We present a streamlined axiom system of special relativity in firs-order logic. From this axiom system we ``derive'' an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

In this essay, I examine the curved spacetime formulation of Newtonian gravity known as Newton–Cartan gravity and compare it with flat spacetime formulations. Two versions of Newton–Cartan gravity can be identified in the physics literature—a ‘‘weak’’ version and a ‘‘strong’’ version. The strong version has a constrained Hamiltonian formulation and consequently a well-defined gauge structure, whereas the weak version does not (with some qualifications). Moreover, the strong version is best compared with the structure of what Earman (World enough and spacetime. (...) Cambridge: MIT Press) has dubbed Maxwellian spacetime. This suggests that there are also two versions of Newtonian gravity in flat spacetime—a ‘‘weak’’ version in Maxwellian spacetime, and a ‘‘strong’’ version in Neo-Newtonian spacetime. I conclude by indicating how these alternative formulations of Newtonian gravity impact the notion of empirical indistinguishability and the debate over scientific realism. r 2004 Elsevier Ltd. All rights reserved. (shrink)

The philosophical debate whether the epistemological and conceptual structure of science is better characterized as hierarchical or as holistic cannot be decideda priori. A case study on general relativity should help to clarify our representation of this section of physics. For this purpose Sneed's model-theoretic approach is used to reconstruct the structure of relativity. The proposed axiomatization of general relativity takes into account approximations and utilizes local models for a realistic view on the functioning of the theory. A central objective (...) of the paper is to give an explication of the approximative empirical claim of general relativity, which is designed to identify the empirical content of the theory in a weak hierarchical form. (shrink)

Intuitively, a classical field theory is background-independent if the structure required to make sense of its equations is itself subject to dynamical evolution, rather than being imposed ab initio. The aim of this paper is to provide an explication of this intuitive notion. background-independence is not a not formal property of theories: the question whether a theory is background-independent depends upon how the theory is interpreted. Under the approach proposed here, a theory is fully backgroundindependent relative to an interpretation if (...) each physical possibility corresponds to a distinct spacetime geometry; and it falls short of full background-independence to the extent that this condition fails. The notions of geometrization, physical possibility, and gauge equivalence play important roles. An elementary notion of gauge equivalence is employed which does not depend upon the Hamiltonian or Lagrangian frameworks and whose connection with interpretative issues is clear. (shrink)

There is a widespread impression that General Relativity, unlike Quantum Mechanics, is in no need of an interpretation. I present two reasons for thinking that this is a mistake. The first is the familiar hole argument. I argue that certain skeptical responses to this argument are too hasty in dismissing it as being irrelevant to the interpretative enterprise. My second reason is that interpretative questions about General Relativity are central to the search for a quantum theory of gravity. I illustrate (...) this claim by examining the interpretative consequences of a particular technical move in canonical quantum gravity. (shrink)

Comprehensive coverage of the special theory (frames of reference, Lorentz transformation, relativistic mechanics of mass points, more), the general theory ...

Radically reoriented presentation of Einstein's Special Theory and one of most valuable popular accounts available derives relativity from Newtonian ideas, ...

The existence of a definite tangent space structure (metric with Lorentzian signature) in the general theory of relativity is the consequence of a fundamental assumption concerning the local validity of special relativity. There is then at the heart of Einstein's theory of gravity an absolute element which depends essentially on a common feature of all the non-gravitational interactions in the world, and which has nothing to do with space-time curvature. Tentative implications of this point for the significance of the vacuum (...) solutions in general relativity, and for the issue of quantising gravity, are briefly examined. (shrink)

The purpose of this paper is to evaluate the `Lorentzian Pedagogy' defended by J.S. Bell in his essay ``How to teach special relativity'', and to explore its consistency with Einstein's thinking from 1905 to 1952. Some remarks are also made in this context on Weyl's philosophy of relativity and his 1918 gauge theory. Finally, it is argued that the Lorentzian pedagogy---which stresses the important connection between kinematics and dynamics---clarifies the role of rods and clocks in general relativity.

The ambition of this volume is twofold: to provide a comprehensive overview of the field and to serve as an indispensable reference work for anyone who wants to work in it. For example, any philosopher who hopes to make a contribution to the topic of the classical-quantum correspondence will have to begin by consulting Klaas Landsman’s chapter. The organization of this volume, as well as the choice of topics, is based on the conviction that the important problems in the philosophy (...) of physics arise from studying the foundations of the fundamental theories of physics. It follows that there is no sharp line to be drawn between philosophy of physics and physics itself. Some of the best work in the philosophy of physics is being done by physicists, as witnessed by the fact that several of the contributors to the volume are theoretical physicists: viz., Ellis, Emch, Harvey, Landsman, Rovelli, ‘t Hooft, the last of whom is a Nobel laureate. Key features - Definitive discussions of the philosophical implications of modern physics - Masterly expositions of the fundamental theories of modern physics - Covers all three main pillars of modern physics: relativity theory, quantum theory, and thermal physics - Covers the new sciences grown from these theories: for example, cosmology from relativity theory; and quantum information and quantum computing, from quantum theory - Contains special Chapters that address crucial topics that arise in several different theories, such as symmetry and determinism - Written by very distinguished theoretical physicists, including a Nobel Laureate, as well as by philosophers - Definitive discussions of the philosophical implications of modern physics - Masterly expositions of the fundamental theories of modern physics - Covers all three main pillars of modern physics: relativity theory, quantum theory, and thermal physics - Covers the new sciences that have grown from these theories: for example, cosmology from relativity theory; and quantum information and quantum computing, from quantum theory - Contains special Chapters that address crucial topics that arise in several different theories, such as symmetry and determinism - Written by very distinguished theoretical physicists, including a Nobel Laureate, as well as by philosophers. (shrink)

Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing that it is a non-sequitur. I conclude that the Effective version (...) of the Church–Turing Thesis is unaffected by SAD computation. (shrink)

The author proposes to add another dichotomy to the list of essential tensions proposed by Professor Duda, namely beauty and ugliness. Physicists believe that only beautiful theories describe the world correctly, and that General Relativity is one of the most beautiful physical theories. The author explains why physicists regard this theory as beautiful.

