional world and what is ultimately real is the worldtube of the meter stick as shown in Fig. 1.3. We have a clear alternative—the meter stick is either the three-dimensional object of our perceptions (Fig. 1.2) or it is a four-dimensional object (Fig. 1.3)—the meter stick’s worldtube. Now we will see why the thought experiment clearly demonstrates that length contraction of a meter stick would be impossible if the meter stick existed as a three-dimensional body (not a worldtube). Assume that an ordinary meter stick (Fig. 1.2) is at rest with respect to an observer . A. Another meter stick at rest in another observer’s (observer . B’s) reference frame and moves relative to the first one at a distance .1 mm above it. Let us assume that at event . M the middle point of . B’s meter stick is instantaneously above the middle point of . A’s meter stick. Assume also that lights are installed inside . A’s meter stick, which can change their color simultaneously at any instant in . A’s frame (Fig. 1.4). At the event of the

16

Early versions of such a visualisation of Minkowski’s explanation of length contraction were discussed in [16, 17]. 17 Even if the technology now makes an actual experiment possible, hardly anyone would support it because it is certain that what is described here would be measured.

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Fig. 1.3 The worldtube of the meter stick; in fact, this is a simplification of its worldtube, but it conveys the main idea—that the meter stick exits en bloc at all moments of its “life.”

Fig. 1.4 . A’s meter stick changes colors simultaneously in . A’s reference frame

meeting . M all lights are simultaneously red in . A’s frame. At all previous moments all lights were green. At all moments after the meeting all lights would be blue. The worldtube of the meter stick with changing colors is depicted in Fig. 1.5. When . A and . B meet instantaneously at event . M (Fig. 1.6) this event is present for both of them. At that moment all lights of . A’s meter stick will be simultaneously red for . A. In other words, the present meter stick for . A is red (that is, all parts of . A’s meter stick, which exist simultaneously for . A at . M, are red). All moments before . M, when all lights of the meter stick were green, were past for . A, whereas all moments when the meter stick will be blue are in . A’s future. Imagine now that. B’s meter stick contains cameras, instead of lights, at every point along its length. At the event of the meeting . M all cameras take snapshots of the parts of . A’s meter stick which the cameras face. All snapshots are taken simultaneously in . B’s reference frame. Even without looking at the pictures taken by the cameras it is clear that not all pictures will show a red part of . A’s meter stick, because what is simultaneous for . A is not simultaneous for . B. When the picture of . A’s meter stick is assembled from the pictures of all cameras it would show two things as depicted in Fig. 1.7—(i) . A’s meter stick photographed by . B is shorter, and (ii) only the middle part of the picture of . A’s meter stick is red; half is green and the other half is blue.

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Fig. 1.5 The worldtube of . A’s meter stick when its color changes. Again, this is a simplification of its worldtube, but it conveys the main idea—that . A’s meter stick exits en bloc at all moments of its proper time

Fig. 1.6 At event . M the middle points of . B’s meter stick is instantaneously above the middle point of . A’s meter sticks

Fig. 1.7 Relativistically contracted meter stick measured by observer . B at event . M—its most striking feature is not that it is shorter, but the fact that it contains at once as present for . B the green part (comprised of parts of the meter stick which existed at different moments in . A’s past), the red part (common present for . A and . B) and the blue part comprised of parts of the meter stick which will exist at different moments in . A’s future

So what is past (green), present (red), and future (blue) for . A exists simultaneously as present for . B. But this is only possible if (i) observers . A and . B have different three-dimensional spaces (which is impossible in a three-dimensional world), and (ii) if . A’s meter stick is a real four-dimensional object (worldtube) as shown in Fig. 1.8. The instantaneous space of . B corresponding to event . M (the inclined white rectangle) intersects the worldtube of the meter stick at an angle (different from the angle at which . A’s space, represented by the horizontal red rectangle, intersects it) and the resulting three-color cross section is what is measured by . B—a different three-dimensional meter stick, which is shorter18 than the meter stick measured by . A. As Minkowski’s explanation of length contraction, further visualized by this thought experiment, is of extreme importance for understanding of the profound physical meaning of length contraction (as a manifestation of the reality of the 18

In Fig. 1.8 the inclined “cross section,” which represents the different three-dimensional meter stick measured by . B, appears longer, not shorter, because, as indicated above, a fact in the pseudoEuclidean geometry of spacetime is represented on the Euclidean surface of the paper.

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Fig. 1.8 The worldtube of . A’s meter stick with different colors must be a real four-dimensional object in order that . B’s three-dimensional space intersects it at a different angle—in the threecolor cross section outlined with the white line. At event . M, when the middle points (the mark .50 cm) of . A’s and . B’s meter sticks instantaneously coincide, . A’s three-dimensional space intersects the worldtube of . A’s meter stick in the red cross sections, whereas . B’s three-dimensional space intersects it in the three-color cross section

worldtube of the contracted body), consider also the spacetime diagram (Fig. 1.9) which depicts the worldtubes of . A’s and . B’s meter sticks involved in the thought experiment. In addition to the meter sticks’ worldtubes, the instantaneous (threedimensional) spaces of observer . A and observer . B, corresponding to event . M, are also given in the spacetime diagram. In order to represent properly what is happening in the pseudo-Euclidean geometry of spacetime on the Euclidean geometry of the page, . A’s instantaneous space is orthogonal to . B’s time axis .t B , whereas . B’s instantaneous space is orthogonal to . A’s time axis .t A . The formal justification for this choice is the invariance of the interval in spacetime .s 2A = s B2 or c2 t A2 − x 2A = c2 t B2 − x B2

.

(only one—the .x—dimension of space is considered). We can formally write c2 t A2 + x B2 = c2 t B2 + x 2A ,

.

which resembles the expression for the invariance of length in Euclidean space as given in two coordinate systems with orthogonal axes .(ct A , x B ) and .(ct B , x A ). With the help of this trick length contraction is correctly represented in the diagram— . B’s instantaneous space (perpendicular to . A’s time axis .ct A ) at event . M intersects the worldtube of . A’s meter stick and the resulting three-color cross-section . B ' B ' (. A’s meter stick measured by . B) is shorter than the proper length of . A’s meter stick . A A measured by . A (the red cross-section resulting from the intersection of . A’s instantaneous proper space at event . M and the worldtube of . A’s meter stick) and also shorter than the proper length . B B of . B’s identical meter stick measured by . B.

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17 ctA

ctB

xB B

A M

B′ B

B′ A

xA

Worldtube of

Worldtube of

Fig. 1.9 The thought experiment schematically given in Fig. 1.8 is properly represented here in terms of the worldtubes of the . A’s and. B’s meter sticks. Observer . A’s instantaneous proper space at event. M intersects the worldtube of. A’s meter stick and the resulting three-dimensional cross-section . A A represents the proper length of . A’s meter stick (measured by . A), whose color is red at event . M. Observer . B’s instantaneous proper space at event . M intersects both the worldtube of . A’s meter stick in the three-dimensional (three-color) cross-section . B ' B ' , which is relativistically contracted (as measured by . B at event . M), and the worldtube of . B’s meter stick in the three-dimensional cross-section . B B, which is the proper length of . B’s meter stick (measured by . B at event . M)

It should be emphasized again that no length contraction would be possible if (i) A and . B did not have different three-dimensional spaces, and (ii) the meter stick’s worldtube did not exist as a four-dimensional object. Otherwise, if reality were a three-dimensional world, i.e., if there existed a single three-dimensional space shared by observers . A and . B and . A’s meter stick were a three-dimensional object, both observers would measure the same three-dimensional meter stick (the same set of simultaneously existing parts of the meter stick), which would mean that the observers would share the same (absolute) class of simultaneous events in a clear contradiction with relativity. Needless to say that anyone who questions Minkowski’s (the accepted) spacetime explanation19 of length contraction should offer an alternative account that is in strict agreement with the existing experimental evidence, and should demonstrate how relativity of simultaneity (and therefore length contraction and time dilation) would be possible in a three-dimensional world.

.

19

As one of the greatest achievements of spacetime physics is the replacement of the postulates of three-dimensional physics with .100% explanations with length contraction being such an explanation. In the next chapters we will see more such .100% explanations.

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Employing Minkowski’s ideas to other spacetime situations also convincingly demonstrates that (i) the experiments that confirmed the twin paradox effect would be impossible if the worldtubes of the twins were not real [5, Chap. 5], and (ii) the profound physical meaning of the unavoidable vicious circle, which led Einstein to the conventionality of simultaneity and also to the conventionality of the one-way velocity of light (which demonstrated that it is impossible to determine experimentally either the distant simultaneity of two events or the one-way-velocity of light), is that reality is an absolute four-dimensional world (spacetime) [5, 16, 20, 21, p. 159]. When Minkowski decoded the profound physical message hidden in the failed experiments to detect absolute (uniform) motion—observers in relative motion have different times and spaces which is possible in an absolute four-dimensional world— then the physical meaning of both relativity of simultaneity and conventionality of simultaneity became clear: as reality is a four-dimensional world which is not divided into three-dimensional spaces (i.e., into classes of simultaneous events since space is defined as a class of simultaneous events) and times, simultaneity is both relative (to a given reference frame) and conventional (in a single reference frame depending on the choice of .ε20 ); in other words, all events of spacetime exist equally (in the sense used by Minkowski) and therefore it is indeed a matter of choice (of reference frame or of .∈) which class of events to regard as simultaneous. That is why, it is a contradiction in terms to say that simultaneity is relative (meaning there is no preferred class of events) but is not conventional (meaning there is a preferred class of events). When the conventionality of simultaneity is defined in terms of the dimensionality of the world (strictly following Minkowski’s approach)—a three-dimensional space or world is a class of simultaneous events—it is clear that, if reality is a threedimensional world, simultaneity is both absolute and non-conventional; only in a four-dimensional world (spacetime) simultaneity is both relative and conventional. As experiments would be impossible if reality were not a four-dimensional world (as we saw in this chapter), it is an experimental fact that simultaneity is both relative and conventional. That is why no theoretical argument (even theorem) [18] against the conventionality of simultaneity can challenge an experimental fact. And, not surprisingly, that theoretical argument was properly addressed [19].

Summary The concept of spacetime represents a real four-dimensional world with time as the fourth dimension as Minkowski advocated: (i) his general argument demonstrates that all failed experiments to detect absolute motion since Galileo’s time, including the Michelson-Morley experiment, would not be possible (i.e., they would detect it) if spacetime were not real, (ii) his specific argument demonstrates that the experiments In the literature on whether or not simultaneity is conventional .∈ reflects the choice whether the back and forth speed of light between two points is the same or different. As nothing moves in spacetime since there are only worldlines of light, for example, there, we are free to make such a choice when we describe spacetime in the ordinary three-dimensional language.

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References

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that confirmed length contraction would not be possible if (a) two observers, measuring the length of a meter stick, did not have different three-dimensional spaces, and (b) the worldtube of the meter stick were not a real four-dimensional object. Obviously, anyone who refuses to accept the spacetime view of the world must face and challenge the arguments that the relativistic experimental evidence would be impossible if spacetime did not exist, staring with Minkowski’s arguments.21 This is how science works.

References 1. D.F. Wallace, Everything and More: A Compact History of Infinity (Norton, New York, 2003), p.259 2. P.L. Galison, Minkowski’s space-time: from visual thinking to the absolute world. Hist. Stud. Phys. Sci. 10, 85–121, 98 (1979) 3. V. Petkov, On the reality of spacetime. Found. Phys. 37, 1499–1502 (2007) 4. A.H. Anastassov, The Theory of Relativity and the Quantum of Action (4-Atomism), Doctoral Thesis, University of Sofia, 1984; Self-Contained Phase-Space Formulation of Quantum Mechanics as Statistics of Virtual Particles, Annuaire de l’Universite de Sofia “St. Kliment Ohridski,” Faculte de Physique 81 (1993), pp. 135–163. http://spacetimecentre.org/vpetkov/ 4-atomism.pdf 5. V. Petkov, Relativity and the Nature of Spacetime, 2nd edn. (Springer, Heidelberg, 2009). ((Ch. 5)) 6. H. Poincaré, Sur la dynamique de l’électron. Rendiconti del Circolo matematico Rendiconti del Circolo di Palermo 21, 129–176 (1906). Translated in: [7] 7. V. Petkov (ed.), The Origin of Spacetime Physics, 2nd edn. Foreword by A. Ashtekar. (Minkowski Institute Press, Montreal, 2023) 8. T. Damour, Once Upon Einstein, Translated by E. Novak (A. K. Peters, Wellesley, 2006), p. 52 9. V. Petkov, Physics as Spacetime Geometry in A. Ashtekar, V. Petkov (eds), Springer Handbook of Spacetime (Springer, Heidelberg, 2014), Chapter 8, pp. 141–163 10. H. Minkowski, Space and time, new translation, in Spacetime: Minkowski’s Papers on Spacetime Physics, ed. by H. Minkowski. Translated by Gregorie Dupuis-Mc Donald, Fritz Lewertoff and Vesselin Petkov. Edited by V. Petkov (Minkowski Institute Press, Montreal, 2020) 11. A. Einstein, On the Electrodynamics of Moving Bodies. New publication of the English translation in: [7] 12. H.A. Lorentz, The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat, 2nd edn. (Dover, Mineola, New York, 2003), p. 57; see also his comment on p. 321 13. N.D. Mermin, What’s bad about this habit? Phys. Today 62(5), 8 (2009) 14. H. Weyl, Space-Time-Matter, ed. by V. Petkov (Minkowski Institute Press, Montreal, 2021), p. 218 15. G.F.R. Ellis, R.M. Williams, Flat and Curved Space-Times (Oxford University Press, Oxford, 1988) 16. V. Petkov, From Illusions to Reality: Time, Spacetime and the Nature of Reality (Minkowski Institute Press, Montreal, 2013). ((Ch. 5))

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After challenging Minkowski’s arguments, philosophers may also wish to challenge philosophical arguments for the reality of spacetime, for example the brilliant analyses of Graham Nerlich [22, 23] and Anguel Stefanov [24].

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17. V. Petkov, Spacetime and reality: facing the ultimate judge, in Space, Time and the Limits of Human Understanding. ed. by S. Wuppuluri, G. Ghirardi (Springer, Heidelberg, 2017), pp.137–148 18. D. Malament, Causal theories of time and the conventionality of simultaneity. No.us ˆ 11, 293– 300 (1977) 19. S. Sarkar, J. Stachel, Did malament prove the non-conventionality of simultaneity in the special theory of relativity? Phil. Sci. 66(2), 208–220 (1999) 20. V. Petkov, Simultaneity, conventionality and existence. British J. Phil. Sci. 40, 69–76 (1989) 21. V. Petkov, Conventionality of simultaneity and reality, in The Ontology of Spacetime II, ed. by D. Dieks (Elsevier, Amsterdam, 2008); Philosophy and Foundations of Physics Series, vol. 4, pp. 175–185 22. G. Nerlich, What Spacetime Explains: Metaphysical Essays on Space and Time (Cambridge University Press, Cambridge, 1994) 23. G. Nerlich, Einstein’s Genie: Spacetime out of the Bottle (Minkowski Institute Press, Montreal, 2013) 24. A. Stefanov, Space and Time: Philosophical Problems (Minkowski Institute Press, Montreal, 2020)

Chapter 2

Inertial and Accelerated Motion in Spacetime Physics

Probably the concept of motion—seemingly the least doubted of our perceptual experience—might have been among the observations of ancient thinkers that gave rise to the proverb “The darkest place is under the lantern.” For centuries the greatest minds have been struggling to understand what people have always thought they were fully familiar with—motion. The Eleatics (fifth century BC) held a completely counter-intuitive view on motion, which is amazingly similar to the spacetime view of the world. According to the Eleatic school of thought the analysis of the observed motion and change revealed internal contradictions in these phenomena which led the Eleatics to conclude that these were nothing more than mere illusions and that the true reality was an eternal existence [1, 2]. Aristotle (384 BC—322 BC) held a down-to-earth view on motion that reflected people’s everyday experience which he summarized in the first sentence in Book VII of his Physics [3]: “Everything that is in motion must be moved by something.” In a brilliant analysis1 of real and thought experiments Galileo (1564—1642) [5] disproved Aristotle’s view of motion and arrived at two important pieces of knowledge which we now call Galileo’s principle of inertia—bodies in uniform motion move on their own (do not need a mover)—and Galileo’s principle of relativity—uniform motion, i.e., motion by inertia, cannot be detected by mechanical experiments. Galileo first noticed that rest and uniform motion, i.e., motion with constant velocity,2 cannot be distinguished experimentally (postulated in his principle of relativity). Newton (1643—1727) insisted that both acceleration and space are absolute because acceleration can be experimentally detected by the resistance a particle offers to its acceleration, and is therefore absolute, which implies that space is also absolute due to the apparently self-evident assumption that any acceleration is with respect to space. However, Newton’s view raised the obvious question “If absolute acceleration 1

For a summary of his analysis see [4, Chap. 2]. Constant velocity, not constant speed, because both the magnitude (the speed) and the direction of the velocity should be constant. 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8_2

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implies an absolute space, why cannot uniform (inertial) motion and rest with respect to this space be also detected experimentally?” After Michelson-Morley experiment and especially after Einstein’s 1905 special relativity, where absolute space was declared superfluous [6], the status even of acceleration became unclear—on the one hand, it is absolute (frame-independent) because accelerated motion is experimentally detectable, but, on the other hand, if one cannot talk about absolute space in relativity, with respect to what a particle accelerates? After Einstein dethroned the absolute space the confusion over the nature of motion was completed and its understanding started to resemble a “motion mess”3 — rest, uniform motion, accelerated motion with respect to what? Einstein’s declaration that motion was relative did not explain anything—when bodies move relative to one another they move in something; those who seem to believe that space does not exist as an entity (and should be understood as a non-entity representing relations between material objects) have never been able to answer even the ancient objection4 —if nothing separates two distant objects why don’t they touch each other? At least Einstein followed Galileo’s example and generalized Galileo’s principle of relativity by postulating that uniform (inertial) motion cannot be detected at all (not only by mechanical experiments) [6]: The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

Einstein did not state explicitly that uniform motion cannot be discovered, but this assertion is implicitly there—the laws are the same in two (or all) inertial systems precisely because uniform motion cannot be discovered; otherwise, if the laws were not the same, that would mean that uniform motion is detectable. In his lecture “Space and Time” Minkowski outlined a novel approach to physical phenomena, which constituted a new physics—he proposed to move a step even further than the newly introduced by him four-dimensional physics and to represent it as spacetime geometry because it explains5 practically all mechanical phenomena as we will see below and in the next chapters. As we have seen in the previous chapter, Minkowski had successfully decoded the profound physical message hidden in the failed experiments to detect absolute uniform motion which stated: the world is four-dimensional with time as the fourth dimension (spacetime). Then he certainly realized that the four-dimensional physics describing this world was in fact spacetime geometry since all particles which appear to move in space are in reality a forever 3 Had the Eleatics been alive they would have been certainly pleased to see that more contradictory features of motion were revealed. 4 For modern arguments that space is an entity see the deep analysis of Graham Nerlich [8]. 5 Minkowski’s novel approach constituted a very rare event in fundamental physics—it provided, as we will see, full (.100%) explanations of most mechanical phenomena. By contrast, Galileo and Einstein had to postulate ideas deduced from experiments without being able to explain them. Nevertheless, both made enormous contributions not only to physics but also to the art of doing physics—particularly their method of exploring the internal logic of (especially fundamental) physical ideas by analyzing real and thought experiments.

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given web of the particles’ worldlines in spacetime. This realization led Minkowski to initiate a program of geometrizing physics [9, pp. 58–59]: The whole world presents itself as resolved into such worldlines, and I want to say in advance, that in my understanding the laws of physics can find their most complete expression as interrelations between these worldlines.

The first results of the employment of this program by Minkowski himself to the issues of the experimental detectability of accelerated motion and the undetectability of uniform (inertial) motion and rest were spectacular. Minkowski dealt with the “motion mess” in a “mightily revolutionary”6 way: • There is no such thing as motion in spacetime7 —what we perceive as moving particles (where—in what space?) are timelike worldlines (worldtubes for spatially extended objects) in spacetime. • Minkowski himself demonstrated that his program of geometrizing physics provided an immediate explanation of why there is no “motion mess” in spacetime by explaining what kind of worldlines correspond to the everyday concepts of rest, uniform motion and accelerated (non-uniform) motion (see Fig. 2.1) [9, p. 62]: A straight worldline parallel to the .t-axis corresponds to a stationary substantial point, a straight line inclined to the .t-axis corresponds to a uniformly moving substantial point, a somewhat curved worldline corresponds to a non-uniformly moving substantial point.

Minkowski clearly explained (Fig. 2.1) why the states of rest and uniform motion are indistinguishable [9, p. 62]: With appropriate setting of space and time the substance existing at any worldpoint can always be regarded as being at rest.

As a straight timelike worldline represents uniform, i.e., inertial motion, it immediately becomes clear why experiments have always failed to distinguish between a state of rest and a state of uniform motion—in both cases a body is a straight worldline as seen in Fig. 2.1 (the green and blue worldlines) and there is clearly no distinction between two straight lines. In the figure, if the time axis of a reference frame is chosen8 along the green worldline, the body represented by this worldline appears to be at rest in the green reference frame (with green space and time), whereas 6

To use his own expression from the draft of his 1908 lecture as quoted in the previous chapter. Fortunately, good books and textbooks make this counter-intuitive feature explicit—see, for example [10]: “The objective world merely exists, it does not happen; as a whole it has no history” and [11]: “There is no dynamics in spacetime: nothing ever happens there. Spacetime is an unchanging, once-and-for-all picture encompassing past, present, and future.” 8 Although there are no such things as space and time in spacetime, we can still describe it in terms of the inadequate concepts space and time (“inadequate” namely because there is nothing in the physical world that corresponds to these two separate concepts) imposed on us by the way we perceive the world. We can choose the time axis along any timelike straight worldline and all bodies whose worldlines are inclined with respect to the chosen time axis (i.e., to the selected straight worldline) will appear in motion relative to the body represented by the chosen worldline and in the body’s proper space. 7

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Fig. 2.1 There is no “motion mess” in spacetime. The straight green and blue worldlines represent uniform (inertial) motion—if the time axis is along the green worldline, the body represented by it is perceived as being at rest, whereas the body represented by the blue worldline is perceived as moving uniformly in the green space (corresponding to the green time); if the time axis is chosen along the blue worldline, the blue worldline represents a body perceived to be at rest, whereas the green worldline represents a uniformly moving body in the blue space (corresponding to the blue time). The curved red worldline represents an accelerated body. The sharp physical distinction between uniform (inertial) motion and accelerated motion is manifested in the sharp geometrical distinction—straightness and curvature of the worldlines representing inertial and accelerated motion, respectively

the body represented by the blue worldline appears to move by inertia (uniformly) with respect to the green reference frame (i.e., in the green space or, more precisely, in the proper space9 of the body represented by the green worldline), because the blue worldline is inclined with respect to the green worldline; if the green and blue worldlines were parallel, the bodies represented by them would appear at rest with respect to each other and would share the same proper space. If the time axis of a reference frame is chosen along the blue worldline, the body represented by that worldline will appear to be at rest in the newly chosen reference frame, whereas the first body (represented by the green worldline) will appear to be uniformly moving with respect to the body represented by the blue worldline (since the green worldline is inclined to the new time axis, i.e., inclined to the blue worldline), i.e., it will be 9

Again, there is no such thing as space in spacetime, but like a time axis can be chosen along a timelike straight worldline, a space (a set of simultaneous events) corresponding to the chosen time axis can be also chosen. That is it does not really matter whether or not the chosen space is orthogonal to the chosen time axis; this is the meaning of conventionality of simultaneity briefly discussed at the end of the previous chapter.

