Time-Warping & the Geometry of Space+Time: Newton-Einstein GeneralRelativity

The interaction of space+time with matter+energy: A prelude to general relativity

Copyright Avi Rabinowitz 2004 Air1@nyu.edu

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A Textbook for Undergraduate Physics Majors and others

WATCH THE ACCOMPANYING VIDEOS: "INTRODUCTORY GENERAL RELATIVITY: EINSTEIN'S BEAUTIFUL& MAGICAL RECASTING OF NEWTONIAN GRAVITY": FOR BEGINNERS & SCIENCE/PHYSICS STUDENTS

Preface

The approach taken in this book as per Einstein & Cartan, Wheeler & Feynman: When one develops the some of the deeper aspects underlying Newtonian gravity one arrives first at a beautiful geometric theory of Newtonian gravity, and then eventually at general relativity (GR). The geometric richness inherent in Newtonian gravity theory was however appreciated only after Einstein developed GR.

Elie Cartan and other mathematicians and physicists fleshed out the differential geometry underpinnings of Einstein’s theory, and then employed post-GR hindsight to develop a complete differential geometric theory of “Neo-Newtonian” gravity. Seeing that intermediate theory – between Newtonian gravity and Einstein’s GR - makes it clear that “many of the features of Einstein's general theory, often thought to be unique, actually have their counterpart in a suitably reformulated Newtonian general theory."[1]

Although this book contains some original material, it is essentially a pedagogical prelude to GR using the Neo-Newtonian formulation to introduce many concepts ordinarily associated with GR but in the familiar context of Newtonian dynamics and gravity, making them simpler to absorb. [As a result, though the book’s primary purpose is an introduction to GR, large parts of it are effectively a pedagogical introduction to a heuristic version of that intermediate theory.][2]

If one wishes to get to the newer GR theory, why start with the older Newtonian view, or the intermediate path of 'NeoNewtonian' theory? The answer is provided by John Wheeler[3] regarding his method of approaching GR, an approach he says was inspired by that of his student Richard Feynman:

“Why not go to GR directly? There are two reasons for pursuing the question of inertia [AR: and of gravity] ….within the framework of the older views ……First, this approach enables one to see the quasi-Newtonian antecedents ……. one sees Einstein’s GR in terms of continuity with the past, not a radical and inexplicable break. (AR: my italics & bold). Second, the more ways one has to look at an important part of physics, the deeper the understanding one acquires of that topic. ….. new views have a rich tissue of connection and correspondence with old views, a correspondence which helps one understand not only what in the new framework is new but what is old.”

Indeed, I hope that the reader of this book, much of which is inspired by the approach of Wheeler [and therefore of Feynman] to the subject[4], will see how close the Newtonian theory of inertia and gravity is to Einstein’s theory of general relativity, and so “sees Einstein’s GR in terms of continuity with the past, not a radical and inexplicable break”.

Hermann Bondi, in his popular-level book "Relativity and Common Sense" expressed a similar philosophy with regards to developing the ideas of special relaivity. The blurb on book cover claims that he "derives relativity from Newtonian ideas, rather than in opposition to them":

As a reviewer pointed out, "one cannot derive relativity from Newtonian mechanics. But Newtonian concepts can be used to advantage."

Bondi himself, writing in the book, is more cautious/precise than the publisher's blurb: "one begins to see the theory not as a revolution, but as a natural consequence and outgrowth of all the work that has been going on in physics since the days of Isaac Newton and Galileo."

We are of course presenting GR rather than SR, but the philosophy is similarly "to see the theory not as a revolution, but as a natural consequence and outgrowth of all the work that has been going on in physics since the days of Isaac Newton and Galileo."

Einstein’s early model of gravity, and the difference between this book and its companion volume: Two meanings of the word ‘introduction’ are relevant here. The introduction to a book is a prelude to the material in the book. If the book’s level is introductory but it covers all the theory at that level, it is also ‘an introduction’.