This column is a milestone. It's the 100 th Alternate View column that I've written for Analog over a period of 16 years beginning in 1983. I was on a sabbatical in Berlin when Stan recruited me to write the column after Jerry Pournelle, my predecessor as AV columnist, decided to step down. The AV columns are a soapbox that was too attractive to pass up, and I've used them to promote an interst in science and to feed cutting-edge science (...) ideas, primarily in the areas of physics and astrophysics, to the readers and writers of science fiction. (shrink)

My interest in the physics of space station gravity developed because last year Vonda McIntyre was writing a book with a space station setting, and she asked my advice. The book, Barbary, is about a teenager who leaves Earth to live in a space station with spin-generated gravity. I helped Vonda in a very minor way by identifying the physical effects that the heroine would experience in that environment. What's it like to ride an elevator in a space station? How (...) would a ball game look if it were played there? If you woke up in a strange location, what simple tests would tell if you were in a rotating space station rather than at rest on the ground? And so on ... I found that there are some interesting side-effects of artificial gravity, perhaps well known to NASA experts but obscure to the rest of us. And I was surprised to find that some recent SF hasn't been too accurate in describing the space habitat environment. (shrink)

Curiously, in some ways gravity is also the strongest force in the universe. It always adds, never subtracts, and can build up until it overwhelms all other forces.. In normal stars gravity is balanced by heat energy from fusion reactions in the star's core. Eventually, however, the hydrogen and heavier elements fueling these reactions are used up, gravity takes over, and the star collapses in on itself. The result is a supernova explosion, which converts a sizable fraction of the star's (...) mass-energy to light, neutrinos, and gravity waves. (shrink)

Wormholes are shortcuts through space time, constructs of general relativity (GR) that appear to offer a physics foundation for faster than light travel and even for travel back in time. They first appeared in the physics literature in 1935, when Albert Einstein and his colleague Nathan Rosen discovered that implicit in general relativity is a tunnel like structure in the topology of space time connecting two separated regions. Einstein and Rosen were actually trying to explain fundamental particles like electrons and (...) proton. They suggested that if lines of electric flux were threaded through such a structure, the flux would be trapped and one end would appear to be an isolated positive charge and the other end would appear to be a negative charge. Later, however, general relativity was used to calculate the masses of such “particles” and it was realized that they would be have a mass of at least a few micrograms, far heavier than the mass of an electron or proton. (shrink)

Based on an analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of (...) the bundle of Lorentz metrics over spacetime. (shrink)

I argue that, contrary to the recent claims of physicists and philosophers of physics, general relativity requires no interpretation in any substantive sense of the term. I canvass the common reasons given in favor of the alleged need for an interpretation, including the difficulty in coming to grips with the physical significance of diffeomorphism invariance and of singular structure, and the problems faced in the search for a theory of quantum gravity. I find that none of them shows any defect (...) in our comprehension of general relativity as a physical theory. I conclude by comparing general relativity with quantum mechanics, a theory that manifestly does stand in need of an interpretation in an important sense. Although many aspects of the conceptual structure of general relativity remain poorly understood, it suffers no incoherence in its formulation as a physical theory that only an ‘interpretation’ could resolve. *Received November 2007; revised February 2009. †To contact the author, please write to: Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pittsburgh, PA 15260; e‐mail: erik@strangebeautiful.com . When science starts to be interpretive it is more unscientific even than mysticism. (D. H. Lawrence, “Self‐Protection”). (shrink)

In his general theory of relativity (GR) Einstein sought to generalize the special-relativistic equivalence of inertial frames to a principle according to which all frames of reference are equivalent. He claimed to have achieved this aim through the general covariance of the equations of GR. There is broad consensus among philosophers of relativity that Einstein was mistaken in this. That equations can be made to look the same in different frames certainly does not imply in general that such frames are (...) physically equivalent. We shall argue, however, that Einstein's position is tenable. The equivalence of arbitrary frames in GR should not be equated with relativity of arbitrary motion, though. There certainly are observable differences between reference frames in GR (differences in the way particles move and fields evolve). The core of our defense of Einstein's position will be to argue that such differences should be seen as fact-like rather than law-like in GR. By contrast, in classical mechanics and in special relativity (SR) the differences between inertial systems and accelerated systems have a law-like status. The fact-like character of the differences between frames in GR justifies regarding them as equivalent in the same sense as inertial frames in SR. (shrink)

The main aim of our paper is to show that interpretative issues belonging to classical General Relativity (GR) might be preliminary to a deeper understanding of conceptual problems stemming from on-going attempts at constructing a quantum theory of gravity. Among such interpretative issues, we focus on the meaning of general covariance and the related question of the identity of points, by basing our investigation on the Hamiltonian formulation of GR. In particular, we argue that the adoption of a peculiar gauge-fixing (...) within the canonical reduction of ADM metric gravity may yield a new solution to the debate between substantivalists and relationists, by suggesting a \emph{tertium quid} between these two age-old positions. Such a third position enables us to evaluate the controversial relationship between entity realism and structural realism in a well-defined case study. After having indicated the possible developments of this approach in Quantum Gravity, we discuss the structuralist and holistic features of the class of spacetime models that are used in the above mentioned canonical reduction. (shrink)