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moving (to the left) in the blue space (the proper space of the body represented by the blue worldline). Minkowski’s program of regarding physics as spacetime geometry also provides a self-evident explanation of the puzzling question that has been worrying deep thinkers since the time of Galileo when he disproved Aristotle’s view of motion (“Everything that is in motion must be moved by something”) and postulated that uniform motion needs no mover. Galileo merely stated the experimental fact (captured in Galileo’s principle of inertia) that a uniformly moving body moves on its own (we now call it by inertia). This piece of experimental evidence constituted another “contribution” to the “motion mess.” In science we strive to explain everything in terms of cause and effect, but saying that a body moves by inertia is nothing more than placing a label, because if a body moves on its own, there is no cause for its uniform motion. Saying that only accelerated motion requires a cause (a mover) does not explain what makes it so different from uniform motion (in addition to the known difference) given that they both are motions—both an accelerating body and a uniformly moving body appear to move in space, but since Newton’s time until Einstein’s 1905 special relativity physicists believed that, while both bodies do appear to move in space, inexplicably, only an accelerating body’s motion is experimentally detectable through the body’s resistance to its acceleration. Einstein postulated (on the basis of the experimental evidence) that absolute space was superfluous, but did not explain why and our senses continued to stubbornly keep the puzzle alive—both an accelerated body and a uniformly moving body move in space, but only the accelerated body needs a mover. Stated in one sentence: our senses suggest that bodies move in something— space—and if space is indeed regarded as an entity, where bodies exist and move, our intuition makes us think that the natural state of bodies is being at rest (as Aristotle believed) and any motion in space should require a mover. Minkowski explained the profoundly deep physical meaning of the non-existence of absolute space (there is no absolute, i.e., single space in the world because the hidden physical message of the failed experiments to detect absolute motion was that all observers in relative motion have their own spaces, which implied that the world is four-dimensional) and the puzzle disappeared. Here is the explanation in terms of the geometry of spacetime:10 There is no motion in spacetime—our everyday experience of all moving objects we perceive, which so convincingly seems to suggest that all those objects are really moving, is nothing more than a manifestation of the forever-given network of the worldlines of those bodies. Consider the first two worldlines (green and blue) in Fig. 2.1 and assume that the vertical green worldline now represents an observer who watches the uniform motion of a body represented by the blue worldline, which is 10

Let me stress it again that worldlines (or worldtubes) are not just graphical representations; those who are tempted to think they are nothing more than abstract concepts should read again the first chapter to understand why the ultimate judge in physics—the experimental evidence—declared that the true reality is spacetime. Those, who are tempted to disagree with the ultimate judge’s ruling, know the way—disprove the arguments given in the chapter starting with Minkowski’s arguments. Again, because there are many examples in the literature—ignoring arguments is not how science, especially physics, works.

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Fig. 2.2 The straight green and blue worldlines represent the uniform relative motion of an observer (represented by the green worldline) and a body (represented by the blue worldline). The body is seen by the observer as moving by inertia, i.e., on its own, without a mover. But in reality there are simply two inclined straight worldlines in spacetime and therefore the perceived motion by inertia (on its own) is fully (.100%) explained in terms of the geometry of spacetime

inclined (Fig. 2.2). The past light cone, whose apex is at event .t0 on the green worldline, intersects the blue worldline at event . A; in the ordinary everyday language this means that at the moment (event).t0 the green observer sees the blue three-dimensional body (the three-dimensional cross section resulting from the intersection of the past light cone and the blue worldline) at a distance .x0 from himself. The past light cone at event .t1 (represented in the figure only by the yellow arrow) intersects the blue worldline at event . B, which means that at the moment (event) .t1 the green observer sees the blue body at a greater distance.x1 from himself. He incorrectly (inadequately) interprets these observations in a sense that the same11 three-dimensional blue body recedes uniformly on its own, without a mover. Therefore, the (three-dimensional) blue body, which is seen by the green observer to be moving on its own by inertia, is in reality a straight (timelike) worldline that is inclined to the straight (timelike) green worldline of the observer. The two straight worldlines, which represent the relative uniform motion of the (three-dimensional) green observer and the (three-dimensional) blue body, are forever given in spacetime. As in the higher reality of spacetime there is no such thing as motion, therefore, naturally, there is no cause (no mover) for something (motion) that does not exist there.

11

As seen in Fig. 2.2 at the moments.t0 and.t1 the observer does not see the same three-dimensional body—he sees two different three-dimensional cross sections (two different three-dimensional bodies) at events . A and . B.

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What about accelerated motion which has been regarded as caused by a mover? Does the same argument (no motion in spacetime) apply to it to conclude that it does not need a mover either? The argument explains that neither a uniformly moving body no an accelerating body move “in space” because they both are merely forever given worldlines in spacetime. As we have seen in Fig. 2.1 what makes uniform (inertial) and accelerated motion sharply physically distinct (unlike inertial motion, accelerated motion is experimentally detectable) is the sharply geometrical distinction of their worldlines—the worldline of a uniformly moving body is straight, whereas the worldline of an accelerating body is curved (deformed). Minkowski pointed out that “Especially the concept of acceleration acquires a sharply prominent character” [9, p. 66]. There is no cause for the straightness of the worldline of a uniformly moving body (so one may say “That is why there is no mover in the perceived inertial motion of the body”), but there is a cause for the deformation of the worldline of the accelerating body (so one may say “That is why there is a mover in the perceived accelerated motion of the body”)—something statically caused 12 the deformation of the worldline, i.e., in three-dimensional language, something (a mover) is causing the acceleration of the body. Figure 2.3 represents the mutual deformation of the worldlines of two elastic balls—in three-dimensional language, initially the balls move uniformly and approach each other (their worldlines are straight); then they collide (instantaneously decelerating each other, being instantaneously at rest with respect to each other, and mutually accelerating13 ) and finally recede from each other moving uniformly. The two worldlines mutually deform each other at the event of the collision—in threedimensional language, as each ball is prevented by the other from moving by inertia, it resists its deceleration by exerting an inertial force on the other ball; it is these inertial forces that are the static causes for the deformation of the two worldlines. We will discuss the nature of the inertial force in the next chapter, but even here it is evident that in spacetime those forces are restoring forces that statically resist the mutual deformation of the two worldlines (like the restoring force in a deformed three-dimensional metal rod). Now we can see how the last element of the “motion mess”—the absoluteness of acceleration given that there is no absolute space—is explained in terms of the geometry of spacetime. We saw that the physically absolute acceleration of a body (because it is experimentally detectable) is represented in spacetime by an absolute geometrical property—the curvature (i.e., the deformation) of the worldline of the accelerated body. So, Minkowski’s program of regarding physics as spacetime geometry produced an important and impressive result—an unforeseen resolution of the debate over the status of acceleration14 : 12

A possible way to try to visualize static deformation is to think of a metal construction, which contains deformed metal components—those components are deformed, but the whole construction is static. 13 This acceleration is a complex phenomenon caused by the elastic nature of the balls. 14 Minkowski’s program of geometrizing physics does provide a spectacular resolution of this centuries-old debate, but nevertheless it is worth explicitly mentioning that Minkowski’s approach decisively disproved Mach’s ideas of relativity of motion, including of acceleration (that initially

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Fig. 2.3 The blue and red worldlines represent two elastic balls which approach each other moving uniformly, then collide (mutually decelerating and accelerating), and recede by again moving uniformly. Each ball is the cause (mover) for the change of uniform motion of the other ball

Acceleration is absolute not because it is with respect to an absolute space, but because it is a manifestation of an absolute geometrical feature of a timelike worldline—its curvature or rather its deformation. In flat spacetime curvature means deformation, but in curved spacetime it is the deformation of a timelike geodesic worldline that manifests itself as an absolute μ (curved-spacetime) acceleration .a μ = d 2 x μ /dτ 2 + [αβ (d x α /dτ )(d x β /dτ ). The natural curvature of worldlines in curved spacetime, caused by the spacetime curvature, manifests itself as a geodesic deviation, which in turn gives rise to relative acceleration. This acceleration reflects two facts—that there are no straight worldlines and no parallel or rather congruent worldlines in curved spacetime. This acceleration is not absolute, but relative since it involves two geodesic worldlines (which are influenced Einstein), because one can still come across confusions in the literature caused by Mach’s scientific philosophy. According to Mach, if there were a single body in the Universe, one could not say anything about its state of motion—whether it is moving by inertia (uniformly) or accelerating— because only motion relative to other bodies makes sense according to Mach. It follows from what we discussed in this chapter that in spacetime such a hypothetical situation is exceedingly clear— a single body in the Universe is either a timelike geodesic (straight in flat spacetime) worldline (which means that the body moves by inertia) or a deformed timelike worldline (which means that the body accelerates). Mach also held that the Ptolemaic and the Copernican models of our planetary system are equivalent because he believed that rotation and translation were equivalent. Again, it is exceedingly clear that rotation and translation are sharply distinct in spacetime since the worldline of a body moving translationally is either a straight worldline (in flat spacetime) or a geodesic worldline (in curved spacetime) when the body moves uniformly, or a deformed worldline when the body accelerates translationally, whereas the worldline of a rotating body (around a center) is a helix. That is why, the Ptolemaic and the Copernican systems are sharply distinct—in reality, the fact that the planets’ worldlines are helixes around the worldline of the Sun demonstrates that it is the Copernican system that adequately represents the solar system.

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Fig. 2.4 The worldlines.a and.b represent relative acceleration in curved spacetime, which involves two worldlines (i.e., two bodies which move relative to each other without resisting their motion). The deformed worldline .c represents absolute acceleration in spacetime, which involves a single worldline (i.e., a single body which accelerates absolutely and resists its acceleration)

naturally curved but not deformed), whereas absolute acceleration involves a single non-geodesic worldline (which is deformed). The motion of a relatively accelerating body cannot be experimentally detected, because it does not resist its relative (geodesic) acceleration (its worldline is geodesic, i.e., not deformed). Figure 2.4 shows the two types of acceleration in curved spacetime. The timelike geodesic worldlines .a and .b represent two bodies which are in a state of relative acceleration. The timelike non-geodesic worldline .c is deformed which means that the body it represents is absolutely accelerating and resists its acceleration (which makes absolute acceleration in curved spacetime also detectable).

Summary Since ancient times any serious attempt to understand the nature of motion, which looked self-evident to all people over the centuries (with very few exceptions) led to unanticipated difficulties and revealed a number of puzzles: • the Eleatics’, particularly Zeno’s, arguments taken seriously • the experimental detectability of accelerated, but not of uniform motion, given that they both are motions in space • the status of accelerated motion—relative or absolute and if absolute in what sense • finally, Einstein declaration in 1905 that absolute space was superfluous, which completed the “motion mess.”

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Employing his own radical program of geometrizing physics, not only did Minkowski manage to rise above the “motion mess,” but provided counter-intuitive, provokingly novel and spectacular (unprecedented.100%) explanations15 of the above puzzles.

References 1. J. Barnes, Early Greek Philosophy, 2nd edn. (Penguin Books, London, 2001), Part II 2. J. Barnes, The Presocratic Philosophers (Routledge, London, New York, 1982), Chap. X 3. Aristotle, Physics, in Great Books of the Western World, vol. 7, ed. by M.J. Adler (Encyclopedia Britannica, Chicago, 1993) 4. V. Petkov, Relativity and the Nature of Spacetime, 2nd edn. (Springer, Heidelberg, 2009) 5. G. Galileo, Dialogue Concerning the Two Chief World Systems—Ptolemaic and Copernican, 2nd edn. (University of California Press, Berkeley, 1967). The Second Day 6. A. Einstein, On the Electrodynamics of Moving Bodies. New publication of the English translation in: [7] 7. V. Petkov (ed.), The Origin of Spacetime Physics, Foreword by A (Ashtekar (Minkowski Institute Press, Montreal, 2020) 8. G. Nerlich, The Shape of Space, 2nd edn. (Cambridge University Press, Cambridge, 1994) 9. H. Minkowski, Space and time, new translation, in Spacetime: Minkowski’s Papers on Spacetime Physics, ed. by H. Minkowski. Translated by Gregorie Dupuis-Mc Donald, Fritz Lewertoff and Vesselin Petkov. Edited by V. Petkov (Minkowski Institute Press, Montreal, 2020) 10. H. Weyl, P. Pesic, (eds.), Mind and Nature Selected Writings on Philosophy, Mathematics, and Physics (Princeton University Press, 2009), p. 135 11. R. Geroch, General Relativity: 1972 Lecture Notes (Minkowski Institute Press, Montreal, 2013), pp.4–5 12. E. Mach, The Science of Mechanics, 6th edn. (La Salle, Illinois, 1960)

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The resistance a body offers to its acceleration (i.e., the resistance the deformed worldline of an accelerated body offers to its deformation) is not fully (.100%) explained in this chapter, but this open question since Newton’s time (and certainly even before that) will be addressed in the next chapter.

Chapter 3

Origin and Nature of Inertia in Spacetime Physics

As pointed out in the previous chapter the brilliant analyses1 of real and thought experiments carried out by Galileo [2], which resulted in disproving Aristotle’s view of motion, led him not only to what we now call Galileo’s principle of relativity (uniform motion, i.e., motion by inertia, cannot be detected by mechanical experiments) but those analyses first led him to what we now call Galileo’s principle of inertia (which was essential for disproving Aristotle’s view of motion)—bodies preserve their state of rest or of uniform motion (they continue to move on their own without a mover until prevented from doing so). Newton adopted Galileo’s principle of inertia as the First Law in his Principia [4, p. 416]: Every body perseveres in its state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed.

We saw in the previous chapter how employing Minkowski’s program of geometrizing physics provided a full (.100%) explanation of the puzzling phenomenon of inertial motion (motion without a mover)—“Every body perseveres in its state of being at rest or of moving uniformly straight forward”—which both Galileo and Newton had to postulate. Einstein had adopted and further developed Galileo’s research strategy of postulating pieces of knowledge extracted from the experimental evidence (Galileo’s two principles) and analyzing real and thought experiments; Einstein himself based his special relativity on two postulates—the (generalized) Galileo’s principle of relativity and the constancy of the speed of light (both were fully explained by Minkowski as we saw in the first chapter). An indication that Einstein might have analyzed Galileo’s principle of inertia in depth is his apparent dissatisfaction with it [3, p. 41]: The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration. 1

For a summary of Galileo’s analyses see [1, Chap. 2].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8_3

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This seems to suggest that Einstein had read more Mach’s anti-absolute-space ideas than Newton’s Principia. Otherwise, he would have analyzed thoroughly the two aspects of inertia which Newton identified by analyzing the experimental evidence at that time—(i) a body moves uniformly on its own (Galileo’s principle of inertia adopted as Newton’s Law 1 quoted above) and (ii) when “compelled to change its state by forces impressed” a body that is moving uniformly on its own resists the change of its state as Newton explained [4, p. 404]: Inherent force of matter is the power of resisting by which every body, as far as it is able, perseveres in its state either of resting or of moving uniformly straight forward.

Had Einstein read at least the above quote, it would have become clear to him that Newton did not appear to have seen any weakness of the principle of inertia or an argument in a circle; on the contrary, the two aspects of inertia (deduced from the experimental evidence) make it completely clear (without any Machian references to distant masses) that a mass moves without acceleration (by inertia) if and only if it does not resist its motion. If Einstein had analyzed the principle of inertia by using his famous thought experiments involving lifts (as he did when he formulated his principle of equivalence) he would have certainly realized that from inside of such a lift (without looking outside) it can be determined whether the lift moves uniformly (by inertia) or accelerates. Instead of a lift, Einstein could have even used, without any modifications, Galileo’s famous ship experiment with which Galileo demonstrated what we call Galileo’s principle of relativity—that rest and uniform (inertial) motion cannot be distinguished by performing mechanical experiments [2, pp. 186–187]: Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated

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during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained in it, and to the air also.

If the ship accelerates its acceleration can be easily discovered from within the ship without looking outside and regardless of whether or not the ship is far from other bodies (and regardless of whether or not other bodies exist at all)—e.g., the droplets and the smoke will be deflected toward the stern. Perhaps the only useful influence Mach’s ideas had on Einstein was that they led him to the realization that other masses change the properties (geometry) of spacetime, which in turn changes the shape of geodesic worldlines representing inertial motion [3, p. 39]: The principle of inertia, in particular, seems to compel us to ascribe physically objective properties to the space-time continuum.

However, two things should be stressed. First, contrary to the Machian ideas, effectively only the nearby (not the distant) masses determine the shape of a geodesic worldline, which represents a particle moving by inertia (i.e., they determine its inertia). Second, the external masses affect only the first aspect of inertia—how a particle moves (in three-dimensional language) on its own non-resistantly—not the second aspect of inertia—the resistance a particle offers when prevented from moving by inertia. To see why the resistance a particle offers to its acceleration depends only on the particle’s mass, not on external masses (i.e., not on the properties of spacetime which determine only the shape of the particle’s geodesic worldline), let us show the obvious—that inertial forces are real—because this issue still causes confusion. Consider two identical Einstein lifts . A and . B which are at rest with respect to each other and are far away from gravitating masses (Fig. 3.1). In the middle of each lift there is a floating ball at a distance .h from the “floor.” At a given moment . A starts to accelerate along its height. An observer in it sees that the ball there starts to accelerate (to “fall”) toward the floor and can regard the ball’s acceleration as caused by a fictitious inertial force. However, the observer in. A knows it is a fictitious (unreal) force because in reality (here this expression is exact) it is . A that accelerates and the ball “falls” non-resistantly toward the floor; otherwise, if a real force acted on the ball it would resist its acceleration. Also, an observer in . B sees that it is . A’s floor that approaches the ball in . A, whereas the ball has not changed its state and remains at the same place (relative to . B) where it was before . A started to accelerate. When . A’s floor reaches the ball, the floor starts to accelerate it. The ball resists its acceleration by acting back on the floor and exerting a real inertial force on it. What in reality (i.e., in spacetime) happens in . A is illustrated on the spacetime diagram in Fig. 3.2. The vertical brown band represents the worldtube of the ball, whereas the curved blue line represents the curved worldline of . A’s floor. The floor’s worldline is curved because . A accelerates (and acceleration is represented by a curved worldline), whereas the worldtube of the ball is straight outside of the red circle, which means that the ball is not accelerating.

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Fig. 3.1 Two lifts . A and . B, with a floating ball in each of them, are initially at rest relative to each other. When . A starts to accelerate an observer inside it can say that a fictitious inertial force accelerates the ball toward the floor (however, an observer in . B sees that the ball in . A have not changed its state and remains at the same place relative to . B). But when the ball hits the floor, the floor starts to accelerate the ball and it resists its acceleration by exerting a real inertial force on the floor

As seen in the figure (in the red circle) the floor’s worldline converges to, touches and curves (deforms)2 the worldtube of the ball, which in the ordinary threedimensional language means that . A’s floor starts to accelerate the ball—it is exactly then when the ball starts resisting its acceleration and exerts a real inertial force .Fin on the floor (the proof that the inertial force is real is the deformation on the floor caused by .Fin ). Also, it is the reality of inertial forces that makes accelerated motion experimentally detectable. As we saw in the previous chapter, Minkowski’s program of geometrizing physics provided a complete (.100%) explanation of the first aspect of inertia—what is the physical meaning of the observation that a body moving by inertia appears to move on its own. However, the second aspect of inertia—the resistance a body offers to its acceleration—which is the essence of the phenomenon of inertia, has been for centuries and still remains a mystery. The origin and nature of inertia (and therefore of mass 2

The everyday language I use is incorrect if taken literally (not as a description of the spacetime diagram)—nothing “converges to, touches and curves” in spacetime; everything is all there en bloc like on a film strip.

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Fig. 3.2 The floor of the accelerating lift . A shown in Fig. 3.1 is represented by the curved blue worldline, whereas the ball in . A apparently falling toward the floor is represented by the brown worldtube. It is clearly seen here that in reality, i.e., in spacetime, it is the floor’s curved worldline that approaches (converges to) the ball’s straight worldtube. “When” the floor’s worldline reaches the ball’s worldtube, it “starts” (in a static sense) to curve (i.e., to deform) it (what you see is all there at once in spacetime like on a film strip). The real inertial force .Fin which the ball exerts on the floor appears to originate from the deformation of the ball’s worldtube

as well because mass is defined as the measure of the resistance a particle offers to its acceleration) is one of the deepest open questions in fundamental physics. Minkowski made a gigantic step toward revealing the origin and nature of inertia by showing that a curved (deformed) timelike worldline in spacetime represents an accelerated particle and explained why “Especially the concept of acceleration acquires a sharply prominent character” [5, p. 66]—because the acceleration of a particle is proportional to the curvature of its worldline [5, p. 69]. Linking the acceleration of a particle with the curvature (deformation) of its worldline (rather worldtube) leads to an almost self-evident conjecture [1, Chap. 9] as seen in Fig. 3.2 (in the red circle)—the deformation of the ball’s worldtube does appear to give rise to a static restoring force (statically resisting the worldtube’s deformation), which we interpret as the inertial force .Fin . It should be stressed that it is the deformation of a worldline (not its curvature) that seems to be responsible for an inertial force, because, as explained in the previous chapter, it is the deformation of a worldline that manifests itself as an absolute acceleration in both flat and curved spacetime; a curved worldline in flat spacetime is always deformed, but in curved spacetime a geodesic worldline is naturally curved due to the curvature of spacetime and is not deformed. The worldline of an absolutely accelerated particle in curved spacetime is not geodesic since it is additionally curved, i.e., it is deformed.