Einstein’s early attempt to describe gravity as spacetime curvature was really a theory of time warping, without any warping of space, and that model – usually now referred to as the ‘entwurf’ - is the GR-equivalent of the model developed in the bulk of the book. The focus is more of an introduction to a gravity theory intermediate between that of Newton and of Enstein, Specifically, a theory which highlights or is based on one of the essential concepts of GR – the time-warping which underlies the phenomena included in the category “Newtonian gravity”.

The full theory of GR and comparing this book to its companion volume: A time-warping theory is however only an introduction to GR. For completeness, the Einstein equations are presented (albeit in a heuristic manner) in the final chapter; in principle this convert the book into an introduction to GR. Therefore while the full theory of GR is not the subject of the bulk of the book, it is indeed its hidden subtext, as indicated in the subtitle by the word GRavity or General RelAtiVITY.

The companion volume[5] covers GR at an introductory level, but is nevertheless comprehensive. It starts out with a presentation of time-warping, then of spatial curvature, then SR, and then unites them all into GR, and develops the Einstein field equaitions and presents various solutions, but does all of this at a very introductory level. This book in contrast presents only the time-warping material, and does so at a more introductory level, so it is more introductory than that book, and also only covers the topics in the beginning parts of that book.

This book is very useful as a preparation for jumping into a full treatment of GR, as for example presented in the companion volume; readers of this book will see clearly the conundrums which occur when one takes Newtonian gravity theory seriously, and how the NeoNewtonian idea of a curved spacetime resolves them; as a result the more advanced interested students will likely be motivated to undertake the study of a fully-consistent curved spacetime theory, namely GR.

Physics students are often not exposed to theories of space and time in their undergraduate or even graduate careers, a lacuna which is almost incomprehensible, especially as our theory of space and time is based on theories of inertia and gravitation, which are topics dealt with even in high school treatments of physics.

The material in this text is not about esoteric philosophical issues but basic fundamental aspects of our existence: space, time, matter-energy, and all within the context of a quantitative but introductory text for physics undergraduates.

Although readers are assumed to have encountered some aspects of special relativity in their introductory physics course, we introduce SR in this book only at the very end. We also do not present a formal NeoNewtonian theory such as Cartan’s, and therefore until the final chapter where the Einstein equations are introduced any combination of space and time we speak of is a naïve one, and we symbolize this by calling it not “spacetime” as in SR or Cartan’s theory but rather “space+time” (abbreviated here as "s+t").

As the title indicates, this book can be seen equally as:

1 an introduction to the theory of s+t geometry;

2 a heuristic (intuitive but not mathematically rigorous) introduction to “Neo-Newtonian gravity theory”, specifically recasting of the Newtonian theories of inertia and of gravity in terms of the warping or curvature of time.

3 an introduction to some of the central aspects of Einstein's theory of gravity, ie General Relativity (GR), but within the much more familiar framework of Newtonian gravity, which we term Newton-Einstein gravity theory, and far more limited in its scope than GR.

Our inclusive use of the term "gravity": Long ago the term 'gravity' refered to only one phenomenon - that all things are pulled towards the ground; those that are on the ground feel this as 'weight' and objects not held by something fall to the ground immediately. (When things were more carefully measured it could be specified that the amount of weight and of speed decreased with distance from the Earth's center, and so the conception of gravity became a little more sophisticated, but it described the same phenomenon.) It was also found by the ancients that one could interpret the motions of the celestial lights as the orbits of bodies in the sky and there were theories relating the orbital period to various factors. No one imagined that these two phenomena - 'gravity' and 'orbital theory' - were related, nor was there any reason to relate them.