It is generally acknowledged that the requirement that the laws of a spacetime theory be covariant under a general coordinate transformation is a restriction on the form but not the content of the theory. The prevalent view in the physics community holds that the substantive version of general covariance – exhibited, for example, by Einstein’s general theory of relativity – consists in the requirement that diffeomorphism invariance is a gauge symmetry of the theory. This conception of general covariance is explained (...) and confronted by two challenges. One challenge claims, in effect, that substantive general covariance is not deserving of the name since, just as it is possible to rewrite any spacetime so that it satisfies formal general covariance, so it is also possible to rewrite the theory so that it satisfies the proffered version of substantive general covariance. The other challenge claims that the proffered version of substantive general covariance is not strong enough to guarantee the intended meaning of general covariance. Both challenges are discussed in terms of concrete examples. It is argued that both challenges fail but, at the same time, that they help to clarify what is at stake on the seemingly never ending dispute over the nature and status of general covariance. (shrink)

Except for a few brief periods, Einstein was uninterested in analysing the nature of the spacetime singularities that appeared in solutions to his gravitational field equations for general relativity. The existence of such monstrosities reinforced his conviction that general relativity was an incomplete theory which would be superseded by a singularity-free unified field theory. Nevertheless, on a number of occasions between 1916 and the end of his life, Einstein was forced to confront singularities. His reactions show a strange asymmetry: he (...) tended to be more disturbed by (what today we would call) merely apparent singularities and less disturbed by (what we would call) real singularities. Einstein had strong a priori ideas about what results a correct physical theory should deliver. In the process of searching through theoretical possibilities, he tended to push aside technical problems and jump over essential difficulties. Sometimes this method of working produced brilliant new ideas-such as the Einstein-Rosen bridge-and sometimes it lead him to miss important implications of his theory of gravity-such as gravitational collapse. (shrink)

The philosophical literature on time and change is fixated on the issue of whether the B-series account of change is adequate or whether real change requires Becoming of either the property-based variety of McTaggart's A-series or the non-property-based form embodied in C. D. Broad's idea of the piling up of successive layers of existence. For present purposes it is assumed that the B-series suffices to ground real change. But then it is noted that modern science in the guise of Einstein's (...) general theory poses a threat to real change by implying that none of the genuine physical magnitudes countenanced by the theory changes its value with time. The aims of this paper are to explain how this seemingly paradoxical conclusion arises and to assess the merits and demerits of possible reactions to it. (shrink)

It is argued that the main problem with "the problem of the direction of time" is to figure out what the problem is or is supposed to be. Towards this end, an attempt is made to disentangle and to classify some of the many issues which have been discussed under the label of 'the direction of time'. Secondly, some technical apparatus is introduced in the hope of producing a sharper formulation of the issues than they have received in the philosophical (...) literature. Finally, some tentative suggestions about the central issues are offered. In particular, it is suggested that entropy and irreversibility are much less crucial to the central issues than most philosophers would have us believe. This suggestion is not made because of any firm conviction of its correctness but rather because it helps to focus the discussion on some basic but long neglected assumptions which underlie traditional approaches. (shrink)

This paper presents a critical examination of claims advanced by several philosophers to the effect that 'time travel' represents a physical possibility and that the interpretation of certain actually observed phenomena in terms of 'time travel' is both legitimate and advantageous. It is argued that (a) no convincing motivation for the introduction of the time travel hypothesis has been presented; (b) no coherent and interesting sense of 'going backward in time' has been supplied which makes 'time travel' compatible with Special (...) Relativity; (c) even the conceptual possibility of 'time travel' is an unsettled and somewhat nebulous question. (shrink)

Some Philosophical Prehistory of General Relativity As history, my remarks will form rather a medley. If they can claim any sort of unity (apart from a ...

Inflationary cosmology won a large following on the basis of the claim that it solves various problems that beset the standard big bang model. We argue that these problems concern not the empirical adequacy of the standard model but rather the nature of the explanations it offers. Furthermore, inflationary cosmology has not been able to deliver on its proposed solutions without offering models which are increasingly complicated and contrived, which depart more and more from the standard model it was supposed (...) to improve upon, and which sever the connection between cosmology and particle physics that initially made the inflationary paradigm so attractive. Nevertheless, inflationary cosmology remains a promising research program, not least because it offers an explanation of the origin of the density perturbations that seeded the formation of galaxies and other cosmic structures. Tests of this explanation are underway and may settle the issue of whether inflation played an important role in the early universe. (shrink)

Spacetime substantivalism leads to a radical form of indeterminism within a very broad class of spacetime theories which include our best spacetime theory, general relativity. Extending an argument from Einstein, we show that spacetime substantivalists are committed to very many more distinct physical states than these theories' equations can determine, even with the most extensive boundary conditions.

The standard theory of computation excludes computations whose completion requires an infinite number of steps. Malament-Hogarth spacetimes admit observers whose pasts contain entire future-directed, timelike half-curves of infinite proper length. We investigate the physical properties of these spacetimes and ask whether they and other spacetimes allow the observer to know the outcome of a computation with infinitely many steps.