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Three facts strongly suggest the conjecture that the worldtube of an absolutely accelerated body statically resists its deformation and this resistance manifests itself as the body’s inertia: • a body resists its absolute acceleration by exerting an inertial force on the obstacle that prevents it from moving by inertia • the worldtube of such a body is deformed • the worldtube of a three-dimensional body is real3 and, like a three-dimensional rod, resists its deformation. The static resistance arising in a deformed worldtube (and the involved fourdimensional stress) can be traced down to the self-forces acting on the accelerated constituents (elementary particles) of the accelerated body represented by that worldtube, which have contributions from electromagnetic, weak and strong interactions [1, Chap. 9]. Had Minkowski lived longer he might have been thrilled to realize that inertia is another manifestation of the reality of spacetime, i.e., of the four-dimensionality of the world (like length contraction, known at his time, and like time dilation and the twin paradox [1, Chap. 5]). At the end of the previous chapter we acknowledged the fact that only the resistance a body offers to its acceleration (i.e., the static resistance the deformed worldline of an accelerated body offers to its deformation) was not fully (.100%) explained in the chapter. Now, if the conjecture that naturally follows when Minkowski’s program of geometrizing physics is proved correct, it will provide a full explanation of the origin of inertia. This explanation of the origin of inertia suggests that the resistance a body offers to its acceleration (the second aspect of inertia) originates from the body itself (or, more precisely, statically exists in the body’s deformed worldtube), that is, it depends only on the body’s mass, not on external masses. This conclusion is independently supported by experiment—the experimental equivalence of inertial mass and passive gravitational mass captured in Einstein’s equivalence principle. To see why it is indeed an experimental fact that the resistance with which a body opposes its acceleration arises in the body itself and does not depend on the properties of spacetime, let us first demonstrate that Minkowski’s program of geometrizing physics also fully explains the physical meaning of the equivalence principle. Consider two Einstein lifts (the type he used in his thought experiments that helped him arrive at his equivalence principle)—as shown in Fig. 3.3 lift . A accelerates vertically along the . y-axis with an acceleration .a, whereas lift . B is at rest on the Earth’s surface. An observer in . A and an observer in . B release identical balls from the middle of the lifts. The ball in . A appears4 to accelerate downwards with an 3

Those tempted to question this fact should return to Minkowski’s argument for the reality of the worldtube of a relativistically contracted body in Chap. 1 to determine whether they are indeed questioning an experimental fact. 4 Einstein had been probably impressed by the equivalence of the two lifts . A and . B because it would “explain” (if a postulate—the equivalence principle in this case—can explain anything) the

3 Origin and Nature of Inertia in Spacetime Physics

37

Fig. 3.3 Lift . A accelerates with an acceleration .a = g. Lift . B is on the Earth’s surface. Observers in . A and . B appear to be unable to determine experimentally whether they are accelerating or at rest on the Earth’s surface from inside of their lifts, without looking outside. Two identical balls are dropped in the lifts. The ball in . A falls with an acceleration .a = g in . A. The second ball falls with an acceleration .g in . B. The mass of the ball .m in measured in . A is equal to the mass of the ball .m g measured in . B. Einstein postulated these experimental facts as his equivalence principle

acceleration .a, whereas the ball in . B falls with an acceleration .g (the magnitude of A’s acceleration is .a = g). When the observers in . A and . B measure the mass of their balls they confirm the experimental fact that the inertial mass .m in (measured in . A) and the passive gravitational mass .m g (measured in . B) are equal, i.e., .m in = m g . No matter what kind of experiments the observers in . A and . B will perform, it is impossible from within the lifts to determine which lift is accelerating and which is at rest on the Earth’s surface.5 Like he did in his special relativity (when he postulated the failure of experiments to detect absolute motion as his relativity principle, adding to it the postulate of the constancy of the speed of light), Einstein did the same thing in his general relativity— he postulated the failure of experiments to detect (in the situation depicted in Fig. 3.3) which lift is accelerating and which is on the Earth’s surface as his equivalence principle.

.

experimental fact that all bodies, regardless of their masses, fall with the same acceleration in the Earth’s gravitational field—as bodies of different mass appear to fall toward . A’s floor with the same acceleration (because in reality it is the floor that accelerates and approaches them), by the equivalence principle similar bodies should fall with the same acceleration in . B as well. 5 Here again the same restrictions apply (as those imposed by Einstein in his thought experiments)— the dimensions of the lifts should be small enough in order that the gravitational field in . B is homogeneous (and the trajectories of balls falling in . B are parallel).

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3 Origin and Nature of Inertia in Spacetime Physics

Instead of following Einstein’s line of thought I will follow Minkowski’s program of geometrizing physics. As we saw in Fig. 3.1 the ball only appears to accelerate in lift . A; it is . A that accelerates in reality,6 whereas the ball (before hitting the floor) is not subject to an acceleration, i.e., its worldtube is not deformed as seen in Fig. 3.2. The situation in . B is indeed identical to that in . A—the ball in . B only appears to accelerate. In reality it is not subject to an absolute acceleration because it does not resist its fall—this is an experimental fact7 (that is why “in reality”!). Although in spacetime physics it is completely clear that no absolute acceleration is involved in the fall of the ball (which explains why the ball does not resist its fall) and that the ball’s acceleration is relative (caused by geodesic deviation) as explained in the previous chapter, this point still appears to cause confusion. I think the best way to have a full understanding of what is happening in the lifts . A and . B is to understand what is happening in reality, i.e., in spacetime. The blue worldlines of . A’s and . B’s floors and the brown worldtubes of the two balls are depicted in Fig. 3.4. The left-hand part of the figure shows what happens in . A (which was already pictured in Fig. 3.2)—the worldtube of the ball is straight (not deformed) outside of the red circle, which means that the ball is not accelerating; “when” (in a static sense) the ball hits the floor (red circle) its worldtube is deformed and statically resists its deformation by exerting the inertial force .Fin on the floor. The right-hand side of Fig. 3.4 shows that virtually the same things happen in . B— the worldtube of the ball is geodesic (not deformed) outside of the red circle, which means that the ball falling in . B is not absolutely accelerating (because its absolute, μ curved-spacetime, acceleration is zero: .d 2 x μ /dτ 2 + Γαβ (d x α /dτ )(d x β /dτ ) = 0); “when” (in a static sense) the ball hits the floor (red circle) its worldtube is deformed and statically resists its deformation by exerting the inertial force.Fg on the floor. That .Fg is indeed an inertial force is clearly seen in Fig. 3.4, but this fact also obviously follows from the experimental fact that the falling ball does not resist its fall—as it does not resist its motion, it moves by inertia and when the floor opposes its inertial motion, the ball resists its deceleration by exerting the inertial force .Fg on the floor. Traditionally, .Fg (the weight of the ball) has been called gravitational force (the instant when the ball hit the floor is neglected exactly like in . A) and as Rindler [6] put 6

Two facts demonstrate that there is no relativity here:

• accelerated motion is absolute because it is experimentally detectable through the resistance with which a body opposes its acceleration—the ball in . A does not resist its “fall” in . A; it is the lift . A that resists its acceleration • accelerated motion is absolute because it is represented by an absolute geometrical feature—the curvature (i.e., the deformation) of the worldline of the accelerated body; the worldtube of the ball in . A is not deformed. 7

Even if it were not an experimental fact, Einstein’s thought experiments would have revealed that there would be full equivalence between the lifts. A and. B only if the ball falling in. B does not resist its fall exactly as the ball in . A does not resist its apparent fall; even an experiment could have been performed to test that prediction of the proposed by Einstein equivalence principle by attaching an accelerometer to the ball falling in . B (which measures an eventual resistance a body offers to its motion).

3 Origin and Nature of Inertia in Spacetime Physics

39

Fig. 3.4 What is happening in. A is given in the left-hand part of the figure—it was already depicted in Fig. 3.2. The right-hand part of the figure shows what is happening in. B—the worldtube of the ball is not deformed (it is geodesic) before it hits the floor (in ordinary three-dimensional language, the falling ball moves by inertia), which explains why the ball does not resist its fall; when the ball hits the floor (the red circle) it is prevented from moving by inertia and it exerts an inertial force .Fg on the floor, which (after the moment of the collision) has been traditionally called gravitational force (the weight of the ball). Like in . A this real inertial force appears to originate from the deformation of the ball’s worldtube

it “ironically, instead of explaining inertial forces as gravitational .. . . in the spirit of Mach, Einstein explained gravitational forces as inertial,” which naturally explains why “there is no such thing as the force of gravity” [7]. Figure 3.4 provides full explanation of Einstein’s equivalence principle—why the balls falling in . A and . B only appear to accelerate, why the inertial force .Fin is equal to the gravitational force .Fg (because .Fg turned out to be inertial too) and why the inertial mass of the ball .m in in . A is equal to the passive gravitational mass .m g of the ball in . B (because both masses turned out to be inertial—.m in and .m g are the measures of the resistance the balls in . A and in . B offer to their equal accelerations .a = g, i.e., decelerations, in the spacetime regions marked by the red circles in the figure). I believe it is now completely clear: the experimental fact of the equivalence of inertial mass and passive gravitational mass constitutes experimental proof that the resistance a body offers to its acceleration (the second aspect of inertia) does not depend on the properties of spacetime—as seen in Fig. 3.4 (and as mass is the measure of the resistance a body offers to its acceleration) .m in can be measured in . A (in flat spacetime) by measuring .Fin , whereas .m g can be measured in . B (in curved spacetime) by measuring the same inertial force .Fg . As .Fin = Fg and .a = g, therefore .m in = m g . In other words, the worldtube of an absolutely accelerated body offers the same resistance in flat and curved spacetime as proved by the equivalence .Fin = Fg and the equivalence of inertial mass and passive gravitational mass; so that resistance originates in the body itself (in its worldtube) and is not determined by the properties of spacetime.

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3 Origin and Nature of Inertia in Spacetime Physics

Employing Minkowski’s program of geometrizing physics leads to another clarification, which is essential for the issue of whether gravitational phenomena are caused by gravitational interaction or are merely manifestations of the non-Euclidean geometry of spacetime. As .Fin and .Fg are real inertial forces, it is obvious that they do work and the deformations on the floors of . A and . B is caused by these forces, which means that it is the inertial energy of the balls that transforms into deformation energy; so far this energy has been called kinetic but, despite that it is related to the motion of a body, it has not been explicitly stated that the true physical meaning of the origin of that energy is the body’s inertia. The qualitative argument that kinetic energy is actually inertial energy has a straightforward quantitative counterpart. That inertial energy—the work done by inertial forces—is equal to kinetic energy is easily demonstrated by an example depicted in Fig. 3.5. At moment .t = t1 a ball travels at constant “initial” velocity .vi towards a huge block of some plastic material; we can imagine that the block is mounted on the steep slope of a mountain. The ball hits the block, deforms it and is sharply decelerated. At moment .t = t2 the block stops the ball, that is, the ball’s final velocity at .t2 is .v f = 0 (the block’s mass is effectively equal to the Earth’s mass, which ensures that .v f = 0). According to the standard explanation it is the ball’s kinetic energy . E k = (1/2)mvi2 which transforms into deformation energy. But a proper physical explanation demonstrates that the energy of the ball, which is transformed into deformation

Fig. 3.5 A massive plastic block is deformed when hit by a ball moving by inertia. Traditionally, it is stated that the ball’s kinetic energy converts into a deformation energy. However, the deep physical explanation is that the ball’s energy is inertial energy since the deformation is caused by the work done by the real inertial force with which the ball resists its deceleration (i.e., its being prevented from moving by inertia); as we saw above that inertial force is a manifestation of the static resistance with which the ball’s worldtube opposes its deformation by the worldtube of the plastic block

3 Origin and Nature of Inertia in Spacetime Physics

41

energy, is its inertial energy . E i , because the ball resists its deceleration .ax∗ and it is the work .W = Fin Δx (equal to . E i ) done by the inertial force . Fin = max∗ that is responsible for the deformation of the plastic material. Using the relation between.vi ,.v f ,.ax∗ and the distance.Δx in the case of deceleration v2 = vi2 − 2ax∗ Δx

. f

and taking into account that .v f = 0 we find a∗ =

. x

vi2 . 2Δx

Then for the ball’s inertial energy . E i we have .

E i = W = Fin Δx = max∗ Δx =

1 2 mv . 2 i

Therefore the inertial energy of the ball is indeed equal to what has been descriptively (lacking physical depth) called kinetic energy. By the same reasoning it becomes evident (Fig. 3.6) that what has been traditionally called kinetic energy of a ball falling toward the Earth’s surface is indeed inertial energy by nature. The same direct calculations show that the falling ball’s energy that transforms into deformation energy is in fact inertial (not kinetic) energy—equal to

Fig. 3.6 A massive plastic block is deformed when hit by a falling ball which moves by inertia (its worldtube is geodesic). Traditionally, it is stated that what converts into a deformation energy is the ball’s kinetic energy. However, the deep physical explanation is that the falling ball’s energy is inertial energy since the deformation is caused by the work done by the real inertial force with which the ball resists its deceleration (i.e., the deformation of its geodesic worldtube)

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3 Origin and Nature of Inertia in Spacetime Physics

the work done by the inertial force with which the ball acts on the plastic block that prevents it from moving by inertia (i.e., prevents it from falling): .

E i = W = Fin Δz = maz∗ Δz =

1 2 mv . 2 i

Summary Newton deduced two aspects of inertia from the experimental evidence at his time: • a body moves uniformly on its own (Newton’s first law) • when “compelled to change its state by forces impressed” a body that is moving uniformly on its own resists the change of its state. In the previous chapter we saw how Minkowski’s program of geometrizing physics provided full explanation of inertial motion (motion without a mover). In this chapter we continued to employ Minkowski’s program which (particularly his arguments that the timelike worldtubes of three-dimensional bodies are real fourdimensional objects) led us to the natural conjecture that the resistance a particle offers to its acceleration originates from the static resistance its worldtube offers to its deformation (that static resistance can be traced down to the self-forces acting on accelerated elementary constituents of the body which have contributions from electromagnetic, weak and strong interactions). This conjecture immediately explains that the force of weight (traditionally regarded as gravitational force) is in spacetime physics an inertial force (making it exceedingly clear why there is no gravitational force in general relativity)—the worldtube of a particle at rest on the Earth’s surface is deformed and resists its deformation (i.e., as the falling particle’s worldtube is geodesic the particle moves by inertia (non-resistantly) and when it hits the ground the particle resits its being prevented from moving by inertia and an inertial force acts on the ground)—that resistance manifests itself as the force of weight of the particle. That conjecture also automatically explains the equivalence of inertial mass and (passive) gravitational mass: they are the same thing—inertial mass: an accelerating particle resists its acceleration (i.e., its worldtube resists its deformation) and the measure of this resistance is the particle’s inertial mass; the worldtube of a particle on the Earth’s surface is deformed and resists its deformation, i.e., the particle is prevented from moving by inertia (prevented from falling) and the measure of that resistance is the particle’s inertial mass (in curved spacetime), misleadingly called in the past (passive) gravitational mass. Minkowski’s program of geometrizing physics also led to a clarification that is essential for the proper understanding of gravitational phenomena in spacetime physics: what so far has been called kinetic energy of a body is in reality inertial energy (the work done by inertial forces) because the true physical meaning of the origin of that energy is the body’s inertia.

References

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Another result of employing Minkowski’s program of geometrizing physics is not only fully explaining the physical meaning of the principle of inertia (i.e., of inertial motion as shown in the previous chapter) but also freeing its understanding from any Machian references to distant masses.

References 1. V. Petkov, Relativity and the Nature of Spacetime, 2nd edn. (Springer, Heidelberg, 2009) 2. G. Galileo, Dialogue Concerning the Two Chief World Systems—Ptolemaic and Copernican, 2nd edn. (University of California Press, Berkeley, 1967), The Second Day 3. A. Einstein, The Meaning of Relativity: Four Lectures Delivered at Princeton University, May, 1921, in Relativity: Meaning and Consequences for Modern Physics and for our Understanding of the World, ed. by A. Einstein. With an Introduction by Tian Y. Cao. Edited by Vesselin Petkov (Minkowski Institute Press, Montreal, 2021) 4. I. Newton, The Principia: Mathematical Principles of Natural Philosophy. A New Translation by I. Bernard Cohen and A. Whitman assisted by Julia Budenz (University of California Press, Berkeley, 1999) 5. H. Minkowski, Space and time, new translation, in Spacetime: Minkowski’s Papers on Spacetime Physics, ed. by H. Minkowski. Translated by Gregorie Dupuis-Mc Donald, Fritz Lewertoff and Vesselin Petkov. Edited by V. Petkov (Minkowski Institute Press, Montreal, 2020) 6. W. Rindler, Essential Relativity: Special, General, and Cosmological (Springer, Heidelberg, 1997), p.244 7. J.L. Synge, Relativity: The General Theory (Nord-Holand, Amsterdam, 1960), p.109

Chapter 4

Relativistic Mass

The theory of spectra however gives us today, with the fine structure of the hydrogen lines, a much more accurate means for determining the velocity dependence of the electron mass. This leads to a complete confirmation of the relativistic formula, which can thus be considered as experimentally verified. Wolfgang Pauli [1]

I think the present status of relativistic mass in spacetime physics should not be silently tolerated. On the one hand, the physics community is divided1 —some over-confident colleagues (mostly particle physicists) firmly reject the concept of relativistic mass (e.g., in papers entitled “The Virus of Relativistic Mass in the Year of Physics” [15]), whereas what appears to be the majority of physicists continue to regard it as an integral part of spacetime physics2 including even in introductory textbooks and books published in the last several years (e.g., [17–26]). On the other hand, both mass and relativistic mass are equally supported by the experimental evidence—since mass is defined as the measure of the resistance a particle offers to its acceleration (which is the accepted definition based on the experimental evidence) and since it is also an experimental fact that a particle’s resistance to its acceleration increases as the particle’s velocity increases, it does

1

Despite the discussions in the American Journal of Physics [2–5], Physics Today [6–9] and the The Physics Teacher [10–14]. 2 Despite that during the last three decades physicists have witnessed (or rather endured), as Max Jammer put it, “what has probably been the most vigorous campaign ever waged against the concept of relativistic mass” [16]. For the controversy over the concept of relativistic mass in recent years see [16] and the Editor’s Appendix “On Relativistic Mass” in [27]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8_4

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follow that the particle’s mass increases when its velocity increases.3 Therefore the concept of relativistic mass also reflects an experimental fact—the increasing resistance a particle offers when accelerated to velocities close to that of light. At the turn of the 20th century the classical electron theory had already predicted that the electromagnetic mass of the electron increases as its velocity increases before the theory of relativity. In 1905 Einstein [28] derived two velocity-dependent masses of the electron m Longitudinal mass = (√ )3 1 − v2 /c2 .

Transverse mass

m =√ 1 − v2 /c2

and remarked “that these results as to the mass are also valid for ponderable material points” [28]. After the publication of this paper Einstein never explicitly stated what he thought of the relativistic (velocity-dependent) mass. Hardly in a 1948 letter to Lincoln Barnett (quoted in [2]) he commented on relativistic mass: It is not proper to speak of the mass . M = m/(1 − v 2 /c2 )1/2 of a moving body, because no clear definition can be given for . M. It is preferable to restrict oneself to the “rest mass” .m. Besides, one may well use the expression for momentum and energy when referring to the inertial behavior of rapidly moving bodies.

As the quote in [2] is not an exact translation (and appears to suggest that Einstein was explicitly against the concept of relativistic mass . M), the translation given here is by Ruschin [9] who pointed out the potentially misleading translation. A scan of Einstein’s letter in German is included in [6]. The quote from Einstein’s letter was given here for two reasons—(i) to show that Einstein had reservations about the concept of relativistic mass, and (ii) to state the fact that his reservations had been used as an argument against the use of relativistic mass (see, for example, [6]). As the ultimate judge in physics is the experimental evidence, quoting an authority (even as an additional argument4 ) rather suggests that the arguments against relativistic mass cannot survive a closer scrutiny. 3 It cannot be stated that it is sufficient to say it is the particle’s energy that increases with its velocity, because the crucial experimental fact is the increasing resistance the particle offers to its acceleration and the measure of this resistance is the particle’s mass. It is this resistance, which ensures that a particle cannot be accelerated to or greater than the velocity of light, because the particle’s resistance (i.e., its relativistic mass) approaches infinity as its velocity approaches the velocity of light. 4 It is clear that the quote from Einstein does not constitute an argument:

• Einstein himself did not give any argument against relativistic mass (obviously, “because no clear definition can be given for . M” is not an argument); he only expressed his reservations without providing any adequate justification • the verdict of the ultimate judge cannot be overruled by an authority; moreover, Einstein himself wrote “To punish me for my contempt for authority, fate made me an authority myself” [30].

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What constitutes an unprecedented situation in physics is that the rejection of the concept of relativistic mass is based on open rejections of • the accepted definition of mass (the measure of the resistance a body offers to its acceleration) without specifying what definition of mass is used • the overwhelming experimental evidence of the increased resistance a particle offers to its acceleration when its velocity approaches .c. In his book Einstein’s Theory of Relativity Max Born explicitly warned about the danger of improper understanding of mass in relativity [31]: In ordinary language the word mass denotes something like amount of substance or quantity of matter, these concepts themselves being defined no further... In physics, however, as we must very strongly emphasize, the word mass has no meaning other than... the measure of the resistance of a body to changes of velocity.