Then, based on work by Galileo, Kepler and others, Newton found that there's a higher-level theory which unites the two. That the two phenomena of 'falling' and orbit are one and the same was convincingly demonstrated by the fact that one equation, F = GMm/r² , includes both the Earthly gravity formula and the equations of celestial orbits. The higher level theory was also called gravity, or Newtonian gravity. Furthermore he found that the newer theory related to all matter in the universe - specifically that all particles attract each other - and was not just about the pull of large bodies, eg of the Earth on the objects on it, or about the pull of the sun on the planets. Though this theory has the same name - gravity - it was hardly the same as the old limited theory of gravity, which applied only to downward pull on objects near the surface of the Earth; thus the term 'gravity' became far more inclusive.

Later gravity was found (by Laplace etc) to also give rise to a theory of the galactic structure and was applied to a theory of cosmology, ie not only of how bodies orbit and how they fall, but also a theory of the structure of the universe as a whole. Later when 'the universe' was found to contain far more than one galaxy, gravity theory was again applied to formulate a newer theory of cosmology. However the same old word was used for these theories - 'gravity' came over time to mean something more and more general. For the pupose of this discussion here we'll use a capital letter here for this newer theory (including cosmology), calling it Gravity as opposed to gravity.

Later it was found that Gravity is an aspect of a higher theory, Einstein's theory of General Relativity (GR). This theory contains many new phenomenon which have nothing to do with the motion of objects (falling or orbiting) or cosmology, and is based on the concept of a curved spacetime, and so it is a far more general theory than 'Gravity'. Nevertheless GR professionals often use the term 'gravity' to mean GR, and MTW's classic tome is titled simply "Gravitation" rather than "Curved Spacetime" or "General Relativity".

For the moment let's call Einstein's GR theory 'GRavity'. Thus 'GRavity' contains within it the old theory of objects falling to the Earth, the old theory of orbits, the complete Newtonian Gravity theory which included the latter two as well as cosmology, and much more. [We can say that GRavity has various 'projections': gravity, cosmology, spatial curvature etc.]

This convention of terms - gravity, Gravity, GRavity - was just made up here for the purpose of clarifying the terminological distinction, and in the book I don't adopt it, I simply use the ordinary form, 'gravity', and expect that the reader is able to understand which is meant.

[The reader can keep in mind that although I write about 'gravity', the book will present mostly a theory of Gravity but with the intent of stretching its boundaries in order to arrive at the outlines of a theory of GRavity.]

New understanding vs new predictions; our model; neo-Newtonian gravity vs general relativity

The model of gravity theory presented in this book does not give rise to new predictions, but rather to deeper understandings. It is also a pointer to a deeper theory (GR) which does give rise to new predictions. Newtonian gravity theory contains various rich concepts which are ‘wasted’ in the theory. For example the existence of a potential allows one to solve problems more easily but is not necessary for solving them; similarly the equivalence principle (EP) is very nice but is not necessary to the theory. These two are however fundamental and indispensable features of the model we present.

Thus until the final chapter in which GR itself is briefly introduced, the model presented here aims to maximally exploit the deep and rich features of Newtonian gravity theory while remaining in a ‘neo-Newtonian’ context[6].

What Constitutes a ‘theory of space and time' as opposed to other physical theories?

When we attempt to create a theory we need to know what it is that we are expecting to describe, or to be able to predict. For example, a theory of gravity attempts to explain, or describe quantitatively, the known phenomena such as things falling or orbiting, and relates it to mass. A theory of electricity tries to explain electrical flow, lightning etc and relates it to charged particles.

What is it that we seek in ‘a theory of s+t’? Are we attempting to explain (or describe) known phenomena? We will see that what we are creating is a theory of the geometry of s+t and its relation to physics, or dynamics, more specifically to inertia and gravity, and the reader will better understand what THAT means by the time they’ve completed the book.

Neo-Newtonian Gravity vs General Relativity

General Relativity is a theory of the curvature of spacetime. One aspect of curved spacetime, in certain circumstances, produces what ordinary people call gravity. It turns out to be related to what could be called ‘time warping’. Another produces space curvature, which (to all but GR specialists) has absolutely no relation to ordinary gravity and was not suspected to exist by Newton (but which Gauss and Riemann postulated before Einstein). Yet another aspect of curved spacetime leads to space-expansion, and there are yet other aspects.