We address the question of whether it is possible to operate a time machine by manipulating matter and energy so as to manufacture closed timelike curves. This question has received a great deal of attention in the physics literature, with attempts to prove no-go theorems based on classical general relativity and various hybrid theories serving as steps along the way towards quantum gravity. Despite the effort put into these no-go theorems, there is no widely accepted definition of a time machine. (...) We explain the conundrum that must be faced in providing a satisfactory definition and propose a resolution. Roughly, we require that all extensions of the time machine region contain closed timelike curves; the actions of the time machine operator are then sufficiently “potent” to guarantee that closed timelike curves appear. We then review no-go theorems based on classical general relativity, semi-classical quantum gravity, quantum field theory on curved spacetime, and Euclidean quantum gravity. Our verdict on the question of our title is that no result of sufficient generality to underwrite a confident “yes” has been proven. Our review of the no-go results does, however, highlight several foundational problems at the intersection of general relativity and quantum physics that lend substance to the search for an answer. (shrink)

We discuss the possibility to build and operate a time machine, a device that produces closed timelike curves (CTCs). We specify the spacetime structure needed to implement a time machine and assess attempted no-go results against time machines in classical general relativity, semi-classical quantum gravity, quantum field theory on curved spacetime, and in Euclidean quantum gravity. Such no-go theorems for time machines would show that, under physically reasonable conditions, CTCs cannot develop in spacetimes initially free of these pathologies. Our review (...) indicates that an investigation of the prospects of achieving no-go results has not been entirely successful in establishing such generality. At the same time, the pursuit of chronology protection results has proved to be a fruitful way to probe the foundations of classical GTR and the interface between general relativity and quantum field theory. (shrink)

The aim of this book is to give an account of Einstein's work without introducing anything very technical in the way of mathematics, physics, or philosophy.

The remarkable connections between gravity and thermodynamics seem to imply that gravity is not fundamental but emergent, and in particular, as Verlinde suggested, gravity is probably an entropic force. In this paper, we will argue that the idea of gravity as an entropic force is debatable. It is shown that there is no convincing analogy between gravity and entropic force in Verlinde’s example. Neither holographic screen nor test particle satisfies all requirements for the existence of entropic force in a thermodynamics (...) system. As a result, there is no entropic force in the gravity system. Furthermore, we show that the entropy increase of the screen is not caused by its statistical tendency to increase entropy as required by the existence of entropic force, but in fact caused by gravity. Therefore, Verlinde’s argument for the entropic origin of gravity is problematic. In addition, we argue that the existence of a minimum size of spacetime, together with the Heisenberg uncertainty principle in quantum theory, may imply the fundamental existence of gravity as a geometric property of spacetime. This provides a further support for the conclusion that gravity is not an entropic force. (shrink)

We analyze the possible implications of spacetime discreteness for the special and general relativity and quantum theory. It is argued that the existence of a minimum size of spacetime may explain the invariance of the speed of light in special relativity and Einstein’s equivalence principle in general relativity. Moreover, the discreteness of spacetime may also result in the collapse of the wave function in quantum mechanics, which may provide a possible solution to the quantum measurement problem. These interesting results might (...) have some important implications for a complete theory of quantum gravity. (shrink)

General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its Riemannian geometry. I review this picture of `Geometrodynamics' and comment on various developments after Einstein.

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All change involves temporal variation of properties. There is change in the physical world only if genuine physical magnitudes take on different values at different times. I defend the possibility of change in a general relativistic world against two skeptical arguments recently presented by John Earman. Each argument imposes severe restrictions on what may count as a genuine physical magnitude in general relativity. These restrictions seem justified only as long as one ignores the fact that genuine change in a relativistic (...) world is frame-dependent. I argue on the contrary that there are genuine physical magnitudes whose values typically vary with the time of some frame, and that these include most familiar measurable quantities. Frame-dependent temporal variation in these magnitudes nevertheless supervenes on the unchanging values of more basic physical magnitudes in a general relativistic world. Basic magnitudes include those that realize an observer's occupation of a frame. Change is a significant and observable feature of a general relativistic world only because our situation in such a world naturally picks out a relevant class of frames, even if we lack the descriptive resources to say how they are realized by the values of basic underlying physical magnitudes. (shrink)

All change involves temporal variation of properties. There is change in the physical world only if genuine physical magnitudes take on different values at different times. We defend the possibility of change in a general relativistic world against two skeptical arguments recently presented by John Earman. Each argument imposes severe restrictions on what may count as a genuine physical magnitude in general relativity. These restrictions seem justified only as long as one ignores the fact that genuine change in a relativistic (...) world is frame-dependent. We argue on the contrary that there are genuine physical magnitudes whose values typically vary with the time of some frame, and that these include most familiar measurable quantities. Frame-dependent temporal variation in these magnitudes nevertheless supervenes on the unchanging values of more basic physical magnitudes in a general relativistic world. Basic magnitudes include those that realize an observer’s occupation of a frame. Change is a significant and observable feature of a general relativistic world only because our situation in such a world naturally picks out a relevant class of frames, even if we lack the descriptive resources to say how they are realized by the values of basic underlying physical magnitudes. (shrink)

The topics of gravitational field energy and energy-momentum conservation in General Relativity theory have been unjustly neglected by philosophers. If the gravitational field in space free of ordinary matter, as represented by the metric g ab itself, can be said to carry genuine energy and momentum, this is a powerful argument for adopting the substantivalist view of spacetime.This paper explores the standard textbook account of gravitational field energy and argues that (a) so-called stress-energy of the gravitational field is well-defined neither (...) locally nor globally; and (b) there is no general principle of energy-momentum conservation to be found in General Relativity. I discuss the nature and justification of the zero-divergence law for ordinary stress-energy, and its possible connection with the failure of General Relativity to realise Mach's principle. (shrink)