Let me stress it again— the accepted definition of mass in physics, reminded by Born, reflects both the pre-relativistic experimental fact that a particle resists its acceleration5 and the relativistic experimental fact that a particle offers an increasing resistance to its acceleration as its velocity approaches .c.6 What makes the concept of relativistic mass an integral part of spacetime physics is that it is this increasing resistance (i.e., its increasing mass) that prevents a particle from reaching the velocity of light—a particle’s mass will increase and will approach infinity when the particle’s velocity approaches .c, thus preventing it from reaching .c. The rejection of the overwhelming experimental evidence of the increased resistance a particle offers to its acceleration when its velocity approaches .c is no less inexplicable. It seems the outright rejection of the concept of relativistic mass have prevented those who opposed this concept from realizing the profound role of relativistic mass in spacetime physics and properly interpreting the experimental evidence proving the velocity dependent mass. Perhaps the best summary of both the role of relativistic mass in spacetime physics and that experimental evidence was given by Feynman [32, p. 15–9]7 : What happens if a constant force acts on a body for a long time? In Newtonian mechanics the body keeps picking up speed until it goes faster than light. But this is impossible in relativistic mechanics. In relativity, the body keeps picking up, not speed, but momentum, which can continually increase because the mass is increasing. After a while there is practically no acceleration in the sense of a change of velocity, but the momentum continues to increase.

5

The measure of that resistance is the particle’s mass. The measure of that increasing resistance is the particle’s relativistic (velocity dependent) mass. So, Einstein was incorrect when he wrote “no clear definition can be given for . M,” i.e., for the relativistic mass. 7 As seen from his clear explanation one may reasonably wonder whether the campaign against relativistic mass would have been even launched, if Feynman had been alive at that time. 6

48

4 Relativistic Mass Of course, whenever a force produces very little change in the velocity of a body, we say that the body has a great deal of inertia, and that is exactly what our formula for relativistic mass says (see Eq. 15.108 )—it says that the inertia is very great when .v is nearly as great as .c.

Feynman did not say it explicitly, but his explanation is clear—the physical meaning of “the body has a great deal of inertia” is “the body offers a great deal of resistance to its acceleration;” therefore the body has increasing mass that does not allow it to move as fast as light. This increase of mass with velocity has been repeatedly confirmed by experiments designed to test Einstein’s prediction and continuously (on a daily basis) confirmed by particle accelerators. Feynman specifically discussed the experimental confirmation of the velocity dependence of mass by particle accelerators: To deflect the high-speed electrons in the synchrotron that is used here at Caltech, we need a magnetic field that is 2000 times stronger than would be expected on the basis of Newton’s laws. In other words, the mass of the electrons in the synchrotron is 2000 times as great as their normal mass, and is as great as that of a proton!

Again, Feynman did not state it explicitly, but his explanation is clear—a 2000 times stronger magnetic field is needed because high-speed electrons offer 2000 times greater resistance to their acceleration than slowly moving electrons; that is why the mass of high-speed electrons is 2000 times greater than the mass of slowly moving electrons. As particle accelerators unambiguously demonstrate that the concept of relativistic mass reflects an experimental fact, the rejection of this concept amounts to rejection of the experimental evidence. Even before the accelerators, the early experiments to test the prediction of Einstein’s 1905 paper that the mass of a particle depends on its velocity conclusively confirmed it. The velocity dependence of the electron mass was confirmed in 1908 by Bucherer [33] and in 1916 by Guye and Lavanchy [34]. The proton relativistic mass variation was confirmed in 1958 [35]. Bucherer measured the ratio of charge to mass (.e/m) for .β-ray electrons and showed that at high velocities, comparable to the velocity of light, the masses of the electrons depended on their velocities. This experiment allowed only two interpretations—that either .e or .m varies in the ratio .e/m—and independent experiments (see [36]) ruled out the interpretation that the electron charge decreases as its velocity increases. Therefore, the Bucherer experiment would be impossible if the mass of electrons did not increase as their velocities increase. That is why, rejection of the relativistic increase of the electron mass means rejection of Bucherer’s experimental result. The crushing experimental evidence that mass increases with velocity makes the rejection of this experimental fact truly inexplicable. Moreover, those who reject the concept of relativistic mass made, effectively, only two somewhat relevant objections against this concept. Despite that this chapter should end here, let us examine 8

√ Equation 15.10 reads .p = mv = m 0 v/ 1 − v 2 /c2 .

4 Relativistic Mass

49

these objections to demonstrate that they are indeed untenable (without referring to the ruling of the ultimate judge—that the concept of relativistic mass reflects an experimental fact). The double argument of Taylor and Wheeler [37, pp. 250–251] goes as follows: The concept of ‘relativistic mass’ is subject to misunderstanding [.. . .]. First, it applies the name mass—belonging to the magnitude of a 4-vector—to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself.

It is true that the magnitude of the four-momentum is proportional to the rest (proper) mass .m 0 , whereas the four-momentum’s time component is proportional to the relativistic mass .m. But, the situation is exactly the same with respect to proper and coordinate time—the magnitude of the displacement four-vector.Δx (connecting two events on a timelike worldline) is proportional to the proper time .Δτ , whereas the coordinate time .Δt is the time component of the four-vector .Δx. So, if we cannot talk about relativistic mass, by the same argument we should talk only about proper time, which is an invariant, and deny the name ‘time’ to the coordinate time, which is frame-dependent (thus denying the relativistic time dilation because it is the coordinate time that “dilates,” i.e., that changes relativistically like the relativistic mass which also changes relativistically). The second part of the argument—“the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself”—directly contradicts the experimental fact of the equivalence of inertial mass and passive gravitational mass as we saw in the previous chapter—the worldtube of an absolutely accelerated body offers the same resistance in both flat and curved spacetime as proved by the equivalence of inertial mass and passive gravitational mass; so that resistance originates in the body itself (in its worldtube) and is not determined by “the geometric properties of spacetime√itself.” Some authors point out that.γ = 1/ 1 − β 2 should not be “attached” to the mass, because it comes from the 4-velocity. That it comes from the 4-velocity is, of course, correct—.γ ensures that the velocity of a particle cannot exceed that of light; in other words, .γ ensures that no 4-velocity vector, which is timelike, can become lightlike or spacelike. But that is kinematics; it says nothing about dynamics. That is, it says nothing about why a particle cannot exceed the velocity of light; in other words, it does not even ask the question of what the mechanism that prevents it from doing so is. That mechanism is suggested by Newtonian mechanics, where (to repeat) mass is defined as the measure of the resistance a particle offers to its acceleration; when Einstein postulated that the velocity of light .c is a limiting velocity, which no particle (with non-zero rest mass) can achieve, it was almost self-evident to assume that a particle would offer an increasing resistance when accelerated to velocities approaching that of light, that is, a particle’s mass will increase and will approach infinity when the particle’s velocity approaches .c. And (i) Einstein’s 1905 special relativity predicted it [28] (that a particle would offer an increasing resistance when

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accelerated to velocities approaching that of light) and (ii) that increasing resistance has been repeatedly confirmed experimentally. There exists indeed a serious difficulty involving the relativistic mass .m—the relativistic generalizations of kinetic energy and Newton’s second law cannot be obtained by merely replacing the classical (Newtonian) mass .m N with .m in the classical expressions. Moreover, in the general case, the relativistic force acting on a particle is not parallel to its acceleration and it also appears that the relativistic mass behaves as a tensor9 because a particle’s resistance to its acceleration is different in different directions; it is greatest along the particle’s velocity (serving as the mechanism that prevents a particle’s velocity from exceeding that of light). However, increased resistance (i.e., increased relativistic mass) is rather only naming the mechanism that prevents a particle from reaching the velocity of light. The origin and nature of the resistance a particle offers when accelerated (an open question in classical physics) and of the increased resistance a particle offers when accelerated to velocities approaching that of light (an open question in spacetime physics) constitute one of the deepest open questions in spacetime physics.

Summary For about three decades physicists have endured, as Max Jammer put it “what has probably been the most vigorous campaign ever waged against the concept of relativistic mass” [16]. It remains a mystery that most of those who waged the vigorous campaign against the concept of relativistic mass were particle physicists given that it was mostly particle accelerators which determined that the resistance particles offer to their acceleration increases as their velocity increases and therefore proved that relativistic mass is an experimental fact. That is why, it is exceedingly clear that the correct definition of mass in physics, as strongly emphasized by Born, is the definition that has been accepted since Newton—mass is the measure of the resistance of a particle to its acceleration. Indeed, both mass (rest mass) and relativistic mass of a particle are defined as the measure of the resistance the particle offers to its acceleration—in classical physics, mass is the measure of the resistance of a particle to its acceleration, whereas, in spacetime physics, mass is the measure of the increasing resistance of a particle to its acceleration when its velocity approaches .c. The campaign against the concept of relativistic mass is doubly unfortunate: • De facto, it rejects the accepted definition of mass (without offering another one) and also incomprehensibly rejects the overwhelming experimental evidence that particles offer an increasing resistance to their acceleration when their velocity approaches .c.

9

An attempt to address this fact was already made by E. B. Rockower in 1987 [38].

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• Most importantly, it tries to make it impossible even to approach one of the deepest open questions in fundamental physics—the origin and nature of the resistance a particle offers when accelerated (an open question in classical physics) and of the increasing resistance a particle offers when accelerated to velocities approaching .c (an open question in spacetime physics).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

W. Pauli, Theory of Relativity (Pergamon Press, New York, 1958), p.83 C.G. Adler, Does mass really depend on velocity, dad? Am. J. Phys. 55, 739 (1987) T.R. Sandin, In defense of relativistic mass. Am. J. Phys. 59, 1032 (1991) L.B. Okun, Mass versus relativistic and rest masses. Am. J. Phys. 77, 430 (2009) E. Hecht, Einstein on mass and energy. Am. J. Phys. 77, 799 (2009) L.B. Okun, The concept of mass. Phys. Today 42, 31 (1989) L.B. Okun, Putting to rest mass misconceptions. Phys. Today 43, 15, 115, 117 (1990) W. Rindler, Putting to rest mass misconceptions. Phys. Today 13 (1990) S. Ruschin, Putting to rest mass misconceptions. Phys. Today 15 (1990) E. Hecht, There is no really good definition of mass. Phys. Teach. 44, 40 (2006) E. Hecht, Einstein never approved of relativistic mass. Phys. Teach. 47, 336 (2009) A. Hobson, The definition of mass. Phys. Teach. 48, 4 (2010) E. Hecht, On defining mass. Phys. Teach. 49, 40 (2011) R.L. Coelho, On the definition of mass in mechanics: why is it so difficult? Phys. Teach. 50, 304 (2012) L.B. Okun, The virus of relativistic mass in the year of physics, in Gribov Memorial Volume: Quarks, Hadrons, and Strong Interactions, Proceedings of the Memorial Workshop Devoted to the 75th Birthday of V. N. Gribov (World Scientific Publishing, Singapore, 2009), pp. 470–473 M. Jammer, Concepts of Mass in Contemporary Physics and Philosophy (Princeton University Press, Princeton, 2000), p.51 H.D. Young, R.A. Freedman, Sears and Zemansky’s University Physics with Modern Physics, 15th edn. (Pearson Education Limited, Harlow, 2020) Ø. Grøn, Introduction to Einstein’s Theory of Relativity: From Newton’s Attractive Gravity to the Repulsive Gravity of Vacuum Energy, 2nd end. (Springer, 2020) W.-G. Boskoff, S. Capozziello, A Mathematical Journey to Relativity: Deriving Special and General Relativity with Basic Mathematics (Springer, 2020) A. Romano, M.M. Furnari, The Physical and Mathematical Foundations of the Theory of Relativity: A Critical Analysis (Birkhuser, 2019) H. Günther, V. Mller, The Special Theory of Relativity: Einstein’s World in New Axiomatics (Springer, 2019) A.K. Raychaudhuri, Classical Theory of Electricity and Magnetism: A Course of Lectures (Springer, 2022) G. Kunstatter, S. Das, A First Course on Symmetry, Special Relativity and Quantum Mechanics, 2nd edn. (Springer, 2022) F. Rahaman, The Special Theory of Relativity: A Mathematical Approach, 2nd edn. (Springer, 2022) Y. Shapira, Linear Algebra and Group Theory for Physicists and Engineers, 2nd edn. (Springer, 2023) R. Oloff, The Geometry of Spacetime: A Mathematical Introduction to Relativity Theory (Springer, 2023) A. Einstein, Relativity, ed. by V. Petkov (Minkowski Institute Press, Montreal, 2018)

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28. A. Einstein, On the Electrodynamics of Moving Bodies. New publication of the English translation in: [29, p. 99] 29. The Origin of Spacetime Physics, Foreword by A. Ashtekar. Edited by V. Petkov. (Minkowski Institute Press, Montreal, 2020) 30. B. Hoffman, Albert Einstein: Creator and Rebel (Viking, New York, 1972), p.24 31. M. Born, Einstein’s Theory of Relativity (Dover Publications, New York, 1965), p.33 32. The Feynman Lectures on Physics. The New Millennium, vol. 1. (Basic Books, New York, 2010) 33. A.H. Bucherer, Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der LorentzEinsteinschen Theorie. (Measurements of Becquerel rays. The Experimental Confirmation of the Lorentz-Einstein Theory). Physikalische Zeitschrift 9(22), 755762 (1908) 34. C.E. Guye, C. Lavanchy, Vérification expérimentale de la formule de Lorentz-Einstein. Arch. Sc. Phys. et Nat. 42, 286–299, 353–373, 441–448 (1916) 35. V.P. Zrelov, A.A. Tiapkin, P.S. Farago, Measurements of the mass of 660 Mev protons. Sov. Phys. JETP 34(7, 3), 384–387 (1958) 36. W.G.V. Rosser, Relativity and High Energy Physics (Wykeham Publications, London, 1969), p.15 37. E.F. Taylor, J.A. Wheeler, Spacetime Physics: Introduction to Special Relativity, 2nd edn. (Freeman, New York, 1992) 38. E.B. Rockower, A relativistic mass tensor with geometric interpretation. Am. J. Phys. 55, 70 (1987)

Chapter 5

Gravitation

Gravitation as a separate agency becomes unnecessary Arthur S. Eddington [1] An electromagnetic field is a “thing;” gravitational field is not, Einstein’s theory having shown that it is nothing more than the manifestation of the metric Arthur S. Eddington [2]

The situation in gravitational physics is truly unique. On the one hand, general relativity, which identified gravitational phenomena with the non-Euclidean geometry of spacetime, is considered to be the greatest intellectual achievement in fundamental physics together with Minkowski’s discovery of the spacetime structure of the world because without it general relativity, where gravity is curvature of spacetime, would not have been possible. On the other hand, upon closer examination, the accepted version of general relativity, which came from Einstein, resembles rather an incomplete version of “the greatest intellectual achievement in fundamental physics.” At the time when Einstein was approaching the completion of general relativity, it might have occurred to him, but perhaps it looked too revolutionary even to him to state explicitly that a full identification of gravity with the curvature of spacetime implies that there is no gravitational interaction in the physical world. He might have realized before Eddington that “Gravitation as a separate agency becomes unnecessary.” And he smuggled in the theory the concept of gravitational energy and momentum (which is not part of the theory itself) to force general relativity to treat gravitation as a physical interaction. Not only is the present situation in gravitational physics unique, but also Einstein’s path to general relativity is no less unique in the same sense. On the one hand, Mach’s sterile scientific philosophy (utter relativism) did what looked virtually impossible for a sterile doctrine—made an indirect contribution to physics by initially helping © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8_5

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Einstein to realize that relative acceleration had a place in general relativity—e.g., as we saw in Chap. 3, the acceleration of a falling body is apparent, that is, relative (as seen in Fig. 3.3 the acceleration of a ball in lift . B, which is on the Earth’s surface, is apparent or relative (not absolute1 ), exactly like the observed acceleration of a ball inside the accelerating lift . A is apparent). On the other hand, however, Einstein seems to have never fully adopted the main ideas of Minkowski’s spacetime physics. Einstein overcame his initial reaction to Minkowski’s ideas—“Since the mathematicians have invaded the relativity theory, I do not understand it myself any more” [3]—and later (in his revolutionary 1916 paper [4]) credited Minkowski for preparing the ground for general relativity2 : The generalization of the theory of relativity was greatly facilitated by the form given to special relativity by Minkowski, who was the first mathematician to clearly recognize the formal equivalence of the spatial coordinates and the time coordinate, which made it useful for the construction of the theory.

Even in 1916 (as seen in the quote) Einstein did not appear to have realized the depth of what Minkowski did because he regarded Minkowski’s die Welt (the world, i.e., spacetime) as a mathematical space (not as representing a four-dimensional physical world) by writing “the formal equivalence of the spatial coordinates and the time coordinate”3 (italics added). As a really sad result, he appears to have never been able to see clearly or perhaps to accept the truly revolutionary nature of his own general relativity. He even did not seem to believe or perhaps was not willing to accept that general relativity geometrized gravitation (probably because he had been aware that a full geometrization of gravitation implies that there is no gravitational interaction) as he wrote in a letter to Lincoln Barnett on June 19, 1948 [8]: I do not agree with the idea that the general theory of relativity is geometrizing Physics or the gravitational field.

Judging by this quote, it seems Einstein had chosen to look at the mathematical formalism of general relativity as pure mathematics (e.g., assuming that the Riemann 1 The distinction between relative and absolute acceleration in curved spacetime is illustrated in Fig. 2.4. 2 Strangely, the first page of Einstein’s 1916 paper, containing this recognition of Minkowski’s work, did not appear in the 1923 English translation The Principle of Relativity [5] which seems to have been a translation of the paper published in the 1923 German collection Das Relativitätsprinzip [7] (where the first page was also missing) without consulting the original publication of Einstein’s paper in Annalen der Physik. The missing page was translated by André Michaud and Fritz Lewertoff and is included in the new English publication of Einstein’s paper in The Origin of Spacetime Physics [6]. 3 Also, Einstein stated that Minkowski only reformulated his special relativity—“the form given to special relativity by Minkowski”—a view that is widely shared. However, there is enough evidence which indicates that Minkowski had arrived independently at the physics of flat spacetime, but Einstein and Poincaré published first while Minkowski had been obtaining the foundational elements of flat spacetime physics by developing its four-dimensional formalism. Minkowski had told Max Born that he “did not publish them because he wished first to work out the mathematical structure in all its splendour;” for more details see Appendix B.

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curvature tensor is nothing more than a mathematical description and does not represent real spacetime curvature) and to regard gravitation as a physical interaction involving exchange of gravitational energy and momentum. Not surprisingly, this is effectively the accepted interpretation of general relativity despite that there is no justification for inserting the concept of gravitational energy and momentum in the theory. Not only is there no justification for this concept, but three obvious and independent arguments against that concept (each of which, taken alone, is sufficient to rule it out) have been merely ignored (which itself is unprecedented in physics): • general relativity stubbornly refuses to yield a proper tensorial expression for gravitational energy and momentum which is a clear indication that that foreign concept in general relativity does not represent a real physical quantity • there is no gravitational field as a physical field whose energy is gravitational; at best gravitation can be regarded as a geometrical field, which, however, does not possess energy • the experimental fact that there is no gravitational force in Nature4 demonstrates that there is no gravitational energy either for the obvious reason—gravitational energy should be defined as the work done by gravitational forces. The reason for the present peculiar situation in gravitational physics was briefly discussed in Chap. 1—if the status of spacetime is not explicitly and firmly determined, inevitably there would be a lack of clarity of what gravitational phenomena actually are. Indeed, if spacetime did not represent a real four-dimensional world (and were nothing more than a mathematical space), then, clearly, gravitational phenomena could not be manifestations of the curvature of something that does not exist. I think this might explain why in 1948 Einstein wrote he disagreed that “the general theory of relativity is geometrizing Physics or the gravitational field”—perhaps even in 1948 Einstein had doubts about the reality of spacetime. But at least two comments written after 1948 appear to indicate that what Einstein wrote in that letter might not have been his final view of the true meaning of his general relativity. In 1952 Einstein added a fifth appendix Relativity and the Problem of Space to the fifteenth edition of his book Relativity: The Special and General Theory where his doubts that spacetime represents a real four-dimensional world seem to have been overcome [11]: It appears therefore more natural to think of physical reality as a four-dimensional existence, instead of, as hitherto, the evolution of a three-dimensional existence.

In March 1955 Einstein wrote a letter of condolences to the widow of his longtime friend Besso which provided another indication of what seems to be Einstein’s full realization of the depth of Minkowski’s discovery that reality is an absolute fourdimensional world [12]: 4

As Synge stressed it repeatedly “in relativity there is no such thing as the force of gravity” [3, p. 109]. The theoretical fact that general relativity does not contain the concept of gravitational force is based on the experimental fact that there is no such force—falling bodies do not resist their apparent acceleration, which proves that no force is causing their fall (we will return to this point below).

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5 Gravitation Now Besso has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.

Einstein’s changing views on the nature of spacetime may explain his changing views on the nature of general relativity. But, more than a hundred years since the publication of general relativity, what could explain the tacit toleration of its presents status—still an incomplete version of “the greatest intellectual achievement in fundamental physics”? The situation seems to have been even worsening in recent decades (making the present version even more incomplete), especially with the attempts to create a theory of quantum gravity—by trying to make general relativity more amenable to quantization the attempted reformulations drifted the understanding of gravitational phenomena away from their identification with the non-Euclidean geometry of spacetime; such an example is the Hamiltonian formulation of general relativity—a totally problematic approach to general relativity that should not have been even attempted if spacetime had been regarded as what it actually is—representing a real four-dimensional world. As a result of ignoring the arguments against the artificial introduction of the concept of gravitational energy and momentum in general relativity and further twisting it in order to try to quantize it, it becomes increasingly difficult to identify and clarify misconceptions and confusions such as the two examples discussed in the next chapter—(i) that bodies (modeled as point masses), whose worldlines are geodesic, emit gravitational waves, and (ii) that gravitational waves carry gravitational energy. By contrast, if spacetime is regarded as real, Einstein’s general relativity has, as we will see below, a natural explanation because, for example, the Riemann curvature tensor represents real spacetime curvature —gravitation is completely geometrized since it is fully explained as a manifestation of the curvature of spacetime without assuming the existence of gravitational interaction. This naturally explains why the mathematical formalism of general relativity stubbornly refuses to yield a tensorial expression for the gravitational energy and momentum—because there is no such thing as gravitational energy and momentum in the physical world. In turn, the nonexistence of gravitational interaction naturally explains the unsuccessful attempts to create a theory of quantum gravity5 —simply there is nothing to quantize.6 Then the actual open question in gravitational physics seems to be how matter curves 5

I think, so far, only loop quantum gravity (LQG) makes a relevant effort to unify general relativity (Einstein’s theory of gravitation) and quantum physics, because LQG is rigorously based on the mathematical formalism of general relativity (gravitation = spacetime geometry) without trying to modify it in order to become amenable to quantization. This is precisely what distinguishes LQG from the other approaches to create a theory of quantum gravity—LQG quantizes the (curved) spacetime itself (i.e., employs quantum Riemannian geometry), not gravitational interaction. Also, LQG gets rid of the Big Bang singularity and it seems also of the black hole singularities (see [14]). 6 I find it difficult to understand that the physicists working on quantum gravity seem to have never entertained the possibility that the reason for the unsuccessful attempts to create quantum gravity might be that they have been trying to quantize something that might not exist—gravitational interaction. That possibility might have looked to them not so heretical if they had recalled that Eddington explicitly stated that possibility in the distant 1921—“Gravitation as a separate agency becomes unnecessary” [1].