In GR one starts with a full theory of spacetime curvature and 'projects' it to produce these various phenomena, including gravity – see the downward arrows in the diagram below:

Although one cannot construct an object when seeing only its shadow, seeing many shadows cast by many lights can help. And what helps most of course is knowing in advance what the object itself looks like! J

In this book we proceed in the reverse direction from that in the diagram above (see the upward arrow): we study Newtonian gravity in depth and see heuristically – with knowledge aforehand that GR exists - why it makes sense to build it up into a theory of a warped or ‘curved’ time, and why this should be within the context of the geometry of a curved combination of space and time. Until the final chapter we do not produce a consistent theory of curved spacetime as is done in GR, and so we deal only with gravity and not with curved space and warped time and all the other purely-GR phenomena dealt with by GR.

We do so heuristically, and the result is a fascinating journey resulting in a beautiful (but heuristic) theory. Although the result is sufficient reward in of itself, it also serves as the motivation for subsequently, in the final brief chapter, tackling the more general and complete theory of spacetime curvature, GR. Hopefully this introduction will then motivate students to tackle a more rigorous and comprehensive treatment of GR in other texts, and knowing where all the math is leading they will be less likely to be deterred by its complexity[7].

Why this book avoids SR until the very end: Special Relativity would nowadays be a natural starting point for a geometry of space and time, and as that is the subject of this book one might certainly expect it to play a central role; however for several reasons I have deliberately avoided introducing it until the very end.

There are various ways to arrive at GR. One way is to generalize SR; another is to combine SR with Newtonian gravity, ie by constructing a covariant law of gravity satisfying the principles of SR; a third, lesser-known way is to develop the geometric aspects underlying Newtonian gravity; I avoid SR among other reasons in order to indicate to what degree one can indeed arrive at GR via this third path.[An additional consideration is offered by Havas at the conclusion of his article.]

In this text, SR is left aside except for the final chapter, though not to slight it. On the contrary, acquiring the basic understanding of s+t geometry the book imparts, and seeing what SR can do to further SR develop towards a theory of GR, can provide a motivation for delving into a more rigorous study of SR. [There are of course various classic and excellent treatments of SR, of SR-based NeoNewtonian theory, and SR-based introductions to GR.]

After SR is introduced in the final chapter, readers will see why the idea of a covariant theory based on SR appears as the natural way to a consistent and more elegant approach to gravity and spacetime rather than as some ad-hoc new theory or symmetry principle – however beautiful - or as an outgrowth of electromagnetism.

The book is about spacetime curvature, but as long as SR is left outit is not automatically about “GR”: To some it may seem that the moment one introduces the topic of spacetime curvature, one is necessarily talking about GR, and so it may not be obvious that spacetime curvature as a topic belongs in a treatment of Newtonian gravity even if we call it 'NeoNewtonian' gravity. However, in actuality the conclusion that spacetime is curved is reached via the equivalence principle (EP) which is wholly a Newtonian topic, and so just as the EP is legitimate topic for a book on (Neo)Newtonian gravity, so is spacetime curvature. As pointed out for example in MTW [8], what distinguishes GR from this is the addition of SR: "The equivalence principle is not unique to Einstein's description of the facts of gravity. What is unique to Einstein is the combination of the equivalence principle and local Lorentz geometry."

Penrose[9] makes the same point, stating that Cartan-Newton is all really Newtonian physics, just a different interpretation of it - until one adds in the SR idea that spacetime is Minkowski rather than Euclidean:

"This shows how a concept of 'curvature' for space-time can be used to describe the action of gravitational fields. The possibility of using such a description follows ultimately from Galileo's insight (the principle of equivalence) and it allows us to eliminate gravitational 'force' by free fall. In fact nothing I have said so far requires us to go outside Newtonian theory. This new picture provides merely a reformulation of the theory. However the new physics does come in when we try to combine this picture with what we have learnt from Minkowski's description of special relativity – the geometry of space-time that we no know applies in the absence of gravity. The resulting combination is Einstein's general relativity."