This paper reviews trends in rural/urban under-5 mortality differentials in Sub-Saharan Africa in historical perspective, with particular attention to the case of Kenya. The rural/urban mortality gap has narrowed within the last half-century, but while this was largely due to rapidly falling rural infant and childhood mortality over most of the period, in recent years it has been due primarily to a stalling and even upturn in urban under-5 mortality as urban economic and environmental conditions have sharply deteriorated in rapidly (...) growing cities. Policy attention and resources need to be directed to large urban areas to prevent further deterioration of urban mortality and associated health conditions. (shrink)

This is the chapter on general relativity for the Cambridge Companion to Einstein which I am co-editing with Christoph Lehner.

This paper will serve as the editorial note on Einstein's 1916 review article on general relativity in a planned volume with all of Einstein's papers in Annalen der Physik. It summarizes much of my other work on history of general relativity and draws heavily on the annotation of Einstein's writings and correspondence on general relativity for Vols. 4, 7, and 8 of the Einstein edition.

o an outsider, nothing might seem more ridiculous than the spectacle of grown men and women sitting around a conference table soberly discussing what would happen if a volume of the Encyclopedia Britannica were dropped down a black hole. Yet this very question lies at the heart of the "information paradox," a seeming contradiction to the laws of physics that is causing scientists to re-examine some of their most basic assumptions about how the universe is made.

General Relativity and the Standard Model often are touted as the most rigorously and extensively confirmed scientific hypotheses of all time. Nonetheless, these theories appear to have consequences that are inconsistent with evidence about phenomena for which, respectively, quantum effects and gravity matter. This paper suggests an explanation for why the theories are not disconfirmed by such evidence. The key to this explanation is an approach to scientific hypotheses that allows their actual content to differ from their apparent content. This (...) approach does not appeal to ceteris-paribus qualifiers or counterfactuals or similarity relations. And it helps to explain why some highly idealized hypotheses are not treated in the way that a thoroughly refuted theory is treated but instead as hypotheses with limited domains of applicability. (shrink)

I look at the ‘flavour-oscillation clocks’ proposed by D. V. Ahluwalia and two of his arguments suggesting that such clocks might behave in a way that threatens the geometricity of general relativity (GR). The first argument states that the behaviour of these clocks in the vicinity of a rotating gravitational source implies a non-geometrical element of gravity. I argue that the phenomenon is best seen as an instance of violation of the ‘clock hypothesis’ and therefore does not threaten the geometrical (...) nature of gravitation. Ahluwalia’s second argument, for the ‘incompleteness’ of general relativity, involves the idea that flavour-oscillation clocks can detect constant gravitational potentials. I argue that the purported ‘incompleteness-establishing’ result is in fact one that applies to all clocks. It is entirely derivable from general relativity, does not result in the observability of the potential, and is not at odds with any of general relativity’s foundations. (shrink)

It is well-known that Newton’s theory of gravity, commonly held to describe a gravitational force, can be recast in a geometrical form: Newton- Cartan theory. It is less well-known that general relativity, an apparently geometrical theory, can be reformulated in such a way that it resembles a force theory; teleparallel gravity does just this. This raises questions. One of these concerns theoretical underdetermination. I argue that these theories do not, in fact, represent cases of worrying underdetermination. On close examination, the (...) alternative formulations are best interpreted as postulating the same spacetime ontology. In accepting this, we see that the ontological commitments of these theories cannot be directly deduced from their mathematical form. The geometrical nature of a gravitational theory is not a straightforward consequence of anything internal to that theory as a theory of gravity. Rather, it essentially relies on the rest of nature (the nongravitational interactions) conspiring to choose the appropriate set of inertial frames. (shrink)

The paper shows a new example of nonuniqueness of the solutionto Newtonian equations of motion for infinite gravitational systems. Unlike otherexamples, the gravitational field presents no singularity, nor are the non-gravitational forcesintroduced in the model singular (in particular, there are no collisions). The result is also ofinterest because it points to an interesting limitation of the elementary (Newtonian) formulationof classical mechanics.

The Hamiltonian structure of General Relativity (GR), for both metric and tetrad gravity in a definite continuous family of space-times, is fully exploited in order to show that: i) the "Hole Argument" can be bypassed by means of a specific "physical individuation" of point-events of the space-time manifold M^4 in terms of the "autonomous degrees of freedom" of the vacuum gravitational field (Dirac observables), while the "Leibniz equivalence" is reduced to differences in the "non-inertial appearances" (connected to gauge variables) of (...) the same phenomena. ii) the chrono-geometric structure of a solution of Einstein equations for given, gauge-fixed, initial data (a "3-geometry" satisfying the relevant constraints on the Cauchy surface), can be interpreted as an "unfolding" in mathematical global time of a sequence of "achronal 3-spaces" characterized by "dynamically determined conventions" about distant simultaneity. This result stands out as an important conceptual difference with respect to the standard chrono-geometrical view of Special Relativity (SR) and allows, in a specific sense, for an "endurantist" interpretations of ordinary physical objects in GR. (shrink)

It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein's equation (or of the conservation principle that is, itself, a consequence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein's equation (or the conservation (...) principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). (shrink)