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spacetime (or perhaps, how matter is itself a manifestation of extreme curvature or other properties of spacetime), not how to quantize the apparent gravitational interaction [15]. We can now see how an analysis explicitly based on (i) viewing spacetime as real and (ii) employing Minkowski’s program of regarding physics as spacetime geometry naturally leads to a full geometrization of gravitation which involves no physical interaction. We can do this by imagining that Einstein had fully adopted Minkowski’s flat spacetime physics and his program of geometrizing physics. Then we will try to reconstruct his line of thought (mostly his analyses of thought experiments with lifts—accelerating and at rest on the Earth’s surface) to see where it would have led him. Most probably in November 1907 Einstein had been blessed with an insight that set him on the path toward his theory of general relativity (quoted from [13]): I was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me: “If a person falls freely he will not feel his own weight.” I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.

Einstein had been so impressed by this insight that he called it the “happiest thought” of his life [13]. Then Einstein needed eight years to arrive at his general relativity. Almost certainly during the first years Einstein had been examining physical phenomena caused by acceleration and by gravity; he had been doing that by analyzing thought experiments involving his famous lifts—one accelerating and one on the surface of the Earth (like the ones we discussed in Chap. 3, Fig. 3.3). These analyses helped him discover that there exists a complete equivalence between the phenomena observed in an accelerating lift . A and in a lift . B on the Earth’s surface (Fig. 5.1). For example, it is an (unexplained) experimental fact that different bodies fall with the same acceleration toward the Earth’s surface—like the ones in lift . B in Fig. 5.1 (a small ball and two larger balls one of which is hollow). If identical balls are in the accelerating lift . A, they appear to fall with the same acceleration toward the floor of

Fig. 5.1 Lift . A is accelerating with an acceleration .a = g, whereas lift . B is at rest on the Earth’s surface. There are two identical sets of three balls in each lift—a small ball and two larger balls one of which is hollow

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lift . A as seen by an observer there. But in this case their equal accelerations have a straightforward explanation—in reality it is lift . A that accelerates (as illustrated in Fig. 3.1), whereas the three balls only appear to accelerates; in reality they move by inertia (because they do not resist their motion). So, it is . A’s floor that approaches the three balls and they only appear to an observer in . A to fall toward the floor with the same acceleration. The equal fall of different bodies with the same acceleration as seen by observers in . A and in . B suggests an explanation of the experimental fact that all bodies fall toward the Earth’s surface with the same acceleration—while falling they move by inertia like the balls in . A. This suggested explanation contains the seeds of general relativity, because it implies that no gravitational force is responsible for the fall of the bodies in . B. And, indeed, as mentioned in Chap. 3, it is an experimental fact that falling bodies do not resist their apparent acceleration (e.g. a falling accelerometer measures zero resistance), which proves that no gravitational force is acting on them.7 In other words, this experimental fact demonstrates that falling bodies move by inertia (since they offer no resistance to their fall) confirming the suggested explanation (above) of the fact that all bodies fall toward the Earth’s surface with the same acceleration. It should be emphasized as strongly as possible that a gravitational force would be required to accelerate falling bodies downward if and only if the bodies resisted their acceleration, because only then a gravitational force would be needed to overcome that resistance. Now we can see clearly that the path to the idea that gravitational phenomena are manifestations of the curvature of spacetime would have been open to Einstein had he employed Minkowski’s program of regarding physics as geometry of the real spacetime—the experimental fact that a falling particle accelerates (which means that its worldtube is curved), but offers no resistance to its acceleration (which means that its worldtube is not deformed) can be explained only if the worldtube of the falling particle is both curved and not deformed, which is impossible in the flat Minkowski spacetime where a curved worldtube is always deformed. Such a curved undeformed worldtube can exist only in a non-Euclidean spacetime whose geodesic worldtubes are naturally curved due to the spacetime curvature, but are not deformed. After such a stunning insight Einstein might have seen clearly that all gravitational phenomena are fully explained as manifestations of the curvature of a non-Euclidean spacetime and therefore there is no room for gravitational interaction.8 Indeed, a timelike geodesic (i.e., undeformed) worldtube represents a body moving by inertia and is regarded in spacetime physics as “a natural generalization of Newton’s first law” [10, p. 110], that is, “a mere extension of Galileo’s law of inertia to curved spacetime” [16]. Therefore a body falling toward the Earth’s surface is not subject 7 I think, it would be helpful for the advancement of fundamental physics if this experimental proof that gravitational force does not exist be used to reject any research projects proposing modified theories of gravitation which regard gravity as a force without addressing the experimental fact that falling bodies do not resist their apparent acceleration. 8 After adopting Minkowski’s counter-intuitive (heretical) worldview that spacetime represents a real four-dimensional world, the possibility that there is no gravitational interaction might not have looked to Einstein so heretical any longer.

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to any interaction since it moves by inertia, whose very essence is interaction-free motion because such motion is non-resistant; the motion of a body that participates in interactions involves resistance (the body’s reaction to the action of other bodies). It is obvious why motion by inertia in flat spacetime is interaction-free—the worldtube of a body moving by inertia is straight, i.e., undeformed and the body’s motion is non-resistant; the fact that its worldtube is undeformed (meaning that the body does not accelerate as a result of some interactions) demonstrates that it is not subject to any interactions; that is why the body’s non-resistant motion by inertia is interaction-free. Exactly for the same reasons motion by inertia in curved spacetime is interactionfree—the worldtube of a body moving by inertia is geodesic, i.e., undeformed and the body’s motion is non-resistant; here again, the fact that its worldtube is undeformed demonstrates that it is not subject to any interactions; that is why the body’s nonresistant motion by inertia is interaction-free in curved spacetime as well. This means that no gravitational interaction is involved in the fall of a body toward the Earth’s surface. In the accepted version of general relativity it is implied that the gravitational interaction somehow “hides” in the natural curvature of the geodesic worldtube of a falling body, which is imparted by the curvature of spacetime caused by the Earth. Let us see whether we can find the gravitational interaction in the natural curvature of a geodesic worldtube. The reason a body falls toward the Earth’s surface with an apparent acceleration is that the body’s worldtube is curved, but not deformed, since it has the natural curvature of spacetime induced by the Earth’s mass (represented by the Earth’s stress-energy tensor .Tαβ ), and converges toward the Earth’s worldtube. So what is implied is that the Earth spends some gravitational energy and momentum to curve both spacetime around itself and the worldtubes in that spacetime region and the imparted natural curvature on the worldtube of the falling body constitutes gravitational interaction between the Earth and the body.9 To see that gravitational interaction cannot be found “hiding” in the natural curvature of a geodesic worldtube, let us see what the Einstein equation tells us: .

Rαβ −

8π G 1 R gαβ = 4 Tαβ , 2 c

where . Rαβ is the Ricci curvature tensor, . R is the scalar curvature, .gαβ is the metric tensor, .G is the Newtonian constant of gravitation, .c is the speed of light in vacuum and .Tαβ is the Earth’s stress-energy tensor (in our example). The only thing, relevant to our question, this equation tells us is that there is some link between the geometry of spacetime (represented in the left-hand part of the equation) and matter (represented on the right). General relativity does not reveal the nature of that link. There seem to exist two possible interpretations of this link— 9 I have never seen this stated explicitly, but it has been clearly implied and I do not see how this explanation can be avoided if it is maintained that gravitational interaction is mediated by the spacetime curvature (in general relativity itself, not in the attempts to create a theory of quantum gravity).

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either there exists a (still unknown) mechanism by which mass changes the geometry of spacetime or mass itself is a manifestation of some (still unknown) properties of spacetime. What this equation seems to show clearly, however, is that gravitational interaction is not “hiding” in the natural curvature of a geodesic worldtube, because the Earth curves spacetime around itself (around its worldtube) no matter whether or not the worldtube of a falling body is there; this equation does not provide even a hint that the Earth spends some additional energy and momentum to change the shape of a body’s worldtube from straight (in flat spacetime when the body is far away from the Earth) to naturally-curved when the body starts to fall toward the Earth’s surface. The geodesic worldtubes of different bodies (small and large) in the spacetime region around the Earth’s worldtube have the natural curvature, which spacetime itself has, and are not curved by the Earth. That is why in spacetime physics “a geodesic is particle independent” [16] (which reflects the experimental fact that particles of different masses fall toward the Earth with the same acceleration). It should be specifically stressed that the natural curvature of the body’s worldtube determined by the curved spacetime around the Earth’s worldtube does not constitute gravitational interaction between the Earth and the body because, to repeat it, the Earth curves spacetime around its worldtube no matter whether or not the body’s worldtube is there.10 After this crucial point in the physics of curved spacetime is clarified, all gravitational phenomena are fully explained as effects of the non-Euclidean geometry of spacetime: • Planets orbit the Sun because they move by inertia (non-resistantly) since their worldtubes are geodesic—they are helices around the worldtube of the Sun. • The ball in lift . B (on the Earth’s surface), shown in Fig. 5.2, falls toward the floor non-resistantly, i.e., by inertia, because its absolute acceleration is zero, which means that its worldtube is geodesic, i.e., undeformed; the ball’s fall (i.e. its apparent acceleration) is a manifestation of the curvature of spacetime around the Earth’s worldtube, causing the ball’s geodesic worldtube (which is not deformed but has the natural “curvature” due to the spacetime curvature in that spacetime region) to converge toward the Earth’s worldtube (as shown in Fig. 3.2). Therefore, it is virtually obvious that regarding spacetime as real provides a full (.100%) explanation of the previously unexplained experimental fact that different bodies fall with the same acceleration toward the Earth’s surface—their geodesic worldtubes have the same natural “curvature” caused by the spacetime curvature in the vicinity of the Earth’s worldtube, which manifests itself as the same apparent acceleration; in other words, the worldtubes of the different bodies converge in the same way toward the Earth’s worldtube, which is perceived in . B as falling with 10

For additional and detailed explanations, particularly the four reasons of why “the assumption that the planet’s mass curves spacetime, which in turn changes the shape of the geodesic worldline of a free particle, does not imply that the planet and the particle interact gravitationally” see Sect. 8.2.6 “Are Gravitational Phenomena Caused by Gravitational Interaction According to General Relativity?” in [15].

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Fig. 5.2 Minkowski’s program of regarding physics as geometry of the real spacetime provides a full (.100%) explanation of Einstein’s equivalence principle. The balls in . A and . B only appear to accelerate—in reality they move by inertia (non-resistantly), because their worldtubes are geodesic (not-deformed). The forces measured by the scales in . A and . B are the same because they are both inertial—they arise when the scales in . A and . B prevent the balls from moving by inertia; in reality (in spacetime) the worldlines of the scales in . A and . B statically deform the worldtubes of the balls and the worldtubes resist their static deformation by exerting the inertial forces .Fin = m in a and .Fin = m in g on the worldlines of the scales (and, in turn, deforming the scales’ springs). The inertial mass of the ball in . A is equal to the passive gravitational mass of the ball in . B because they are both inertial—the measure of the resistance the balls offer when they are prevented by the scales from moving by inertia

the same apparent acceleration. Following Minkowski’s approach also provides a complete explanation of that part of the equivalence principle dealing with the falling balls in . A and . B (Fig. 5.2)—the balls move by inertia, i.e., non-resistantly because their worldtubes are geodesic, i.e., undeformed (which means that the acceleration of the ball in . A and the absolute acceleration of the ball in . B are both zero); their apparent accelerations are manifestations of the convergence of the curved worldline of . A’s floor toward the geodesic worldline of the ball in . A and the convergence of the geodesic (but naturally “curved” due to the spacetime curvature) worldtube of the ball in . B toward the Earth’s worldtube (as illustrated in Fig. 3.4). • The ball’s “weight” measured by a scale on . B’s floor is not a gravitational force (as it has been traditionally called), but the inertial force .Fin = m in g, because the scale prevents the ball from moving by inertia (prevents it from falling) and the ball resists its absolute acceleration by exerting the inertial force .Fin on the scale (as a result the scale’s strings are deformed). In reality, i.e., in spacetime, the scale’s worldline statically deforms the ball’s worldtube (which manifests itself as the ball’s absolute acceleration) and it resists its static deformation. Again,

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following Minkowski’s approach provides a full explanation of the other part of the equivalence principle dealing with (i) the equality of the forces measured by the scales in . A and . B and (ii) the equality of the inertial mass of the ball in . A and the passive gravitational mass of the ball in . B—(i) both forces are equal because they are both inertial: the scales in . A and . B prevent the balls from moving by inertia and the balls resist by exerting inertial forces on the scales; as illustrated in Fig. 3.4 the worldtubes of the balls in . A and . B are statically deformed and this equal deformation gives rise to the same inertial forces; (ii) the reason for the equality of the inertial mass and the passive gravitational mass is that they are the same thing—they both are inertial: they are the measure of the resistance the balls offer when the scale prevents them from moving by inertia. • The energy involved in the deformation of the scale’s springs in . B is not gravitational, but inertial (as we saw in Chap. 3), because it is the work done by the inertial force .Fin = m in g. In . A the situation is identical—it is the inertial energy of the ball that transforms into deformation energy of the scale’s springs, because it is the work done by the inertial force .Fin = m in a. It should be stressed that in all cases in which one is tempted to talk about gravitational energy, what is involved there is inertial energy (the work done by inertial forces)—e.g., (i) as shown in [15] it is inertial energy which is transformed into electrical energy in tidal power stations; (ii) as we will see in the next chapter if energy is involved in the detection of gravitational waves (as in the sticky bead argument), it is also inertial.

Summary More than 100 years since the advent of the greatest intellectual achievement in fundamental physics—Einstein’s general relativity—it still looks like an incomplete version. However, following Minkowski’s program of regarding physics as geometry of the real spacetime, demonstrates that gravitational phenomena are not caused by gravitational interaction, but are indeed mere manifestations of the non-Euclidean geometry of spacetime (as general relativity itself demonstrates, without the artificially imposed interpretation that it describes gravitational interaction). As a “side result” of this explanation of the nature of gravitation, Minkowski’s program provides a full (.100%) explanation of Einstein’s equivalence principle. Unfortunately, gravitation is not yet .100% explained—the major open question in spacetime physics seems to be how matter curves spacetime or how some still unknown properties of spacetime manifest themselves as matter.

References

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References 1. A.S. Eddington, The relativity of time. Nature 106, 802–804 (17 February 1921); reprinted in: A.S. Eddington, The Theory of Relativity and its Influence on Scientific Thought: Selected Works on the Implications of Relativity (Minkowski Institute Press, Montreal, 2015), pp. 27–30 2. A.S. Eddington, The Mathematical Theory of Relativity (Minkowski Institute Press, Montreal, 2016), p.233 3. Albert Einstein cited in: A. Sommerfeld, To Albert Einstein’s seventieth birthday, in Albert Einstein: Philosopher-Scientist, ed. by P.A. Schilpp, 3rd ed. (Open Court, Illinois, 1969), pp. 99–105, 102 4. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 49 (1916). New publication of the original English translation ([5]) in [6] 5. A. Einstein, The foundation of the general theory of relativity, in The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, eds. by H.A. Lorentz, A. Einstein, H. Minkowski, H. Weyl. With Notes by A. Sommerfeld. Translated by W. Perrett and G. B. Jeffery (Methuen and Company, Ltd., 1923; reprinted by Dover Publications Inc., 1952) 6. The Origin of Spacetime Physics, 2nd ed. Foreword by A. Ashtekar. Edited by V. Petkov (Minkowski Institute Press, Montreal, 2023) 7. H.A. Lorentz, A. Einstein, H. Minkowski, Das Relativitätsprinzip: Eine Sammlung von Abhandlungen. Mit einem Beitrag von H. Weyl und Anmerkungen von A. Sommerfeld. Vorwort von O. Blumenthal, Fünfte Auflage (Springer Fachmedien Wiesbaden Gmbh, 1923) 8. A letter from Einstein to Lincoln Barnett from June 19 (1948); quoted in [9] 9. D. Lehmkuhl, Why Einstein did not believe that General Relativity geometrizes gravity. Stud. Hist. Phil. Phys. 46, 316–326 (2014) 10. J.L. Synge, Relativity: The General Theory (Nord-Holand, Amsterdam, 1960), p.109 11. A. Einstein, Relativity: The Special and General Theory, New Publication in the Collection of Five Works by Einstein: A (Relativity (Minkowski Institute Press, Montreal, Einstein, 2018), p.109 12. Albert Einstein cited in: Michele Besso, From Wikipedia, the free encyclopedia (http://en. wikipedia.org/wiki/Michele_Besso). Besso left this world on 15 March 1955; Einstein followed him on 18 April 1955 13. A. Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein (Oxford University Press, Oxford, 2005), p.179 14. A. Ashtekar, J. Olmedo, P. Singh, Regular black holes from loop quantum gravity, in Regular Black Holes: Towards a New Paradigm of Gravitational Collapse, ed. by C. Bambi (Springer, 2023), pp. 235–236 15. V. Petkov, Physics as spacetime geometry, in Springer Handbook of Spacetime, eds. by A. Ashtekar, V. Petkov (Springer, Heidelberg, 2014), Chapter 8, pp. 141–163 16. W. Rindler, Relativity: Special, General, and Cosmological (Oxford University Press, Oxford, 2001), p.178

Chapter 6

Gravitational Waves

This chapter and Chap. 7 deal with two fundamental concepts in spacetime physics— gravitational waves and black holes—that do not appear to be based firmly on both general relativity and the general-relativistic experimental evidence. Two of the problems with these concepts are seen even in the abstract of the paper which reported the first detection of gravitational waves [1]: On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. .. . . It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. .. . . These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

One of the reported results is “the first observation of a binary black hole merger.” The problem here is that the existence of black holes (for us as distant observers) is taken for granted (as has been in recent years). The presented in the report justification for the assumption that the compact objects that merged were black holes can hardly be regarded as a firm proof of the existence of black holes [1]: A pair of neutron stars, while compact, would not have the required mass, while a black hole neutron star binary with the deduced chirp mass would have a very large total mass, and would thus merge at much lower frequency. This leaves black holes as the only known objects compact enough to reach an orbital frequency of 75 Hz without contact.

And as we will see in the next chapter, regarding black holes as an accepted fact, given the lack of direct and unambiguous experimental proof, is, de facto, a result of using double standards in physics. The other problem in the abstract is the misunderstanding that the inspiraling members (if modeled as point masses) of a binary system emit gravitational waves. This misunderstanding is best seen in the paper itself [1]: The discovery of the binary pulsar system PSR B1913+16 by Hulse and Taylor and subsequent observations of its energy loss by Taylor and Weisberg demonstrated the existence of gravitational waves. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8_6

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This quote contains the two problems with the understanding of gravitational waves: • that a binary system, whose members are regarded as point masses, emit gravitational waves even before they collide; • that gravitational waves carry gravitational energy. The statement that the observations “by Taylor and Weisberg demonstrated the existence of gravitational waves” does not reflect the actual situation. First, the calculations used in the interpretation of those observations were not rigorously general-relativistic because the weak-field and the Post-Newtonian approximations are used (see, for example, [2]); one can even predict “Gravitational waves from orbiting binaries without general relativity” [3] which also implies that the predicted gravitational radiation (before the members of the binary system coalesce) is not a genuine general-relativistic effect.1 Even a book published this year (2023) has a chapter entitled “Gravitational Waves Sans General Theory of Relativity” whose abstract reads [4]: This chapter will examine how Newton’s static theory, made special relativistic, can predict gravitational waves. And in the Newtonian framework, we arrive at the quadrupole formula for gravitational waves.

Second, the binary system does not possess gravitational energy, because the members of the binary system (regarded as point masses and therefore represented by geodesic worldlines2 ) move by inertia; a “system” whose “members” move by inertia (i.e., interaction-free) does not possess gravitational (or any) energy (classically, the members of the binary system themselves, but not the “system,” possess kinetic energy). Moreover, as shown in Chap. 5 no such energy is present in the proper interpretation of general relativity itself; also, the experimental fact that there is no gravitational force in Nature—falling bodies do not resist their fall—demonstrates that there is no gravitational energy not only in general relativity, but in Nature as well (since no gravitational forces exist that can do work). The two problems with gravitational waves became common misunderstanding after the discovery of the binary pulsar system PSR 1913+16 by Hulse and Taylor in 1974 [5]. It was stated that the decrease of the orbital period of such binary systems is caused by the loss of energy due to gravitational waves emitted by the systems.

1

Gravitational radiation from a binary system is one of the examples when Wheeler’s first moral principle (mentioned on p. xiv)—“Never make a calculation until you know the answer”—could have been of enormous help; it would have shown that the reduction of the orbital period is not caused by a loss of orbital gravitational energy since it is merely an effect of the curvature of spacetime as discussed below. 2 As discussed below when the members of a binary system are regarded as spatially extended bodies weak gravitational waves are generated due to tidal effects in the compact bodies themselves, not due to their relative motion.