Thus, we can get a theory close to GR from Newtonian gravity (though we need to add SR into the mix in order to actually obtain GR). In light of this, it is interesting to go as far as possible without SR [ie without adding in the requirement of the theory being 'local Lorentz'], leaving that step for the very end. Introducing SR earlier would confuse matters since it introduces many concepts which we wish to introduce via gravity instead. [For example, SR as developed via electromagnetism indicates that space and time must be united, but we can see this intuitively without SR via the method of this book. Also, SR introduces a metric, whereas I wish to show that the suggestion of the existence of a metric arises naturally via a deeper understanding of the gravitational potential.] SR also introduces other concepts which are not directly necessary for our purposes [eg the scale factor c and the negative signature].

Since it is beneficial to understand which aspects of the theory, and of GR, are possible to introduce via gravity and which by SR, we keep SR out of the picture until the very end.

Readers are expected to have been introduced to the rudiments of SR via their 'introduction to physics' course. However, to build GR from SR requires a deeper level of SR than most undergraduates possess. This provides another motivation for leaving SR aside until the end, and only employing it mostly heuristically even at that point.

How to use this book

· The book can be read via self-study, or used as a course text.

· A series of lectures based on the text were given by the author to the SPS at NYU; they were videotaped and are available onYoutube, and can provide the student or instructor with an excellent companion to the text.

Topics not included

Of course, as an introduction to GR in the sense mentioned above, we do not intend to be comprehensive, quite the reverse, and so there are no solutions of the Einstein equations, no black holes or gravitational waves etc. Many other texts do this quite well.

And though this book is intended to introduce GR via NeoNewtonian gravity theory, since it is already verging on the too-long, that which exists elsewhere on NeoNewtonian gravity was deliberately left out; this eliminates duplication and leaves room for the Neo-Newtonian material not treated at this level elsewhere. The book is unique, and in order to focus on the unique aspects it is not comprehensive. Included are only the essentials for arriving at the central purposes of the book. Only those topics which are suited to its methodology/approach are presented, and the cases are limited to spherical symmetry.

As examples of special and important topics of Newtonian theory which are not relevant or necessary for the text's purpose of introducing NeoNewtonian gravity, and are omitted: Newtonian black holes, volume conservation, Birkhoff's theorem in Newtonian gravity, and Newtonian cosmology (see more discussion below). These topics are dealt with elsewhere and are in any case better treated as topics in GR. Other texts and web sources have excellent treatments of topics in vector calculus and introductory differential geometry, as well as of relevant discussions of inertia, absolute space etc, and these were also not entirely crucial to our discussion and so were left out.

Why There is Virtually no Treatment of Cosmology in this Book

“Newtonian cosmology”, like Newton-Cartan gravity theory, was developed in hindsight after GR. It would fit perfectly into the frame of the material discussed in this book However, it is not included here for two main reasons:

a) As opposed to the basic Newton-Cartan or "NeoNewtonian theory which is not presently available in an introductory and pedagogical approach, Newtonian cosmology is presented very well in various articles and books on cosmology [See eg MTW, and the article by Tipler {American Journal of Physics 1996}].

b) Cosmology is in any case best presented from the perspective of GR – ie within the context of a curved spacetime (rather than as Newtonian or NeoNewtonian cosmology).

c) GR cosmology which parallels the approach taken in NeoNewtonian theory is indeed presented in the second chapter of the companion volume, where it is shown that cosmology is the flip side of the gravity theory presented in the previous chapter. Readers of the first two chapters of that book can see how both cosmology and Newtonian gravity fit in very tidily into the same framework, ie why cosmology is a theory of gravity, and why they are both theories of spacetime curvature.