Harvey Brown believes it is crucially important that the "geodesic principle" in general relativity is an immediate consequence of Einstein's equation and, for this reason, has a different status within the theory than other basic principles regarding, for example, the behavior of light rays and clocks, and the speed with which energy can propagate. He takes the geodesic principle to be an essential element of general relativity itself, while the latter are better seen as contingent facts about the particular matter (...) fields we happen to encounter. The situation seems much less clear and clean to me. There certainly is a sense in which the geodesic principle can be recovered as a theorem in general relativity. But one needs more than Einstein's equation to drive the theorems in question. Other assumptions are needed. One needs to put more in if one is to get the geodesic principle out. My goal in this note is to make this claim precise, i.e., that other assumptions are needed. (shrink)

This survey article is divided into two parts. In the first (section 2), I give a brief account of the structure of classical relativity theory. In the second (section 3), I discuss three special topics: (i) the status of the relative simultaneity relation in the context of Minkowski spacetime; (ii) the ``geometrized" version of Newtonian gravitation theory (also known as Newton-Cartan theory); and (iii) the possibility of recovering the global geometric structure of spacetime from its ``causal structure".

David Albert claims that classical electromagnetic theory is not time reversal invariant. He acknowledges that all physics books say that it is, but claims they are ``simply wrong" because they rely on an incorrect account of how the time reversal operator acts on magnetic fields. On that account, electric fields are left intact by the operator, but magnetic fields are inverted. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. I argue, to the (...) contrary, that the inversion of magnetic fields makes good sense and is, in fact, forced by elementary geometric considerations. I also suggest a way of thinking about the time reversal invariance of classical electromagnetic theory -- one that makes use of the invariant (four-dimensional) formulation of the theory -- that makes no reference to magnetic fields at all. It is my hope that it will be of interest in its own right, Albert aside. It has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. (The only assumption one needs is temporal orientability.). (shrink)

An interesting difficulty arises if one tries to reconcile Reichenbach's views about "absolute" rotation in general relativity with his commitment to a "causal theory of space-time structure." This difficulty is made precise in the form of a simple theorem about relativistic space-time geometry.

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

We consider the following question within both Newtonian physics and relativity theory. "Given two point particles X and Y, if Y is rotating relative to X, does it follow that X is rotating relative to Y?" As it stands the question is ambiguous. We discuss one way to make it precise and show that, on that reading at least, the answers given by the two theories are radically different. The relation of relative orbital rotation turns out to be symmetric in (...) Newtonian physics, but not in relativity theory. (shrink)

Here, we examine hole-freeness - a condition sometimes imposed to rule out seemingly artificial spacetimes. We show that under existing definitions (and contrary to claims made in the literature) there exist inextendible, globally hyperbolic spacetimes which fail to be hole-free. We then propose an updated formulation of the condition which enables us to show the intended result. We conclude with a few general remarks on the strength of the definition and then formulate a precise question which may be interpreted as: (...) Are all physically reasonable spacetimes hole-free? (shrink)

Within the context of general relativity, we consider one definition of a “time machine” proposed by Earman, Smeenk, and Wüthrich. They conjecture that, under their definition, the class of time machine spacetimes is not empty. Here, we prove this conjecture. †To contact the author, please write to: Department of Philosophy, University of Washington, Box 353350, Seattle, WA 98195‐3350; e‐mail: manchak@uw.edu.

Here, we briefly review the notion of observational indistinguishability within the context of classical general relativity. We settle a conjecture given by Malament (1977) concerning the subject and then strengthen the result considerably. The upshot is this: There seems to be a robust sense in which the global structure of every cosmological model is underdetermined.

Malament-Hogarth spacetimes are the sort of models within general relativity that seem to allow for the possibility of supertasks. There are various ways in which these spacetimes might be considered physically problematic. Here, we examine these criticisms and investigate the prospect of escaping them.

Here we briefly review the concept of "prediction" within the context of classical relativity theory. We prove a theorem asserting that one may predict one's own future only in a closed universe. We then question whether prediction is possible at all (even in closed universes). We note that interest in prediction has stemmed from considering the epistemological predicament of the observer. We argue that the definitions of prediction found thus far in the literature do not fully appreciate this predicament. We (...) propose a more adequate alternative and show that, under this definition, prediction is essentially impossible in general relativity. (shrink)

Here we provide a proof that there exist closed timelike curves in Gödel spacetime with total acceleration less than 2π(9 + 6√3)^1/2. This answers a question posed by David Malament.

In a recent Analysis article, Quentin Smith argues that classical theism is inconsistent with certain consequences of Stephen Hawking's quantum cosmology.1 Although I am not a theist, it seems to me that Smith's argument fails to establish its conclusion. The purpose of this paper is to show what is wrong with Smith's argument. According to Smith, Hawking's cosmological theory includes what Smith calls "Hawking's wave function law." Hawking's wave function law (hereafter, "HL") apparently has, among its consequences, the following claim. (...) (1) The unconditional probability that a universe like this one - i.e., a universe with the metric hij and matter field Φ - should begin to exist is 95%.2 Smith then argues that the theist who accepts HL must also accept that the following sentence was once true.3.. (shrink)

Philosophers of physics should be more attentive to the role energy conditions play in General Relativity. I review the changing status of energy conditions for quantum fields-presently there are no singularity theorems for semiclassical General Relativity. So we must reevaluate how we understand the relationship between General Relativity, Quantum Field Theory, and singularities. Moreover, on our present understanding of what it is to be a physically reasonable field, the standard energy conditions are violated classically. Thus the singularity theorems are unavailable (...) for classical General Relativity. Our understanding of singularities in General Relativity turns on delicate issues of what it is to be a matter field-issues distinct from the content of the theory. (shrink)

In this paper I show that Einstein made essential use of aim-oriented empiricism in scientific practice in developing special and general relativity. I conclude by considering to what extent Einstein came explicitly to advocate aim-oriented empiricism in his later years.