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Since then this has become the accepted interpretation, but it contains the double misunderstanding.3 As the stars in the binary pulsar system PSR 1913+16 had been “modelled dynamically as a pair of orbiting point masses” [6], they had been effectively regarded as geodesic worldlines (not worldtubes), which, however, implies three things that contradict the two main claims—that the inspiraling stars emit gravitational waves before they collide and that the gravitational waves carry gravitational energy: • Bodies whose worldlines are geodesic do not emit gravitational waves; such waves are produced only by bodies that are absolutely accelerating, i.e., by bodies whose worldlines are not geodesic; that is, by bodies whose worldlines are deformed (as the source of gravitational waves is a deformed worldline). The original prediction of gravitational wave emission, obtained by Einstein [7], correctly identified the source of such waves—absolute acceleration (i.e., deformed worldline). Or, in the proper spacetime language, gravitational waves are emitted only by bodies whose worldlines are not geodesic. Einstein considered a spinning rod, or any rotating material bound together by cohesive force. None of the particles of such rotating material (except the centre of rotation) are geodesic worldlines (because they are deformed by the cohesive forces that hold the material together) in spacetime, which means that those particles are undergoing absolute accelerations and therefore emit gravitational waves. But this is obviously not the case with a double-star system if the stars are modeled as point masses, because only relative acceleration (caused by geodesic deviation) is involved in the stars’ motion. • Bodies whose worldlines are geodesic move by inertia without losing energy because the very essence of inertial motion is motion without any loss of energy as discussed in the previous chapter. • The explanation of the decrease of the orbital period of binary pulsar systems given by Hulse and Taylor is effectively pre-relativistic4 —regarding it as caused by a loss of gravitational energy that does not exist either in (rigorously analyzed) general relativity or in Nature. Its proper spacetime (general-relativistic) explanation is the following: the reduction of the orbital period is another manifestation of the curvature of spacetime: the orbital period of the members of a binary system decreases because their helical geodesic worldlines converge; that is, stated in the 3

This interpretation is given even in the 13 October 1993 Press release (published on nobelprize.org) when The Nobel Prize in Physics for 1993 was awarded to Hulse and Taylor (but the reason for awarding the Prize was carefully and correctly worded: “for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation”): “the system is emitting energy in the form of gravitational waves in accordance with what Einstein in 1916 predicted should happen to masses moving relatively to each other” (https://www.nobelprize.org/ prizes/physics/1993/press-release/). The assertion the gravitational emission “should happen to masses moving relatively to each other” is incorrect. We will see below that in 1916 Einstein predicted that gravitational waves are emitted by bodies subject to absolute acceleration, i.e., by bodies whose worldlines are not geodesic, which is not the case with inspiraling members of a binary system. 4 As mentioned above, a prediction of gravitational waves emission by orbiting binaries can be obtained without using general relativity [3, 4].

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ordinary three-dimensional language, the inspiraling stars approach each other while moving by inertia, which leads to a reduction of the orbital period.5 c Therefore the experimental fact of the diminishing of the orbital period of PSR 1913+16, before the collision of the stars, cannot be regarded as evidence either for the emission6 of gravitational waves or for the existence of gravitational energy. And indeed, the correct wording of the official LIGO GW170104 Press Release “LIGO Detects Gravitational Waves for Third Time” (1 June 2017)7 demonstrates that the misconception—the members of a binary system emit gravitational waves before they collide—has been overcome: As was the case with the first two detections,8 the waves were generated when two black holes collided to form a larger black hole.

When the pulsar and its companion are realistically regarded as extended bodies they will emit weak gravitational waves due to tidal effects—as a result of the curvature of spacetime the worldlines of their constituents are not congruent (“parallel”) and the forces that hold them together will deform the worldlines (i.e., the constituents will be subjected to absolute acceleration) and gravitational waves will be emitted. But those gravitational waves are much weaker9 than the gravitational waves emitted when the members of the binary system collide. The confusion over when gravitational waves are emitted effectively started with the discovery of the binary pulsar system PSR 1913+16, but the misconception that gravitational waves carry gravitational energy appears to have become widely accepted after the crucial for gravitational-wave physics Chapel Hill conference in

5

It is truly puzzling that the correct general-relativistic explanation had been ignored by stubborn pre-relativistic thinking from the very beginning [8]: “What mechanisms other than gravitational radiation damping might be invoked to explain the observed change of orbital period? Several possibilities have been considered and discussed in the literature..., but all of them now appear to be either implausible, ad hoc, or of negligible magnitude.” 6 As the latest (2021) review paper [9] demonstrates, unlike the early statements, based on an incorrect model, that a binary system, whose members are regarded as point masses, would emit gravitational waves before the stars collide, recent treatments of binary systems have been using realistic models that, for example, take into account tidal effects that lead to emission of gravitational waves. 7 https://www.ligo.caltech.edu/page/press-release-gw170104 (accessed on 16 March 2021). 8 The reason I used the expression “correct wording” is that the first announcement (see quote at the beginning of this chapter) still mentions emission of gravitational waves before the collision (merger): “It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes” (italics added). And as we saw above the report of the first detection of gravitational waves contains the misconception that gravitational waves are emitted by binary systems, whose members are regarded as point masses, before the merger of the stars. 9 But they become greater immediately before the collision because the tidal effects sharply increase and as a result the worldlines of the constituents of the stars are more deformed, which means that their absolute accelerations increase.

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1957.10 As at the conference Feynman seems to have been the main and decisive defender of the view that gravitational waves carry gravitational energy, let us analyze his arguments and comments to see how the over-confidence of an otherwise brilliant physicist had contributed to a widespread misconception. In the Foreword to Feynman Lectures on Gravitation John Preskill and Kip S. Thorne quote a letter to Victor Weisskopf by Feynman where he recalls the 1957 conference and comments [11]: I was surprised to find a whole day at the conference devoted to this question, and that ‘experts’ were confused. That is what comes from looking for conserved energy tensors, etc. instead of asking ‘can the waves do work?’

It is this comment, particularly the second sentence, that reveals the essence of Feynman’s over-confidence; it can hardly be regarded as doing physics at its best.11 There are two problems in it. First, regarding “That is what comes from looking for conserved energy tensors, etc.,” Feynman seems to underestimate the need for a rigorous analysis of the mathematical formalism of general relativity (and any theory for that matter). As this is a very important,12 but a complex issue, here (in this small book) I will only mention that Minkowski discovered the spacetime structure of the world by doing precisely what Feynman appears to underestimate— by rigorously analyzing the mathematical formalism of Newtonian mechanics. Here is how Minkowski started his analysis in his lecture “Space and Time” [13] (the reason for this long quote is that it says it all): I want to show first how to move from the currently adopted mechanics through purely mathematical reasoning to modified ideas about space and time. The equations of Newtonian mechanics show a twofold invariance. First, their form is preserved when subjecting the specified spatial coordinate system to any change of position; second, when it changes its state of motion, namely when any uniform translation is impressed upon it; also, the zero point of time plays no role. When one feels ready for the axioms of mechanics, one is accustomed to regard the axioms of geometry as settled and probably for this reason those two invariances are rarely mentioned in the same breath. Each of them represents a certain group of transformations for the differential equations of mechanics. The existence of the first group can be seen as reflecting a fundamental characteristic of space. One always tends to treat the second group with disdain in order to unburden one’s mind that one can never determine from physical phenomena whether space, which is assumed to be at rest, may 10

Before the Chapel Hill conference there had been even doubts about the physical reality of gravitational waves apparently because of the obvious problem that the mathematical formalism of general relativity does not contain a proper tensorial expression for gravitational energy and momentum (and how gravitational waves would exist if they would not carry gravitational energy), the lack of global energy conservation in general relativity and maybe because the amplitudes of the predicted by Einstein had been found to be extremely small [10]. 11 In the chapter on relativistic mass I wondered (given Feynman’s brilliant explanation of both the experimental confirmation and the role of relativistic mass in spacetime physics) whether the controversy over (rather the assault on) this concept would have occurred if Feynman had been alive at that time. Now I wonder again whether he would change his view after an intense discussion during a special session at a conference on whether or not gravitational waves carry gravitational energy. 12 For the verification and the proper understanding of a physical theory.

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6 Gravitational Waves not after all be in uniform translation. Thus these two groups lead completely separate lives side by side. Their entirely heterogeneous character may have discouraged any intention to compose them. But it is the composed complete group as a whole that gives us to think.

Second, regarding “instead of asking ‘can the waves do work?’,” Feynman’s last paragraph of his book Feynman Lectures on Gravitation seems to reveal not only over-confidence but perhaps even irritation that “a great many people .. . . worry needlessly at this question” (whether gravitational waves carry gravitational energy), given that, as he implies, the answer to that question should be obvious [12, pp. 219– 220]: What is the power radiated by such a wave? There are a great many people who worry needlessly at this question, because of a perennial prejudice that gravitation is somehow mysterious and different – they feel that it might be that gravity waves carry no energy at all. We can definitely show that they can indeed heat up a wall, so there is no question as to their energy content. The situation is exactly analogous to electrodynamics – and in the quantum interpretation, every radiated graviton carries away an amount of energy .hω.

Before discussing Feynman’s proof that gravitational waves carry gravitational energy, let us address four points in this quote: • Concerning “they feel that it might be that gravity waves carry no energy at all,” we will see below that gravity waves carry no gravitational energy at all and that the question of what kind of energy they carry is more complex; it is not like the electromagnetic energy as Feynman stated in the quote. • Concerning “We can definitely show that they can indeed heat up a wall, so there is no question as to their energy content,” we will see below that when gravitational waves hit a wall, what heats it up is precisely what heats it up when a ball hits it—inertial energy (the work done by the inertial force with which the ball resists its deceleration) as we saw in the previous chapter. • Concerning “The situation is exactly analogous to electrodynamics,” unfortunately that is not true at all. In electrodynamics there is electromagnetic energy—as the energy of a physical field and as the work done by electromagnetic forces, whereas in general relativity there is no physical field (spacetime curvature is not a physical field) and there is no work done by gravitational forces (because there is no such thing as gravitational forces in both general relativity and in Nature). • Concerning “in the quantum interpretation, every radiated graviton carries away an amount of energy .hω,” this is not an argument for the existence of gravitational energy, because the graviton has been, still is, and almost certainly will remain a theoretical entity that does not represent anything in the physical world (because, as we saw in the previous chapter, gravitation does not seem to be a physical interaction that can be quantized). Feynman’s proof13 that gravitational waves carry gravitational energy had been presented at the Chapel Hill conference and is contained in a letter to Weisskopf dated February, 1961. Here is the relevant part of Preskill and Thorne’s Foreword to Feynman Lectures on Gravitation [11, pp. xxv–xxvi]: 13

This seems to be rather a collective proof—see [11, p. xxvi].

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At Chapel Hill, Feynman addressed this issue in a pragmatic way, describing how a gravitational wave antenna could in principle be designed that would absorb the energy “carried” by the wave.. . . Then the letter describes Feynman’s gravitational wave detector: It is simply two beads sliding freely (but with a small amount of friction) on a rigid rod. As the wave passes over the rod, atomic forces hold the length of the rod fixed, but the proper distance between the two beads oscillates. Thus, the beads rub against the rod, dissipating heat.

A careful examination of this argument reveals that inertial, not gravitational, energy is converted into heat. When a gravitational wave changes the shapes of the geodesic worldlines of the beads (and the rod), leaving them geodesic again, the beads follow their changed geodesic worldlines, but the rod prevents them from doing so, i.e., prevents the beads from moving by inertia. As a result the beads resist and exert inertial forces on the rod; it is the work done by these inertial forces, i.e., the inertial energy of the beads, that is converted into heat. The situation described by Feynman is exactly like the situation when a particle away from gravitating masses is prevented from moving by inertia—it exerts an inertial force on the obstacle and it is the work done by the inertial force, i.e., the inertial energy14 of the particle, that is converted into heat dissipating in the obstacle.15 So the energy that is converted into heat when the beads “rub against the rod” is the inertial energy stored in the beads16 which manifests itself through the work done by the inertial forces which the beads exert on the rod when it prevents them from moving by inertia. It is tempting to say that the gravitational waves carry gravitational energy by changing the geometry of spacetime. Such a temptation should be resisted, not only because we saw in the previous chapter that there exist three independent reasons for the non-existence of such energy, but, not less importantly, because one should first ask the question “What is the source of the gravitational energy that is carried by gravitational waves?” To answer this question, let us look again at the correct wording of the official LIGO GW170104 Press Release: As was the case with the first two detections, the waves were generated when two black holes collided to form a larger black hole.

I think it should be completely clear that no gravitational energy is involved in the collision of the two compact bodies. The energy released in the collision is inertial

14

This energy has been traditionally known as kinetic, but, as discussed in the previous chapter, it is more appropriately to call it inertial. I believe it is obvious that this is not a matter of semantics because it reflects the true physical nature of kinetic energy—the work done by inertial forces. 15 In the case of Feynman’s statement above that gravitational wave “can indeed heat up a wall,” the shapes of the particles’ worldlines (comprising the worldtube of the wall) are changed by the gravitational wave, but the forces that hold them together (as a wall) deform the particles’ geodesic worldlines and the arising inertial forces resist those deformations; it is the work done by them that is dissipated as heat in the wall. 16 In spacetime language, it is rather the deformation energy stored in the beads’ deformed worldtubes.

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energy—the inspiraling bodies move by inertia (their worldtubes are geodesic17 ) and when they collide they prevent each other from moving by inertia. Each body resists its absolute acceleration (i.e., resists the deformation of its worldtube) through the arising inertial force in it with which it acts on the other body. The work done by the two inertial forces is the inertial energy involved in the collision. The collision of the two compact bodies is identical to the collision of two bodies in flat spacetime. In this case the bodies, whose worldtubes are straight (geodesic), approach each other because they are “on a collision course.” When they collide they prevent each other from moving by inertia. Each body resists its (negative) acceleration (i.e., resists the deformation of its worldtube) through the arising inertial force in it with which it acts on the other body. The work done by the two inertial forces is the inertial energy involved in the collision. As only inertial energy is involved in the collision of the two compact bodies of the binary system, mentioned in the LIGO Press Release, it is this energy that is responsible for the generation of the wavelike curvature of spacetime (the gravitational waves) that changes the shape of all geodesic worldlines in the spacetime region where that propagation occurs. I hope it is clear that all the talk about generation, propagation of spacetime curvature and gravitational waves comes from the old times of three-dimensional thinking—assuming that a wave is really generated and really travels in the external world. There is no such thing as a propagating wave in spacetime, i.e., no propagation of curvature in spacetime—the spacetime curvature and the additional wavelike curvature are given en bloc in spacetime—noting propagates there and nothing happens there. The “wavelike” geometry of a spacetime region is interpreted in threedimensional language as a wave which propagates in space (exactly like a timelike worldline is interpreted in three-dimensional language as a particle which moves in space). Also, keep in mind that there is no such thing as space in the external world, because spacetime is not divided into a space and a time. To address properly (and overcome) another immediate and misleading reaction “Spacetime is nothing more than an abstract mathematical continuum!” read again Chap. 1 to see clearly that the experimental evidence (captured in the relativity principle at Minkowski’s time and confirming the relativistic effects later) would be impossible if spacetime were nothing more than an abstract mathematical continuum.

17

The geodesic worldtubes of the bodies are helices and their shape is determined by the curvature of spacetime induced by the bodies. As explained in the previous chapter each body curves spacetime regardless of whether or not the other body is there. That is why the converging worldtubes of the bodies do not constitute gravitational interaction; the bodies do not spent any additional energy and momentum to curve the worldtube of the other body.

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Summary Overcoming the two misconceptions involved in the concept of gravitational waves— that the inspiraling stars emit gravitational waves before they collide and that the gravitational waves carry gravitational energy—allows us to state that • No gravitational waves are emitted by a binary system, whose members are modeled as point masses, when they orbit each other before they collide, because their worldlines are geodesic, which means that they move by inertia without any loss of energy. The reduction of the orbital period of the binary system is not caused by a loss of orbital gravitational energy since it is merely an effect of the curvature of spacetime. Gravitational waves are emitted only by bodies that are absolutely accelerating (i.e., whose worldtubes are not geodesic); in other words, using proper spacetime terminology, the source of gravitational waves is the deformation of the worldtube of an absolutely accelerating body. So gravitational waves are emitted when the members of a binary system merge, because during the collision the worldtubes of the compact objects are deformed. The inspiraling compact objects (regarded as spatially extended bodies) do radiate weak gravitational waves (before they collide), which are caused by tidal effects in the bodies themselves (not because of their relative motion)—due to the curvature of spacetime the worldtubes of the compact objects’ constituents are not “parallel” (not congruent), whereas the forces holding the constituents together strive to keep the worldtubes “parallel;” as a result the worldtubes are deformed, which gives rise to gravitational waves. It should be stressed as strongly as possible that in spacetime physics it is impossible to have the following two statements both correct: – The stars of a binary system (modeled as point masses18 ) are represented by geodesic worldlines before they collide and merge, which means that no gravitational waves are emitted because geodesic worldlines represent motion by inertia without any loss of energy, i.e., interaction-free motion. – The stars of a binary system emit gravitational waves before they collide (which appears to be the presently accepted view in the gravity community.) • Gravitational waves do not carry gravitational energy19 because: – there is no such energy in general relativity (no tensorial expression for it, no physical gravitational field that possesses gravitational energy and no gravitational forces that can do work); – the energy, converted into heat when the beads of the Feynman gravitational detector are prevented from moving by inertia, is inertial energy (the energy done by the inertial forces which the beads exert on the rod);

18

As was done in the original 1975 Hulse and Taylor paper [5]. A systematic critical examination of standard arguments for the received view that gravitational waves carry energy-momentum is found in P. Duerr’s publications [14, 15].

19

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– the collision of the two compact members of the binary system, which was the source of gravitational waves detected by LIGO, involved only inertial energy; therefore it is this energy that is responsible for the generation and “propagation”20 of wavelike spacetime curvature (gravitational waves), which changes the shape of all geodesic worldlines (leaving them still geodesic) in the spacetime region where that wavelike spacetime curvature occurs.

References 1. B.P. Abbott et al., (LIGO Scientific Collaboration and Virgo Collaboration), Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102–Published 11 February 2016 2. T. Damour, Binary systems as test-beds of gravity theories, in Physics of Relativistic Objects in Compact Binaries: From Birth to Coalescence, ed. by M. Colpi, P. Casella, V. Gorini, U. Moschella, A. Possenti (Springer, Dordrecht, 2009), pp. 1–42 3. R.C. Hilborn, Gravitational waves from orbiting binaries without general relativity. Am. J. Phys. 86, 186 (2018) 4. A. Kenath, C. Sivaram, Physics of Gravitational Waves: Sources and Detection Methods (Springer, 2023) 5. R.A. Hulse, J.H. Taylor, Discovery of a pulsar in a binary system. Astrophys. J. 195, L51–L53 (1975) 6. J.H. Taylor, J.M. Weisberg, Further experimental tests of relativistic gravity using the binary pulsar PSR 1913+16. Astrophys. J. 345, 434–450 (1989) 7. A. Einstein, Näherungsweise Integration der Feldgleichungen der Gravitation. Sitzungsber. K. Preuss. Akad. Wiss. 1, 688 (1916); Über Gravitationswellen 1, 154 (1918) 8. P.M. McCulloch, J.H. Taylor, L.A. Fowler, Gravitational radiation and the binary pulsar, in Gravitational Radiation, Collapsed Objects and Exact Solutions: Proceedings of the Einstein Centenary Summer School, Held in Perth, Australia, January 1979, ed. by C. Edwards (Springer, Berlin, 1980), pp. 5–11 9. T. Dietrich, T. Hinderer, A. Samajdar, Interpreting binary neutron star mergers: describing the binary neutron star dynamics, modelling gravitational waveforms, and analyzing detections. Gen. Relativ. Gravit. 53, 27 (2021) 10. P.R. Saulson, Josh Goldberg and the physical reality of gravitational waves. Gen. Relativ. Gravit. 43, 3289 (2011) 11. J. Preskill, K.S. Thorne, Foreword to Feynman Lectures on Gravitation [12, p. 26] 12. Feynman et al., Feynman Lectures on Gravitation (CRC Press, Boca Raton, 2018) 13. H. Minkowski, Space and Time, new translation in: Hermann Minkowski, Spacetime: Minkowski’s Papers on Spacetime Physics. Translated by Gregorie Dupuis-Mc Donald, Fritz Lewertoff and Vesselin Petkov. Edited by V. Petkov (Minkowski Institute Press, Montreal, 2020), pp. 57–58 14. P. M. Duerr, It ain’t necessarily so: gravitational waves and energy transport. Stud. Hist. Philos. Mod. Phys. 65, 25–40 (2019). https://doi.org/10.1016/j.shpsb.2018.08.005 15. P.M. Duerr, Fantastic beasts and where (not) to find them: energy conservation and local gravitational energy in general relativity. Stud. Hist. Philos. Mod. Phys. 65, 1–14 (2019). https://doi.org/10.1016/j.shpsb.2018.07.002

20

Of course, nothing propagates or moves in spacetime; simply, there are regions in spacetime whose curvature is wavelike.

Chapter 7

Black Holes

In 1916, shortly after Einstein published his general relativity [1], Schwarzschild published a solution of the Einstein equation [4] that was later understood, mostly due to the works of Finkelstein [5] and Kruskal [6], as describing a region of spacetime whose curvature is so great that nothing—even light—can escape from it. Later this spacetime region with extreme curvature was called a black hole.1 In recent years many physicists have been freely talking about black holes as if their existence was established. The existence of super-compact stellar objects does appear to be an experimental fact, but, I think, it is a misconception to call those objects black holes. Indeed, a recent review “Nomen non est omen: Why it is too soon to identify ultra-compact objects as black holes” of the experimental evidence for the existence of black holes comes to the conclusion, stated in the title, that that evidence is inconclusive [7]: Despite claims that the observed astrophysical black hole candidates are black holes, there is currently no evidence that any one of them possesses a horizon. Both genuine black holes and horizonless UCOs are compatible with observational data, but the observation of PBHs (those bounded by quasilocal and in principle observable horizons) requires never-beforeseen exotic matter and appears to be incompatible with the predictions of semiclassical gravity.

What is not less important than the lack of conclusive experimental confirmation, is the fact that the concept of black hole contains two serious problems. It is this fact, which, I think, indicates that the recent fashion to regard black holes as established physical objects is based on a misconception. The first problem is the well-known fact that the Schwarzschild solution contains a singularity and many relativists feel uneasy about it, because the existence of a singularity in a physical theory is regarded as a clear sign that the theory breaks down in the circumstances where the singularity appears. In general relativity the situation 1 According to different accounts, the term “black hole” had been introduced in the sixties of the last century either by Wheeler or Dicke (Dicke compared that spacetime region to a prison in India called the Black Hole, because no one who entered it left it alive).