Research training: a prescription for a heuristic approach

After reading this book students will see that Newtonian gravity has much unexpected similarity to GR, and that the special case of spherical symmetry allows for many of the similarities to become very pronounced.

The Einstein equations reduce to very simple equations in cases of high symmetry, and for spherical symmetry and certain coordinate choices, they reduce exactly to Newton’s equations of gavity [see discussion of “Minimal Coupling” in this book and in its companion volume]. In a different context Feynman said “I always feel that a simple result ought to be obtained in a simple way”[10]. Usually though, the simpler method presents itself only after the way has been blazed by a more complicated derivation.

Can it be possible to obtain a simple result for high symmetry directly? It would have taken super-human intelligence and insight to have seen - before the work of Einstein - the underlying level at which Newtonian gravity is similar to the ‘higher-level’ theory of GR, and to derive GR from it. However perhaps - as for example Bohr did with the hydrogen atom - a human intelligence, attempting to construct only the spherically symmetric solution of a 'higher-level' theory such as GR could have arrived at elements of it by deeply considering Newtonian or some NeoNewtonian gravity theory.

My hope is that familiarity with the unexpected relationship between Newtonian gravity and GR, and the exact correspondence for spherical symmetry, can help budding researchers in seeking greater depth in known theories and then extrapolating solutions of high symmetry in those theories to create prospective heuristic solutions and equations which are really the projections for high symmetry and specific coordinates of the equations of an as yet unknown higher-level theory

Why we avoid rigorous NeoNewtonian theory: What we develop is a quantitative but heuristic model, which we hope will strike the reader as beautiful. A complete and fully-consistent Neo-Newtonian theory is available in the literature (see note below), but is not presented here not only because it requires heavy mathematics, and because it includes SR, but for another eason. The end result is something like general relativity, but not only is it far less powerful and general, it is uglier and more difficult ['... horrendously complex': See MTW last section of ch 12: p 302 – 303]. No wonder that there is not much interest in it except among some experts. Thus we suffice with a heuristic version of the theory, which takes us from Newtonian gravity to the outlines of GR, and recommend to those with an interest in furthering their understanding of spacetime curvature that the appropriate address is GR rather than a more rigorous version of the heuristic Neo-Newtonian theory presented herein.

Note: In some way the non-SR version of the theory presented by the book is so natural in the Newtonian framework that had Newton seen it, perhaps it would not have seemed strange to him; and yet it is so close to GR that it seems almost natural to fantasize that some great genius would develop rigorous GR after seeing it. Indeed we can imagine that had Newton seen this book he would have been spurred to invent Riemannian mathematics along with calculus, and he would then have invented a type of GR; and had Riemann or Gauss - who lived after Newton and were familiar with his work - seen it they too would have immediately invented some form of GR [see eg Rindler's remarks quoted in the text (note: they are in the first edition of his text, and deleted in a later edition! And see his private communication to me)]. But of course these giants did not see the material presented here because it was developed only after Einstein's work, and readers should not misunderstand the crucial nature of his contribution - it is only in the retrospect of his insight that this Neo-Newtonian model can be constructed.

Quotations From the Pioneers of Quantum Physics

For those readers who will have encountered Quantum Mechanics before reading this book, it will be interesting to see what the pioneers of that field had to say about this one.

Quantum physics was constructed to a large degree by only a few physicists. The major contributions came from about ten, the best-known to students being Heisenberg, Schroedinger, Dirac, Pauli, Bohr, Born, Einstein, Weyl, and later Feynman. It is interesting – although not surprising - that most of these great physicists who contributed to the construction of quantum physics, although they did not really contribute to the construction of GR itself, did write textbooks and articles elucidating GR and investigating specific aspects of it.