In this three-part paper, my concern is to expound and defend a conception of science, close to Einstein's, which I call aim-oriented empiricism. I argue that aim-oriented empiricsim has the following virtues. (i) It solve the problem of induction; (ii) it provides decisive reasons for rejecting van Fraassen's brilliantly defended but intuitively implausible constructive empiricism; (iii) it solves the problem of verisimilitude, the problem of explicating what it can mean to speak of scientific progress given that science advances from one (...) false theory to another; (iv) it enables us to hold that appropriate scientific theories, even though false, can nevertheless legitimately be interpreted realistically, as providing us with genuine , even if only approximate, knowledge of unobservable physical entities; (v) it provies science with a rational, even though fallible and non-mechanical, method for the discovery of fundamental new theories in physics. In the third part of the paper I show that Einstein made essential use of aim-oriented empiricism in scientific practice in developing special and general relativity. I conclude by considering to what extent Einstein came explicitly to advocate aim-oriented empiricism in his later years. (shrink)

The objection that Einstein's principle of general covariance is not a relativity principle and has no physical content is reviewed. The principal escapes offered for Einstein's viewpoint are evaluated.

iinstein oered the prin™iple of gener—l ™ov—ri—n™e —s the fund—ment—l physi™—l prin™iple of his gener—l theory of rel—tivityD —nd —s responsi˜le for extending the prin™iple of rel—tivity to —™™eler—ted motionF „his view w—s disputed —lmost immedi—tely with the ™ounterE™l—im th—t the prin™iple w—s no rel—tivity prin™iple —nd w—s physi™—lly v—™uousF „he dis—greeE ment persists tod—yF „his —rti™le reviews the development of iinstein9s thought on gener—l ™ov—ri—n™eD its rel—tion to the found—tions of gener—l rel—tivity —nd the evolution of the ™ontinuing de˜—te (...) over his viewpointF.. (shrink)

Does quantum mechanics clash with the equivalence principle—and does it matter? Content Type Journal Article Pages 133-145 DOI 10.1007/s13194-010-0009-z Authors Elias Okon, Philosophy Department, UC San Diego, 9500 Gilman Dr., La Jolla CA, 92093, USA Craig Callender, Philosophy Department, UC San Diego, 9500 Gilman Dr., La Jolla CA, 92093, USA Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 1.

Einstein considered general covariance to characterize the novelty of his General Theory of Relativity (GTR), but Kretschmann thought it merely a formal feature that any theory could have. The claim that GTR is ``already parametrized'' suggests analyzing substantive general covariance as formal general covariance achieved without hiding preferred coordinates as scalar ``clock fields,'' much as Einstein construed general covariance as the lack of preferred coordinates. Physicists often install gauge symmetries artificially with additional fields, as in the transition from Proca's to (...) Stueckelberg's electromagnetism. Some post-positivist philosophers, due to realist sympathies, are committed to judging Stueckelberg's electromagnetism distinct from and inferior to Proca's. By contrast, physicists identify them, the differences being gauge-dependent and hence unreal. It is often useful to install gauge freedom in theories with broken gauge symmetries (second-class constraints) using a modified Batalin-Fradkin-Tyutin (BFT) procedure. Massive GTR, for which parametrization and a Lagrangian BFT-like procedure appear to coincide, mimics GTR's general covariance apart from telltale clock fields. A generalized procedure for installing artificial gauge freedom subsumes parametrization and BFT, while being more Lagrangian-friendly than BFT, leaving any primary constraints unchanged and using a non-BFT boundary condition. Artificial gauge freedom licenses a generalized Kretschmann objection. However, features of paradigm cases of artificial gauge freedom might help to demonstrate a principled distinction between substantive and merely formal gauge symmetry. (shrink)

The problem of finding a covariant expression for the distribution and conservation of gravitational energy-momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann's realization that there are infinitely many gravitational energy-momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors (...) of a given type, such as the Einstein pseudotensor, in _every_ coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau-Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether's theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many. (shrink)