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becomes additionally complicated when trying to have a consistent understating of the term “singularity” because “the general covariance of relativity theory creates serious difficulties in formulating a suitable definition of a singularity in this theory” [8]. Einstein himself had been firmly against any attempts to regard singularities as existing in the physical world [9]: The essential result of this investigation is a clear understanding as to why the “Schwarzschild singularities” do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The “Schwarzschild singularity” does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light. .. . . The problem quite naturally leads to the question, answered by this paper in the negative, as to whether physical models are capable of exhibiting such a singularity.

Regardless of the recent tendency of considering the existence of black holes as experimentally confirmed, the singularity problem has never stopped worrying deepthinking relativists. The authors of a recent (2016) attempt to free general relativity of singularities stated their motivation clearly [10]: We believe that no acceptable physical theory should have a singularity (!), not even a coordinate singularity of the type discussed above! The appearance of a singularity shows the limitations of the theory.

Even a more recent (2023) volume2 “Regular Black Holes: Towards a New Paradigm of Gravitational Collapse” is devoted to the question of how spacetime singularities can be eliminated [11]: One of the most outstanding and longstanding problems in General Relativity is the inevitability of spacetime singularities in physically relevant solutions of the Einstein Equations. At a spacetime singularity, predictability is lost and standard physics breaks down. It is widely believed that the problem of spacetime singularities can be solved within a theory of quantum gravity.

The expectation that a theory of quantum gravity should get rid of singularities is also clearly stated by Ashtekar, Olmedo and Singh in their contribution to this volume [12]: There is general agreement in the gravity community that black hole singularities of classical general relativity (GR) offer excellent opportunities to probe physics beyond Einstein. However, as of now, there is no consensus on the fate of black hole singularities in full quantum gravity. Indeed, there is an ongoing debate even on a central question in the subject: Will singularities of classical GR be naturally resolved in full quantum gravity, or will they persist? As the very name of this Volume suggests, in many circles an affirmative answer is taken to be a necessary condition for the viability of a proposed quantum gravity theory.

The second problem is no less worrying. The Schwarzschild solution implies that there exist two spacetime regions inside and outside the Schwarzschild sphere of radius .r = 2m (where .m is the gravitating mass), which “do not join smoothly on 2

Non-singular (or singularity-free) black holes are called regular black holes.

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the surface .r = 2m.”3 The Schwarzschild solution also implies that there exist two “realities”—one for observers falling together with the constituents of the collapsing body (of mass .m), and another for distant observers far away from the Schwarzschild sphere. This situation is perhaps most clearly formulated by Dirac [14]: We see that the Schwarzschild solution for empty space can be extended to the region.r < 2m. But this region cannot communicate with the space for which .r > 2m. Any signal, even a light signal, would take an infinite time to cross the boundary .r = 2m, as we can easily check. Thus we cannot have direct observational knowledge of the region .r < 2m. Such a region is called a black hole, because things may fall into it (taking an infinite time, by our clocks, to do so) but nothing can come out. The question arises whether such a region can actually exist. All we can say definitely is that the Einstein equations allow it. A massive stellar object may collapse to a very small radius and the gravitational forces then become so strong that no known physical forces can hold them in check and prevent further collapse. It would seem that it would have to collapse into a black hole. It would take an infinite time to do so by our clocks, but only a finite time relatively to the collapsing matter itself.

Rigorously and explicitly stated, the Schwarzschild solution of the Einstein equation in general relativity predicts both (i) black holes form for an observer falling with the collapsing body, and (ii) that they will never form for distant observers like us. I am not aware of any attempts to address4 this paradoxical situation despite that it was mentioned as early as 1939 by Oppenheimer and Snyder [15]: The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius.

and in the same paper [15]: The star thus tends to close itself off from any communication with a distant observer; only its gravitational field persists. We shall see later that although it takes, from the point of view of a distant observer, an infinite time for this asymptotic isolation to be established, for an observer comoving with the stellar matter this time is finite and may be quite short.

3 Papapetrou particularly emphasizes the serious anomaly on the Schwarzschild sphere, whose physical meaning, I think, has not been thoroughly examined [13]:

But these geodesics are space-like for .r > 2m and time-like for .r < 2m. The tangent vector of a geodesic undergoes parallel transport along the geodesic and consequently it cannot change from a time-like to a space-like vector. It follows that the two regions .r > 2m and .r < 2m do not join smoothly on the surface .r = 2m. 4 The only comments after my talk “On the asymptomatic formation of black holes” at the Third Minkowski Meeting (11–14 September 2023, Albena, Bulgaria) and during the panel discussion on the nature of black holes just stated that there was no contradiction between the two predictions of the Schwarzschild solution. This is self-evident when those predictions are regarded as mathematics. But they become problematic when it is claimed that they apply to the physical world.

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Using the term “asymptotically”5 is an attempt to suggest that despite that a collapsing body “would take an infinite time” to collapse into a black hole, somehow the black hole would form “asymptotically”!? It seems the fact that the term “asymptotically” is not used in the case of light—“Any signal, even a light signal, would take an infinite time to cross the boundary .r = 2m”—can be only explained as employing double standards in physics. If double standards are not used, then, by the same argument (that black holes somehow form “asymptotically”), it follows that, if light also asymptotically approaches the event horizon (the Schwarzschild sphere), it will reach it and eventually escape (which is not the accepted understanding). In any case, if black holes do form asymptotically, then, by absolutely the same logic, light will asymptotically reach the event horizon and it will be as bright as a star. In fact, the situation with understanding the nature of black holes is even worse, because there exists another instant of using double standards when interpreting the expression “take an infinite time” differently in different situations: • From Dirac’s quote above: “Any signal, even a light signal, would take an infinite time to cross the boundary .r = 2m”—in this case “take an infinite time” is interpreted to mean “never.” • Again from Dirac’s quote above: “It would seem that it would have to collapse into a black hole. It would take an infinite time to do so by our clocks”—in this case, inexplicably, “take an infinite time” is not taken to mean “never” as in the case of light; instead, it is assumed without any justification that black holes exist for distant observers. That is why, it strikes me when physicists confidently talk about black holes perhaps without realizing6 that they have been employing double double standards. The actual situation with the status of black holes in spacetime physics appears to be the following—as we are (and have always been) distant observers it will take an infinite time (for us) for black holes to form, which, in ordinary language, means that black holes will never exist for us. This is the situation without employing double standards, i.e., by explicitly examining both predictions of the same theory (general relativity)—the formation of black holes for observers falling with the constituents of a collapsing body and that they will never form for distant observers. What makes the need to address the present paradoxical situation more urgent is the awarding of the Nobel Prize in Physics for 2020 for work to understand black holes [16]:

5

In addition, taking the term “asymptotically” seriously requires a number of explicit definitions such as “asymptotically existing” (“asymptotic existence”), “asymptotically non-existing,” “asymptotically real” or even more confusing definitions such as “asymptotically alive” and “asymptotically dead”. 6 I believe that the use of double standards has not been intentional, but rather following “an inner voice” which whispers that there is a contradiction (when the Schwarzschild solution is regarded as describing a real physical situation) that should be avoided or at least addressed.

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The Nobel Prize in Physics 2020 was divided, one half awarded to Roger Penrose “for the discovery that black hole formation is a robust prediction of the general theory of relativity”, the other half jointly to Reinhard Genzel and Andrea Ghez “for the discovery of a supermassive compact object at the centre of our galaxy.”

This is another example7 of how the Nobel Committee carefully states the reasons for awarding the Nobel Prize. The term “black hole” was mentioned as a prediction of general relativity, which is indisputably true8 for an observer falling with a collapsing star. There is no mention of experimental discovery of black holes; instead the second half of the Prize was given “for the discovery of a supermassive compact object at the centre of our galaxy.” Penrose also stated the two problems with black holes in 1965 [17]: The body contracts and continues to contract until a physical singularity is encountered at = 0. As measured by local comoving observers, the body passes within its Schwarzschild radius .r = 2m. (The densities at which this happens need not be enormously high if the total mass is large enough). To an outside observer the contraction to .r = 2m appears to make infinite time. Nevertheless, the existence of a singularity presents a serious problem for any interior region.

.r

But he did not seem to have been bothered by the fact that “infinite time” means never and did not comment on the use of double standards in the interpretation of the Schwarzschild solution of the Einstein equation either.

Summary The Schwarzschild solution of the Einstein equation and its interpretation as a prediction of black holes constitutes an unprecedented situation in fundamental physics for two reasons: • A singularity in a theory has been regarded as having physical meaning, that is, has been viewed as representing a real feature of the physical world. • The same theory—general relativity—implies that there exist two “realities”—one for observers falling with a collapsing body (for these observers the collapsing body becomes a black hole for a finite period of time) and another for distant observers 7 The other (relevant) one was mentioned in the previous chapter—the Nobel Prize in Physics for 1993 whose reason for awarding the Prize was also carefully and correctly worded: “for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation.” 8 There was a dramatic development in black hole physics—on 1 December 2023 Roy Kerr posted a very important paper “Do Black Holes have Singularities?” (arXiv:2312.00841) where he demonstrated that “There is no proof that black holes contain singularities when they are generated by real physical bodies.” Therefore, the assertion “black hole formation is a robust prediction of the general theory of relativity” does not seem to be indisputably true if the two defining features of black holes are a singularity and an event horizon, which appears to be the accepted understanding (moreover, Penrose himself in the quote below states it explicitly that a black hole is formed when “physical singularity is encountered”).

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like all of us (for these observers the body will take an infinite time to collapse and therefore a black hole will never form). The reason for emphasizing the two problems with the concept of black holes is to draw the physicists’ attention to them in the hope that more research would be done to find satisfactory explanations of these problems: • What is the justification for assuming that a mathematical singularity should have a counterpart in the physical world? • What is the physical meaning that black holes will form for some observers, but will never come into existence for others?

References 1. A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 49, (1916). New publication of the original English translation ([2]) in [3] 2. A. Einstein, The foundation of the general theory of relativity, in The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, ed. by H.A. Lorentz, A. Einstein, H. Minkowski, H. Weyl. With Notes by A. Sommerfeld. Translated by W. Perrett and G.B. Jeffery (Methuen and Company, Ltd., 1923; Reprinted by Dover Publications Inc., 1952) 3. V. Petkov (ed.), The Origin of Spacetime Physics, 2nd edn. Foreword by A. Ashtekar. (Minkowski Institute Press, Montreal, 2023) 4. K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. K. Preuss. Akad. Wiss. 1, 189 (1916) 5. D. Finkelstein, Past-future asymmetry of the gravitational field of a point particle. Phys. Rev. 110, 965 (1958) 6. M.D. Kruskal, Maximal extension of Schwarzschild metric. Phys. Rev. 119, 1743 (1960) 7. S. Murk, Nomen non est omen: why it is too soon to identify ultra-compact objects as black holes. Int. J. Mod. Phys. D. https://doi.org/10.1142/S0218271823420129 8. R. Geroch, What is a singularity in general relativity? Ann. Phys. 48, 526–540 (1968) 9. A. Einstein, On a stationary system with spherical symmetry consisting of many gravitating masses. Ann. Math., Second Series 40(4), 922–936 (1939) 10. P.O. Hess, M. Schäfer, W. Greiner, Pseudo-Complex General Relativity (Springer, Heidelberg, 2016) 11. C. Bambi (ed.), Regular Black Holes: Towards a New Paradigm of Gravitational Collapse (Springer, 2023), p. 5 12. A. Ashtekar, J. Olmedo, P. Singh, Regular Black Holes from Loop Quantum Gravity in [11], pp. 235–236 13. A. Papapetrou, Lectures on General Relativity (Reidel, Dordrecht, 1974), pp.85–86 14. P.A.M. Dirac, General Theory of Relativity (Princeton, Princeton University Press, 1996), pp.35–36 15. J.R. Oppenheimer, H. Snyder, On continued gravitational contraction. Phys. Rev. 56(455), 456 (1939) 16. https://www.nobelprize.org/prizes/physics/2020/summary/ 17. R. Penrose, Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 18 (1965)

Appendix A

Ancient Logical Arguments Against the Everyday View of Time

Since ancient times thinkers have been suspecting that the perceived flow of time does not reflect a true feature of the world and is nothing more than a stubborn illusion. This suspicion turned out to be an amazing insight—the discovery of the spacetime structure of the world, particularly the experiments which confirmed the predictions of the theory of relativity demonstrated that they would be impossible if reality were what we all perceive—an evolving in time three-dimensional world with a real time flow. The higher reality, glimpses of which we perceive through our senses, turned out to be a four-dimensional world (spacetime) in which all moments of time exist at once as the fourth dimension. More than 25 centuries ago the representatives of the Eleatic school of thought (Parmenides, Zeno, Melissus, and Xenophanes) first explicitly argued that the perceived image of the world coming from our senses might be drastically different from the true reality. Parmenides insisted that an analysis of what we all perceive would unavoidably reveal that many things about the world, which we regard as self-evident, were illusions and that being (what exists) is eternal and unchanging [1, p. 64]: There are very many signs: that Being is ungenerated and imperishable, entire, unique, unmoved and perfect; it never was nor will be, since it is now all together, one, indivisible.

For this reason Parmenides regarded the perceived flow of time (our feeling of a perpetual transformation of the non-existent future into the existent present, and of the existent present into the non-existent past) as self-contradictory (because being is eternal and nothing can come into or go out of existence) and argued that time does not belong to the true reality [1, p. 74]: And time is not nor will be another thing alongside Being, since this was bound fast by fate to be entire and changeless.

As the Eleatic view so openly contradicted our perceptual experience it had been mostly ridiculed since the time of the Eleatics. This attitude prompted Zeno to demonstrate to those who regarded motion and change as self-evident that it is their view, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8

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which naively reflects what comes from our senses, that leads to contradictions. Zeno formulated a number of paradoxes for this purpose such as the Dichotomy—if an object travels from a point A to a distant point B, it has to travel first half of the distance AB, then half of the remaining half, and so on; as the object has to travel an infinite number of such distances (since every distance can be divided into two) and as each of these travels needs some time, the object would need an infinite amount of time to travel the infinite number of distances and would never reach B. Aristotle showed that Zeno had arrived at the paradox, because he explicitly presupposed that space was divisible to infinity, but implicitly assumed that time was not infinitely divisible. If both space and time are infinitely divisible, there is no paradox—if, for example, a distance of one meter is traveled by an object for one second, the object will travel half a meter for half a second and so on, and will not need an infinite amount of time to reach the end point B. In Book VI of his Physics Aristotle wrote about Zeno’s implicit assumption that time is not infinitely divisible [2, Sect. 9]: This is false; for time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles.

However, when Aristotle discussed the nature of time itself in Book IV of his Physics—that of all times (past, present, and future) only the present time (the moment ‘now’) is real—his logical analysis inescapably led him to the opposite conclusion: that the only real moment of time is “the indivisible present ‘now”’ [2, Sect. 13]. Aristotle knew that the duration of ‘now’ could not be zero, because then time would not exist at all. He realized that he had no choice but to assume that the moment ‘now’ is indivisible1 in order to avoid a contradiction in terms—if the moment ‘now,’ which by definition is wholly present, were divisible, it would contain past, present, and future moments [2, Sect. 3]: All time has been shown to be divisible. Thus on this assumption the now is divisible. But if the now is divisible... there will be a part of the now that is past and a part that is future... It is clear, then, from what has been said that time contains something indivisible, and this is what we call the now.

The very fact that Aristotle, one of the greatest thinkers of our civilization who single-handedly created the science of logic, was led by the common-sense view (that only ‘now’ is real) to the inescapable contradiction—the present moment is both divisible and indivisible—implies that that view is wrong. Aristotle seems to have tried to identify the cause of this contradiction. In Book IV of his Physics he appears to have considered the possibility that the contradiction was caused by the seemingly self-evident assumption that the division of time into past, present and future reflected an objective fact in the world and wondered whether that division and the very idea of time might exist only in the mind (or the soul) [2, Sect. 14]: Whether if soul did not exist time would exist or not, is a question that may fairly be asked. 1

But if ‘now’ is indivisible and has some duration (again, it could not be zero since time would not exist at all), then Aristotle’s refutation of Zeno’s Dichotomy fails.

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Sixteen centuries ago Augustine also investigated the nature of time and like Aristotle faced the same paradoxical situation about the duration of ‘now,’ but unlike him explicitly concluded that the division of time into past, present and future did not reflect anything in the physical world (which according to Augustine is an eternal present identical to the Eleatic’s eternal being) and therefore existed only in the mind [3]: What is by now evident and clear is that neither future nor past exists, and it is inexact language to speak of three times – past, present, and future .. . . In the soul there are these three aspects of time, and I do not see them anywhere else.

It is both unfortunate and unhelpful for the advancement of our understanding of the physical world that philosophers and physicists, who claim that time objectively flows, ignore not only the overwhelming experimental evidence for the existence of spacetime (where there is no flow of time), but also the ancient logical arguments (which are now as valid as in ancient times) against the division of time into past, present and future (and therefore against the view that the flow of time exists in the physical world). Science does not advance by ignoring arguments; neither does philosophy .. . .

Appendix B

Minkowski Developed his Own Ideas of Spacetime Physics

The advent of spacetime physics came at the price of four different scientific tragedies involving Hendrik Lorentz, Henri Poincaré, Albert Einstein and Hermann Minkowski whose work laid the foundations of spacetime physics.2 Lorentz’ and Poincaré’s scientific tragedies had the same cause—both Lorentz and Poincaré regarded the new theoretical entities they introduced in physics as pure mathematical abstractions that did not represent anything in the physical world, which prevented them from discovering the spacetime structure of the world. Einstein’s rather subtle scientific tragedy has to do with his unclear and, in some cases, even incorrect views on a number of subjects that might have led to confusions and misconceptions some of which still persist; I think it is in some sense tragic when scientific positions of one of the greatest physicists give rise to confusions and misconceptions that may last for decades. Minkowski’s scientific tragedy is of different kind—he arrived independently at what Einstein called special relativity and at the notion of spacetime, but Einstein and Poincaré published first while Minkowski had been developing the four-dimensional formalism of spacetime physics reported on 21 December 1907 and published in 1908 as a 59-page treatise3 ; he did not publish his results earlier “because he wished first to work out the mathematical structure in all its splendour” (M. Born, see below). Two things appear to indicate that Minkowski alone discovered the spacetime structure of the world. The first indication is the novelty and depth of Minkowski’s approach and theoretical achievements (contained in his two papers included in [4]), which demonstrate that he was developing his own original insights, not interpreting someone’s results. 2 For details see “Editor’s Epilogue: The Quadruple Scientific Tragedy involved in the Discovery of Spacetime Physics” added to the second edition of The Origin of Spacetime Physics [4]. 3 “The Fundamental Equations for Electromagnetic Processes in Moving Bodies;” H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse (1908) S. 53–111. Translated in [5].

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Indeed, in his paper “Space and Time” Minkowski presented his revolutionary ideas of regarding physics as geometry of the discovered by him four-dimensional world and in the 59-page treatise “The Fundamental Equations for Electromagnetic Processes in Moving Bodies” single-handedly developed the four-dimensional mathematical formalism of spacetime physics (neither Einstein nor Poincaré in their 1905 papers were even close to it). In his Foreword to the collections of papers that laid the foundations of spacetime physics The Origin of Spacetime Physics [4] Abhay Ashtekar specifically emphasized the novelty and originality of Minkowski’s longer paper: This is a much more detailed account of Minkowski’s astonishingly deep understanding of how the fusion of space and time into a four-dimensional spacetime continuum leads to a reformulation of electrodynamics. In particular, this paper provides the tensorial formulation of Maxwell’s equations and the action of the Lorentz group on the Maxwell field tensor and the source current. Because of its emphasis on four-dimensional geometry, this discussion of Maxwell’s equations goes distinctly beyond Einstein’s paper on On the Electrodynamics of Moving Bodies. Indeed, Minkowski’s four-dimensional equations are exactly in the same form that we use today, more than a century later!

Also, Minkowski’s way of doing physics is profoundly different from Einstein’s and Poincaré’s ways, which additionally demonstrates Minkowski’s independent path to the discovery of the spacetime structure of the world. For example Einstein used postulates4 —the relativity postulate and the postulate of the constancy of the speed of light—without even attempting to explain them, whereas Minkowski provided explanations (for details, see the first two chapters of this book and [7]). Einstein believed that the essence of his special relativity (and later of his general relativity) was the relativity of physical quantities, whereas Minkowski, as a mathematician, searched for and found the underlying absolute entity (spacetime) that makes possible the very existence of relative quantities5 (as explained in Chap. 1 relativity of space and time is impossible in a three-dimensional world). Einstein talked about relativity of simultaneity, whereas Minkowski showed that if inertial observers in relative motion have different times (as Lorentz conjectured and 4

Minkowski also suggested to use a postulate—the postulate of the absolute world [5, p. 65]: I think the word relativity postulate used for the requirement of invariance under the group is very feeble. Since the meaning of the postulate is that through the phenomena only the four-dimensional world in space and time is given, but the projection in space and in time can still be made with certain freedom, I want to give this affirmation rather the name the postulate of the absolute world (or shortly the world postulate).

.G c

However, the world postulate is a fundamentally different kind of postulate—it simply states the dimensionality of the world as revealed by Minkowski’s rigorous analysis of the experiments captured in the relativity postulate, whereas the relativity postulate states that uniform motion with respect to space cannot be detected, which needs to be explained why; the question of why the world is four-dimensional is a much deeper question, perhaps as fundamental as the question “Why does the world exist?” 5 Mathematicians are well-aware that relative quantities are descriptions of something absolute: “The emphasis on the geometry means an emphasis on the absolutes which underlie relative descriptions” [8].