I remember my own feeling of joyful surprise, and comfort, when I came across the GR writings of these familiar giants, and so in this book and in its companion volume, I employ quotes from, and refer to, books on GR written by Shroedinger, Dirac, Pauli, and Born, as well as Feynman and of course Einstein. [11],[12]

Their presentations are very insightful, and I hope that contemporary undergrads, to whom these names are also so familiar from quantum physics, will find these quotes interesting as well, and will perhaps be motivated at some point to look into their books on this topic.

Further Reading

There is no one book on an advanced technical topic in physics which obviates the need to read another. This book is an excellent way to get into the subject however it is not meant to be read instead of other excellent and more advanced treatments, but rather to provide a path making them enticing and accessible

Some notes scattered throughout the book will direct readers to specific sections of certain books which I believe may be accessible after the preparation of this book especially the extensive material in MTW, in Wheeler's more popular books, the few pages in Penrose's article and book, and some material in Trautman & Kopczynski, as well as specific paragraphs in some of the more advanced, highly technical articles (see Bibliography).

At various juntures in the book presenting a central concept, quotes are inserted from relevant sources presenting that idea, and I also included in an (incomplete/unedited) Appendix a brief analysis/critique of the treatment in some of those sources.

What is done in the companion volume that is related to material in this book: The material in this book is recast from Neo-Newtonian language into the language of GR in the first fifty pages of the author's 450 page "Introduction to GR" text. Although that text can be read independently of this book, beginning students who wish to study GR would be well served to first complete this book, and then coast through the first fifty pages of its companion volume. This would give them a beachead into the subject of GR as treated there.

Bibliography: The bibliography is in some sense part of the book; it presents sources by topic, and can be seen as an outline of the chronological development of the approach upon which the book is based. [Some bibliographic material is still being added; in its final version an alphabetically-ordered bibliography will be compiled as well.]

Our approach as a motivation for GR; preparation for study of a rigorous appraoch: Our NeoNewtonian or Newton-Cartan theory, the geometrical formulation of Newtonian gravity, is far more difficult and less elegant than general relativity. In general relativity, there is a spacetime metric and it determines all of spacetime’s geometrical structure whereas in Neo-Newtonian gravity one needs three separate geometric objects to specify the geometrical structure: the ordinary Euclidean (flat) metric of space, the metric of time (a ‘time function’), and a non-metric connection, ie a connection which is not determined by a metric. What we are doing here is not an attempt to formulate NeoNewtonian theory in a rigorous manner – this is done quite well in numerous articles (see references) – but rather to motivate the idea that the theory we ought to construct, which turns out to be GR, should have not only a non-trivial connection which will reduce to the gravitational acceleration in flat spacetime, as we showed earlier, but also a spacetime metric, and that the connection should therefore be a metic connection.

At that point it is obvious that SR is relevant to all this, since it is a spacetime theory with a metric. SR as is of course founded on the concept of invariance under a group (Lorentz) and treats time and space on the same footing. Thus the combination of our geometrical formulation of Newtonian gravity with SR makes a compelling case that one ought to ty to formulate a covariant 4-d spacetime curvature theory, based on the existence of some larger symmetry group, and which will include gravity, and thus to motivate what is done in GR.

This then allows students to pick up a textbook which begins “Spacetime is a 4-dimensional Riemann manifold endowed with a metric” and to underatand what they mean and why they are doing it. They will also understand why a constraint on the field equations is that they must reduce in the appropriate limit to Newton’s equation of gravity”. Furhermore, after seeing the parallel between free fall worldline deviation and the geodesic deviation on curved surfaces and undertandign the implications of the EP, it will not be a surprise that equations which are built on Riemann curvatures can be made to reduce to a theory of gravity.

[1] Havas [1964: p956 second column].

[2] The theory was developed by "the remarkable French mathematician Elie Cartan (1923)" Penrose [Note 19 p223 referring to p207]; also by Levi-Civita and Weyl. ... sometimes refered to as Cartan-Newton theory... is a "mathematical description of (a) reformulation of Newtonian theory" [Penrose loc cit]. Earman states: The label 'neo-Newtonian' was applied by Sklar (1976). The alternative appellation 'Galilean space-time is justified by…." J Earman "World Enough and Space-Time" P33: referring to Sklar: 1976: "Space, Time and Space-Time". ... or what we for our purposes call 'Neo-Newtonian' or 'Newton-Einstein' gravity theory.