The Anderson-Friedman absolute objects program has been a favorite analysis of the substantive general covariance that supposedly characterizes Einstein's General Theory of Relativity (GTR). Absolute objects are the same locally in all models (modulo gauge freedom). Substantive general covariance is the lack of absolute objects. Several counterexamples have been proposed, however, including the Jones-Geroch dust and Torretti constant curvature spaces counterexamples. The Jones-Geroch dust case, ostensibly a false positive, is resolved by noting that holes in the dust in some models (...) ensure that no physically relevant nonvanishing timelike vector field exists there, so no absolute object exists. The Torretti constant curvature spaces case, allegedly a false negative, is resolved by testing an irreducible piece of the metric, the conformal metric density of weight -2/3, for absoluteness; this geometric object is absolute. A new counterexample is proposed involving the orthonormal tetrad said to be necessary to couple spinors to a curved metric. The threat of finding an absolute object in GTR + spinors is overcome by the use of an alternative spinor formalism that takes a symmetric square root of the metric (with the help of the matrix diag(-1,1,1,1)), eliminating 6 of the 16 tetrad components as irrelevant. The importance of eliminating irrelevant structures, as Anderson emphasized, is clear. The importance of the choice of physical fields is also evident. A new counterexample due to Robert Geroch and Domenico Giulini, however, finds an absolute object in vacuum GTR itself, namely the scalar density $g$ given by the metric components' determinant. Thus either the definition of absoluteness or its use to analyze GTR's substantive general covariance is flawed. Anderson's belief that all absolute objects are nonvariational (that is, not varied in a suitable action principle) and vice versa is also falsified by the Geroch-Giulini counterexample. However, it remains plausible that all nonvariational fields are absolute, so adding nonvariationality as a necessary condition for absoluteness, as Hiskes once suggested, would likely leave no useful work to the Anderson-Friedman condition of sameness in all models. Simply having only variational fields in an action principle (suitably free of irrelevant fields) might be a satisfactory analysis of substantive general covariance, if one exists. This proposal also resembles the suggestion that GTR is "already parameterized," if one decides to parameterize theories by defining the nonvariational fields in terms of preferred coordinates called clock fields. More questions need to be addressed. Which fields should be tested for absoluteness: only primitive fields (which ones?), or all or some (which?) of their concomitants also? Geroch observes that some kinds of geometric objects, such as tangent vectors, scalar densities, and tangent vector densities of non-unit weight, satisfy the condition of sameness in all models if they merely fail to vanish. If these "susceptible" geometric objects can hardly help being absolute, to what degree are they, or the theories harboring them, responsible for this absoluteness? The answer to this question helps to determine the significance of the Geroch-Giulini counterexample. (shrink)

James L. Anderson analyzed the conceptual novelty of Einstein's theory of gravity as its lack of ``absolute objects.'' Michael Friedman's related concept of absolute objects has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4-velocity field of dust in cosmological models in Einstein's theory. Using Nathan Rosen's action principle, I complete Anna Maidens's argument that the Jones-Geroch problem is not solved by requiring that absolute objects not be varied. Recalling Anderson's proscription of (globally) (...) ``irrelevant'' variables that do no work (anywhere in any model), I generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman's intuitions and removes the Jones-Geroch counterexample: some regions of some models of gravity with dust are dust-free, and there is no good reason to have a timelike dust 4-velocity vector there. Eliminating the irrelevant timelike vctors keeps the dust 4-velocity from counting as absolute by spoiling its neighborhood-by-neighborhood diffeomorphic equivalence to (1,0,0,0). A more fundamental Gerochian timelike vector field presents itself in gravity with spinors in the standard orthonormal tetrad formalism, though eliminating irrelevant fields might solve this problem as well. (shrink)

In this paper I wish to make some headway on understanding what \emph{kind} of problem the ``problem of time'' is, and offer a possible resolution---or, rather, a new way of understanding an old resolution. The response I give is a variation on a theme of Rovelli's \emph{evolving constants of motion} strategy (more generally: correlation strategies). I argue that by giving correlation strategies a \emph{structuralist} basis, a number of objections to the standard account can be blunted. Moreover, I show that the (...) account I offer provides a suitable ontology for time (and space) in both classical and quantum canonical general relativity. (shrink)

The apparent underdetermination of the formalism of quantum field theory (QFT) as between a particle and a field interpretation is studied in this paper through a detour over the problem of unifying QFT with general relativity. All we have at present is a partial or approximate unification, QFT in non-Minkowskian spaces. The nature of this hybrid and the problem of its internal consistency are discussed. One of its most striking implications is that particles do not have an observer-independent existence. We (...) trace the ways in which physicists reacted to this at first highly implausible ontological consequence. We conclude that quantum fields rather than particles are after all the basic entities in QFT. (shrink)

Several decades ago, Wheeler and Misner presented a model of electric charge ("charge without charge") based on the topological trapping of electric field lines in wormholes. In this paper, which does not argue for or against the "charge without charge" concept, I describe some generalizations of this model which might serve as topological analogs of color charges and electroweak charges.

This paper sets out to show how Eddington's early twenties case for variational derivatives significantly bears witness to a steady and consistent shift in focus from a resolute striving for objectivity towards “selective subjectivism” and structuralism. While framing his so-called “Hamiltonian derivatives” along the lines of previously available variational methods allowing to derive gravitational field equations from an action principle, Eddington assigned them a theoretical function of his own devising in The Mathematical Theory of Relativity (1923). I make clear that (...) two stages should be marked out in Eddington's train of thought if the meaning of such variational derivatives is to be adequately assessed. As far as they were originally intended to embody the mind's collusion with nature by linking atomicity of matter with atomicity of action, variational derivatives were at first assigned a dual role requiring of them not only to express mind's craving for permanence but also to tune up mind's privileged pattern to “Nature's own idea”. Whereas at a later stage, as affine field theory would provide a framework for world-building, such “Hamiltonian differentiation” would grow out of tune through gauge-invariance and, by disregarding how mathematical theory might precisely come into contact with actual world, would be turned into a mere heuristic device for structural knowledge. (shrink)

Discussions of the intersection of general relativity and thephilosophy of religion rarely take place on the technical levelthat involves the details of the mathematical physics of generalrelativity. John Earman's discussion of theism and generalrelativity in his recent book on spacetime singularities is anexception to this tendency. By virtue of his technical expertise,Earman is able to introduce novel arguments into the debatebetween theists and atheists. In this paper, I state and examineEarman's arguments that it is rationally acceptable to believethat theism and (...) general relativity form a mutually consistent oreven mutually supportive pair. I conclude that each of hisarguments is unsound. (shrink)