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Einstein postulated), they have different (three-dimensional) spaces as well.6 Another indication that Minkowski arrived independently at the conclusion that the times of inertial observers in relative motion have the same status is that he never mentioned relativity of simultaneity, although he had been undoubtedly aware that space is defined in terms of simultaneity—the class of all space points at a given moment of time (i.e. simultaneous with that moment)—and therefore different spaces imply different classes of simultaneous events. The second indication is Max Born’s recollections about Minkowski’s work during a seminar in 1905 and about his conversations with Minkowski. By 1905 Hermann Minkowski was already internationally recognized as an exceptional mathematical talent. At that time he became interested in the electron theory and especially in an unresolved issue at the very core of fundamental physics—at the turn of the nineteenth and twentieth century Maxwell’s electrodynamics had been interpreted to show that light is an electromagnetic wave, which propagates in a light-carrying medium (the luminiferous ether), but its existence was put into question since the Michelson-Morley interference experiments failed to detect the Earth’s motion in that medium. Minkowski’s documented involvement with the electrodynamics of moving bodies began in the early summer of 1905 when he and his friend David Hilbert codirected a seminar in Göttingen on the electron theory. The paper of Minkowski’s student Albert Einstein “On the Electrodynamics of Moving Bodies” [9] was not published at that time; Annalen der Physik received Einstein’s paper on June 30, 1905. Poincaré’s longer paper “On the Dynamics of the Electron” [10], in which Poincaré regarded the Lorentz transformations as rotations in a four-dimensional space with time as the fourth dimension, was not published either; Rendiconti del Circolo matematico di Palermo received Poincaré’s paper on July 23, 1905. Also, “Lorentz’s 1904 paper (with a form of the transformations now bearing his name) was not on the syllabus” [11]. Minkowski’s student Max Born, who attended the seminar in 1905, wrote [12]: We studied papers by Hertz, Fitzgerald, Larmor, Lorentz, Poincaré, and others but also got an inkling of Minkowski’s own ideas which were published only two years later.

Born also recalled what Minkowski had specifically mentioned a number of times during the seminar in 1905 [13]: I remember that Minkowski occasionally alluded to the fact that he was engaged with the Lorentz transformations, and that he was on the track of new interrelationships. 6

As a mathematician Minkowski probably realized that fact immediately and remarked “Neither Einstein nor Lorentz disputed the concept of space” [5, p. 65], i.e., neither of them stated that inertial observers in relative motion should also have different spaces. Einstein discussed relativity of simultaneity, but seems to have not realized in his early papers that a class of simultaneous events forms a three-dimensional space which appears to explain why Einstein mentioned that inertial observers in relative motion have different spaces hardly in the fifth appendix to his popular book Relativity: The Special and General Theory which was added in 1952 [6]: “But it must now be remembered that there is an infinite number of spaces, which are in motion with respect to each other.”

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Again Born wrote in his autobiography about what he had heard from Minkowski after Minkowski’s lecture “Space and Time” given on September 21, 1908 [14]: He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other were pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendour. He never made a priority claim and always gave Einstein his full share in the great discovery.

Minkowski asked Einstein to send him the 1905 paper “On the Electrodynamics of Moving Bodies” hardly on October 9, 1907 [15]. The advent of spacetime physics was indeed marked by four scientific tragedies, but a greater tragedy affected fundamental physics. Given the depth of Minkowski’s discovery of the spacetime structure of the world and its far-reaching consequences, it is indeed tragic that he departed from this world at the age of 44; now fundamental (at least spacetime) physics might look quite different if he had lived longer.

References 1. A.H. Coxon, The Fragments of Parmenides (Parmenides Publishing, Athens, 2009) 2. J. Barnes (ed.), Complete Works of Aristotle, vol. 1 (Princeton University Press, Princeton, 1984) 3. Saint Avrelii Augustine, Confessions (Oxford University Press, New York, 1998), p. 235 4. Editor’s Epilogue, The quadruple scientific tragedy involved in the discovery of spacetime physics, in The Origin of Spacetime Physics, 2nd edn. Foreword by A. Ashtekar. Edited by V. Petkov. (Minkowski Institute Press, Montreal, 2023) 5. H. Minkowski, Spacetime: Minkowski’s Papers on Spacetime Physics. Translated by Gregorie Dupuis-Mc Donald, Fritz Lewertoff and Vesselin Petkov. Edited by V. Petkov (Minkowski Institute Press, Montreal, 2020) 6. A. Einstein, Relativity: The Special and General Theory, new publication in the collection of five works by Einstein: A. Einstein, Relativity (Minkowski Institute Press, Montreal, 2018), p. 103 7. V. Petkov, Physics as spacetime geometry, in Springer Handbook of Spacetime, eds. by A. Ashtekar, V. Petkov (Springer, Heidelberg, 2014), Sect. 8.2 8. E.G. Peter Rowe, Geometrical Physics in Minkowski Spacetime (Springer, London, 2001), p. vi 9. A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik 322(10), 891–921. (Received June 30, 1905; published September 26, 1905). New publication of the English translation “On the Electrodynamics of Moving Bodies” in [1] 10. H. Poincaré, Sur la dynamique de l’électron. Rendiconti del Circolo matematico Rendiconti del Circolo di Palermo 21 (1906), pp. 129–176. Translated in: [1] 11. S. Walter, Minkowski, mathematicians, and the mathematical theory of relativity, in The Expanding Worlds of General Relativity, Einstein Studies, vol. 7, eds. by H. Goenner, J. Renn, J. Ritter, T. Sauer (Birkhäuser, Basel, 1999), pp. 45–86, 46 12. M. Born, Physics in My Generation, 2nd edn. (Springer-Verlag, New York, 1969), p. 101

Appendix B: Minkowski Developed his Own Ideas of Spacetime Physics

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13. Quoted from T. Damour, What is missing from Minkowski’s ‘Raum und Zeit’ lecture. Annalen der Physik 17(9–10), 619–630, 626 (2008) 14. M. Born, My Life: Recollections of a Nobel Laureate (Scribner, New York, 1978), p. 131 15. Postcard: Minkowski to Einstein, October 9, 1907, in The Collected Papers of Albert Einstein, vol. 5, The Swiss Years: Correspondence, pp. 1902–1914, eds. by M.J. Klein, A.J. Kox, R. Schulmann (Princeton University Press, Princeton, 1995), p. 62

Index

A A body that participates in interactions involves resistance, 59 A geodesic is particle independent, 60 Absolute acceleration, 3, 29, 38, 60, 61, 67, 68 Absolute four-dimensional world, 18, 55 Absolute motion, 5, 9, 10 Absolute simultaneity, 10 Absolute space, 22, 27 Abstract four-dimensional mathematical continuum, 10 Abstract mathematical continuum, 72 Accelerated motion, xviii, 3, 24, 27, 29 Accelerating lift, 57 Acceleration, xviii, 22, 27, 28, 34, 45, 50 Accelleration, 58 Accepted definition of mass, vii Apparent acceleration, 58, 59, 61 Apparent gravitational interaction, 4 Aristotle, 21, 25, 82 Ashtekar, 76 “Asymptotic existence”, 78 “Asymptotically alive”, 78 “Asymptotically dead”, 78 “Asymptotically existing”, 78 “Asymptotically non-existing”, 78 “Asymptotically real”, 78 Augustine, 83

B Becoming, xiii “Becoming in spacetime”

contradiction in terms, 5 Being eternal and unchanging, 81 Besso, 55 Binary pulsar systems, 67 Binary systems, viii, 66 Black holes, ix, xxii, xxiii, 65, 68, 75, 78–79 asymptotic formation, ix Block universe, 4 Born, viii, 8, 87, 88

C Cantor, 1 Change of orbital period, 68 Chapel Hill conference, 69 Classical electron theory, 46 Classical mechanics, 2 Classical particle, 4 Concept of motion, 21 Concept of relativistic mass, 45, 46, 49, 50 Concept of spacetime, xvii, 2, 18 Conceptual analyses, xiii Consciousness, 5 Conserved energy tensors, 69 Constancy of the velocity of light, 4 Controversy over relativistic mass, viii Conventionality of simultaneity, 18 Conventionality of the one-way velocity of light, 18 Coordinate singularity, 76 Coordinate time, 49 Copernican system, 28

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 V. Petkov, Seven Fundamental Concepts in Spacetime Physics, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-49730-8

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92 Curvature of something that does not exist, 55 Curvature of spacetime, 58, 59, 66, 67, 73 Curved spacetime, xx, 28, 29, 35, 39, 42, 49 Curved undeformed worldtube, 58 Curved worldline, xviii, 23, 35 D Damour, xii, 7 Deep-thinking relativists, 76 Deformation energy, 41 Deformed worldline, 29, 30, 35, 36, 67 Deformed worldtube, 3, 36 Deterministic behaviour, 4 Deterministic phenomena, 4 Dichotomy, 82 Dicke, 75 die Welt, 1, 8, 9, 54 Different three-dimensional spaces, 13 Dimensionality is a matter of description, 2 Dimensionality of the world incorrect question, 2 Dirac, 77, 78 Distant observer, 77, 78 Distant simultaneity, 18 Distinction between past, present and future is only a stubbornly persistent illusion, 56 Doing physics right, xvi Double standards, 78 Double standards in physics, 65 Duerr, 73 Dynamics, 49 E Eddington, 53, 56 Einstein, viii, xi, xiii, xvii, 3, 9, 18, 22, 25, 28, 31, 32, 37, 39, 46, 49, 53, 55–58, 67, 69, 75, 85, 87, 88 Einstein equation, xxiii, 59, 75, 77, 79 two possible interpretations, 59 Einstein lifts, 36 Einstein’s equivalence principle, 36, 39, 62 Einstein’s general relativity, 62 Einstein’s happiest thought, 57 Eleatics, 21, 22, 29, 81 Eleatic school of thought, 21, 81 Eleatic’s eternal being, 83 Electrodynamics, 70 Electromagnetic energy, 70 Electromagnetic forces, 70

Index Electromagnetic interactions, xix, 36, 42 Electromagnetic mass, 46 Electron, 4, 46 Elementary particles, 36 Equivalence of inertial and passive gravitational mass, 62 Equivalence of inertial mass and passive gravitational mass, 36, 49 Equivalence principle, 36, 37, 61, 62 Essence of length contraction, 13 Event horizon, ix Event solipsism, 5 Evolving in time three-dimensional world, 81 Evolving three-dimensional world, 10 Excluding conceptual analyses, xiii “Exist” as used by Minkowski, 7 Existence absolute, 5 four-dimensional, 55 three-dimensional, 55 “Exists now”, 7 Experimental distinction between a threedimensional world and a fourdimensional world, 2 Experiments do not contradict one another, xii Exploring the internal logic of ideas, xiii External observer, 77

F Fall with the same acceleration, 60 Feynman, viii, 69, 71 Finkelstein, 75 First aspect of inertia, 34 Fitzgerald, 12, 87 Flat spacetime, 35, 49, 59, 60 Flow of time, xiii, 5, 6, 81, 83 Force of gravity, 39 Force of weight, xix, 42 Foundational knowledge, xii, 6 Four-dimensional continuum neither “time” nor “space”, 5 Four-dimensional existence, 55 Four-dimensional formalism, xvii, 54 Four-dimensional mathematical space, 7, 8 Four-dimensional objects, 2, 3 Four-dimensional physics, 22 Four-dimensional stress, 36 Four-dimensional world, 2, 3, 5, 6, 10, 13, 55, 81

Index Four-momentum, 49 Fourth dimension, 81 Four-vector, 49 4-velocity, xx, 49 Free will, 6 Free will is beyond physics, 6 Free will is not beyond physics, 6 Fundamental physics, xi, 6, 35 Future, 56

G Galileo, xi, xiii, xvii, 18, 22, 31 Galileo’s famous ship experiment, 32 Galileo’s law of inertia, 58 Galileo’s principle of inertia, 21, 25, 31 Galileo’s principle of relativity, 8, 21, 22, 31, 32 Galileo’s research strategy, 31 General covariance, 76 General relativity, viii, 3, xxi, 37, 53–56, 59, 65, 69, 73, 75, 76 Geodesic worldlines, 28, 29, 33, 61, 67, 72, 74 Geodesic worldtubes, 41, 58, 60 Geometry of spacetime, 60 Geroch, ix, 23 Gravitation, 70 physical interaction, 53 Gravitation = spacetime geometry, viii Gravitation as a separate agency becomes unnecessary, viii, 53 Gravitational collapse, 76 Gravitational energy, xiii, xv, xxi, xxii, xxiii, 4, 55, 56, 62, 66, 68–71, 73 Gravitational energy and momentum, xiii, xxi, 53 Gravitational field, xxi, 37, 55 homogeneous, 37 not a “thing”, 53 Gravitational field gravitational field, xiii Gravitational forces, xiii, xix, 38, 42, 58, 66, 73 Gravitational interaction, viii, 40, 60, 62 Gravitational phenomena, viii, xx, 42, 53, 55, 62 Gravitational physics, 53, 55 Gravitational radiation, xiii Gravitational radius, 77 Gravitational waves, viii, xiii, xv, xxii, 3, 56, 62, 65, 67, 68, 70, 71, 73 Graviton, 70 Greatest discoveries in physics, xiii

93 Greatest intellectual achievement in fundamental physics, 53

H Hamiltonian formulation of general relativity, 56 Hamiltonian mechanics, xii, 2 Hertz, 87 Higher reality, 1, 8, 81 Hilbert, 87 History of physics, 7 How science works, 19 Hulse, 66, 67 .100% explanation, 4, 17, 22, 26, 30, 34, 62

I Illusions, 81 Inadequate notion of mass, vii Inertia, 34, 59 first aspect, 34 nature of, 34 origin of, 34 phenomenon, 34 second aspect, 34 second aspect of, 36 two aspects of, 42 Inertial energy, xv, xx, xxiii, 40–42, 62, 70, 71, 74 Inertial forces, xix, 27, 33, 38, 39, 42, 62, 71, 72 Inertial forces are real, 33 Inertial mass, xx, 36, 39, 42, 61, 62 Inertial motion, 23, 24, 27, 31, 33, 38, 42 Infinite time, 78, 79 Insignificant unscientific chat, 6 Inter-subjective fact, 5

J Jammer, 50 “Just a description”, 1, 2

K Kinematic relativistic effects, 2 Kinematics, 49 Kinetic energy, xx, 40–42, 50

L Lagrangian mechanics, xii, 2 Lahee, ix

94 Larmor, 87 Length contraction, 2, 10, 12, 14, 17, 36 Lightlike, 49 LIGO, xxii, 68, 72, 74 Local becomings, 5 Local time, 8 Longitudinal mass, 46 Loop quantum gravity, viii Lorentz, xvii, 8, 12, 87 Lorentz transformations, 7, 12

M Mach, 27, 32, 33, 39 Mach’s sterile scientific philosophy, 53 Major open question in spacetime physics, 62 Many spaces, 10 Mass, 35, 60 inertial, xx, 36, 39, 42, 61 longitudinal, 46 passive gravitational, xx, 36, 39, 42, 61 relativistic, 45, 46, 48 transverse, 46 Massive stellar object, 77 Mathematical singularity, 80 Mathematical space, 2 Maxwell, xi Melissus, 81 Meta-theoretical issues, xi Metric tensor, 59 Michelson-Morley experiment, xvii, 8, 10, 12, 18, 22 Michelson-Morley interference experiments, 87 Minkowski, xi, xiii, xvii, 1, 2, 6, 7, 9, 11, 13, 22, 35, 61, 62, 69, 85, 87, 88 Minkowski’s arguments, xiii, xiv, xvii, 1, 2, 6, 19, 25 Minkowski’s double analysis, 8 Minkowski’s explanation of length contraction, vii, 12, 13 Minkowski’s flat spacetime physics , 57 Minkowski’s general argument, 10 Minkowski’s program of geometrizing physics, vii, xiii, xx, 4, 23, 27, 31, 34, 36, 38, 40, 42, 57 Minkowski’s program of regarding physics as geometry of the real spacetime, xxi, 58, 61, 62 Minkowski’s program of regarding physics as spacetime geometry, 25, 27, 57 Minkowski’s specific argument, 2, 12

Index Moment ‘now’, 82 wholy present, 82 Motion by inertia, 21 Motion by inertia is interaction-free, 59 “Motion mess”, 22–25, 27, 30 Muon experiment, 12 Mystical expression, 4 N Natural curvature, 60 Natural curvature of a geodesic worldtube, 60 Naturally curved due to the spacetime curvature but not deformed, 58 Nature, xii, xiii, xvii–xix, xxi, xxii, 55, 70 Nature does not care at all about our personal opinions, 6 Nature of general relativity, 56 Nature of gravitation, 62 Nature of motion, 29 Nature of spacetime, vii, 6, 56 Nerlich, 19, 22 Newton, xi, 25, 30, 50 Newton’s Principia, 31, 32 Newton’s first law, 42, 58 Newton’s second law, 50 Newtonian mechanics, xii, 2, 49, 69 twofold invariance, 69 No gravitational interaction in the physical world, 53 No gravitational interaction is involved in the fall of a body, 59 Nobel Committee, 79 Nobel Prize in Physics, 78 Non-Euclidean geometry of spacetime, xxii, 3, 40, 53, 56, 60, 62 Non-Euclidean spacetime, 58 Non-existent future, 81 Non-existent past, 81 Non-geodesic worldline, 29 Non-resistant motion, 59 Notion of spacetime, 85 O Olmedo, 76 One-way-velocity of light, 18 Open question in classical physics, 50 Open question in spacetime physics, 50 Oppenheimer, 77 Our feeling that time flows, 5

Index P Papapetrou, 77 Parmenides, 81 Passive gravitational mass, xx, 36, 39, 42, 61, 62 Past, 56 Penrose, 79 Perceived flow of time, 81 Phenomenon of inertia, 34 Physical singularity, 79 Physical theories, 6 Physics as spacetime geometry, 27 Physics is radically different in a threedimensional and in a four-dimensional world, 3 Poincaré, xvii, 7, 85, 87 Point masses, viii Postulate of the absolute world, 10 Pre-relativistic thinking, 68 Predetermined probabilistic phenomena, 4 Present, 56 Presentism, 4 Preskill, 69 Principle of inertia, 32, 33 Principle of relativity, 9, 10 Probabilistic behaviour, 4 Probabilistic phenomena, 4 Probabilistic spacetime structure, 4 Proper mass, 49 Proper space, 25 Proper time, 49 Pseudo-Euclidean geometry, 11 Pseudo-tensor, xiii PSR 1913+16, 66, 68 Ptolemaic system, 28 Pulsar, 67 Q Quadruple scientific tragedy, 85 Quantize something that might not exist, 56 Quantum gravity, 56, 76 Quantum mechanics, xi, 4 Quantum physics, viii, 4 R Real four-dimensional objects, 12, 42 Real four-dimensional world, 7, 8, 56 Real spacetime, 4 Reality of spacetime, 11, 12, 19, 36, 55 Regular black holes, 76 Relative acceleration, 28, 29 Relative motion, viii, 13

95 Relative simultaneity, 10 Relative uniform motion, 26 Relativistic effects, 10 Relativistic force, 50 Relativistic mass, vii, 45, 46, 49, 50 experimental confirmation, viii experimental fact, 50 role of in physics, viii Relativity of simultaneity, 12, 17, 86 Relativity postulate, 10 Relativity principle, 72 Relativization of existence, 5 Rest mass, 46, 49, 50 Ricci curvature tensor, 59 Ridgway “Camouflage”, 1 Riemann curvature tensor, 56 Rockower, 50 Rod everyday concept of, 2 worldtube of, 2 Role of mathematics in physics, 7

S Scalar curvature, 59 Schwarzschild, 75 Schwarzschild radius, 79 Schwarzschild singularity, 76 Schwarzschild solution, xxiii, 75–77 Schwarzschild sphere, 77 Science does not advance by ignoring arguments, xiii, 83 Second aspect of inertia, 34 Self-forces, xix, 36, 42 Shut up and calculate, xiv Singh, 76 Singularity, 75, 76, 79 physical, 79 Source of confusion and misconceptions, 2 Source of gravitational waves, 74 “Space and Time”, 8, 22, 69 Spacelike, 49 Spacetime, 1, 81, 83 “just a description”, xii Spacetime curvature, 60, 61, 70 Spacetime geometry, 25 Spacetime language, 67, 71 Spacetime physics, xi, xvii, xix, xx, 1, 2, 4, 12, 17, 38, 42, 45, 50, 54, 78, 85 Spacetime singularities, 76 Spacetime structure of the world, xi, 53, 81 Spacetime view of the world, 19

96 Special relativity, xvii, 22, 37, 54, 85 Speed of light, 10 Static deformation, 61 Static resistance, xix, 36, 42 Status of acceleration, 27 Status of relativistic mass, 45 Status of spacetime, 2–5, 55 Stefanov, 19 Sterility of Poincaré’s scientific philosophy, xii, 7 (Still unknown) properties of spacetime, 60 Strange world, 56 Stress-energy tensor, 59 Strong interactions, xix, 36, 42 Strong tidal effect, viii Stubborn illusion, 81 Stubbornly persistent illusion, 56 Super-compact stellar objects, 75 Supermassive compact object, 79

T Taylor, xx, 49, 66, 67 The actual open question in gravitational physics, 4 The art of doing physicsxi–xiii, 22 “The darkest place is under the lantern”, 21 The greatest intellectual achievement in fundamental physics, 56, 62 Theory of quantum gravity, viii Theory of relativity, xi, xviii, 46, 81 The physics of a three-dimensional world radically different from the physics of a four-dimensional world, xiii This cannot be because it cannot be neither science nor common sense, 6 Thought experiments, xiii, 31, 37, 57 Three-dimensional body, 13, 26 Three-dimensional existence, 55 Three-dimensional language, 9, 27, 68, 72 Three-dimensional object, 2, 12, 13 Three-dimensional physics, 4 Three-dimensional spaces, 13, 16 Three-dimensional thinking, 72 Three-dimensional world, 2, 3, 5, 6, 11, 13, 17 Time

Index coordinate, 49 past, present and future, 83 proper, 49 Time dilation, 10, 12, 17, 36, 49 Time flow, 5 Timelike, 49 Timelike worldline, 23, 26 Transverse mass, 46 True reality, 81 Twin paradox, 18, 36

U Ultimate judge, 25, 49 Ultimate judge in physics, 7 Uniform motion, 22 Uniform motion needs no mover, 25

V Velocity of light, 76 Velocity-dependent mass, 46 Virus of relativistic mass, xx, 45

W Wavelike curvature of spacetime, 72 Wavelike spacetime curvature, 74 Weak gravitational waves, viii Weak interactions, 36, 42 Weisskopf, 69 Weyl, 5 Wheeler, xx, 49, 75 Wheeler’s first moral principle, xiv, 66 Worldline deformation of, 27 Worldlines, 5, 24, 25, 27, 28 Worldtube, 2, 3, 11, 13, 25, 35, 42, 59, 60

X Xenophanes, 81

Z Zeno, 81, 82 Zeno’s arguments, 29, 82