[3] p388 of Ciufolini and Wheeler

[4] References to relevant material in Misner, Thorne & Wheeler, and in Wheeler's other books are provided in the text.

[5] “Introduction to GR: Warped Spacetime, Wormholes & Cosmology”. Although the present book on Newton-Einstein gravity & GR is in a way an introduction to the GR book, each stands independently of the other: they are in this sense companion volumes.

[6] "The theory of local inertial frames in Newtonian theory is quite powerful, and it reaches in a mysterious way much further than one would expect" P.S., private communcation.

"Neo'Newtonian": This term has been taken in the literature to mean a very specific theory, whereas here we mean it generically.

[7] The book’s companion volume, being an introduction to GR in the other sense, does not deal only with Newtonian gravity (time warping) but rather introduces each of the phenomena in the chart above in a separate chapter, showing the relation of ach to spaceitme curvature, and then unifies all the results into one over-arching theory. After that the Einstein equations are set up and solved exactly. Specifically, ii is shown that the individual phenomena previously discussed are arrived at via this one field equation, allowing the student to 'see both the forest and the trees'.

[8] Misner Thorne & Wheeler [MTW] first two sentences of Chapter 12: "Newtonian gravity in the language of curved spacetime.

[9] "The Emperor's new Mind":p207

[10] Feynman 11.4. Of course ‘simple’ needs to be defined: see eg [Misner] p302: “Nature likes theories that are simple when stated in coordinate-free geometric language”. AR: However, perhaps simplest physical sources/situations are simple even in non-covariant form?

[11] When I began to study GR it was something of an outcast or curiosity in physics research; now it is more mainstream, but only as a means of approaching more complicated cosmological models, field theories, and string theory.

[12] Conspicuously absent from the list of quantum physics heroes who also wrote about GR are Heisenberg and Bohr (ie they seem never to have written specifically on GR), as opposed to of course Einstein, who though one of these ten founders of quantum physics, almost single-handedly constructed GR.

See: Some Remarks on Dirac’s Contributions to General Relativity S. DESER Brandeis University Department of Physics Waltham, MA 02454, USA E-mail: deser@brandeis.edu : “ Dirac was a true Martian, a Hungarian (-in-law) one at that. Feynman and Schwinger, neither otherwise overly impressionable, regarded him with awe, and he was right near Einstein and Bohr on Landau’s famous logarithmic rankings. ….Let me begin by setting the historical stage. After its rapid initial successes, GR was very much a stepchild of theoretical physics research for three decades, until the early fifties. 2 Even then, the renaissance to which Dirac’s work belonged was primarily disconnected from the (indifferent or hostile) field and particle theory mainstream. Indeed, this separation was traditional. While GR was understood quickly after its discovery, neither Bohr nor Heisenberg, for example, ever ventured there, although the former did use the equivalence principle against Einstein in a famous debate on quantum mechanics at the 1927 Solvay Conference, and the latter was the first, in the late thirties, to understand why perturbative quantization of theories, with positive dimensional (self-)coupling constant would fail. Pauli of course started life writing a text on GR, but despite continued interest, he never really contributed to it at the “Pauli” level. In later years, Schrodinger did venture into the field with some brilliant pedagogical expositions, but alas mostly into the morass of “unification by nonsymmetic metric” that occupied Einstein’s own late years. Born explicitly wrote that once he understood GR, he vowed never to work on it. Thus (apart perhaps from Jordan and Klein), Dirac was unique among the creators of quantum mechanics to work seriously on GR.”

For a Table of Contents of the book, and more description, see: PREFACE-PRELUDE-NEONN-TIME-WARPING FROM NYU SITE.PDF