Where modern formulations of relatively theory use differentiable manifolds to space-time, Einstein simply used open sets of R 4 , following the then current methods of differential geometry. This fact aids resolution of a number of outstanding puzzles concerning Einstein's use of coordinate systems and covariance principles, including the claimed physical significance of covariance principles, their connection to relativity principles, Einstein's apparent confusion of coordinate systems and frames of reference, and his failure to distinguish active and passive transformations, especially in (...) the context of his hole and point-coincidence arguments. (shrink)

The clock hypothesis of relativity theory equates the proper time experienced by a point particle along a timelike curve with the length of that curve as determined by the metric. Is it possible to prove that particular types of clocks satisfy the clock hypothesis, thus genuinely measure proper time, at least approximately? Because most real clocks would be enormously complicated to study in this connection, focusing attention on an idealized light clock is attractive. The present paper extends and generalized partial (...) results along these lines with a theorem showing that, for any timelike curve in any spacetime, there is a light clock that measures the curve’s length as accurately and regularly as one wishes. (shrink)

Quantum gravity poses the problem of merging quantum mechanics and general relativity, the two great conceptual revolutions in the physics of the twentieth century. The loop and spinfoam approach, presented in this book, is one of the leading research programs in the field. The first part of the book discusses the reformulation of the basis of classical and quantum Hamiltonian physics required by general relativity. The second part covers the basic technical research directions. Appendices include a detailed history of the (...) subject of quantum gravity, hard-to-find mathematical material, and a discussion of some philosophical issues raised by the subject. This fascinating text is ideal for graduate students entering the field, as well as researchers already working in quantum gravity. It will also appeal to philosophers and other scholars interested in the nature of space and time. (shrink)

Substantivalists believe that spacetime and its parts are fundamental constituents of reality. Relationalists deny this, claiming that spacetime enjoys only a derivative existence. I begin by describing how the Galilean symmetries of Newtonian physics tell against both Newton's brand of substantivalism and the most obvious relationalist alternative. I then review the obvious substantivalist response to the problem, which is to ditch substantival space for substantival spacetime. The resulting position has many affinities with what are arguably the most natural interpretations of (...) special and general relativity. I move on to consider and reject two recent antisubstantivalist lines of thought. The interim conclusion is that the best argument for relationalism is an appeal to Ockham's razor. However, for this to be successful there must be genuine relationalist theories that share the theoretical virtues of their substantivalist rivals but without the additional ontological commitment. The bulk of the paper is therefore an investigation of various concrete relationalist proposals. I distinguish three options for the relationalist in the face of the success of Galilean invariant physics and trace how these generalise to relativistic physics. One of the options is particularly promising but, since its basic objects end up being spacetime points, this does not help the prospects of relationalism as traditionally conceived. I end with some reflections on the fate of substantivalism in the aftermath of the Hole Argument, concluding that we have as yet to be given good reasons to abandon the natural, substantivalist interpretation of current physics. (shrink)

It generally agreed that the requirement of formal general covariance is a condition of the well-formedness of a spacetime theory and not a restriction on its content. Physicists commonly take the substantive requirement of general covariance to mean that the laws exhibit diffeomorphism invariance and that this invariance is a gauge symmetry. This latter requirement does place restrictions on the content of a spacetime theory. The present paper explores the implications of these restrictions for interpreting the ideology and ontology of (...) classical general relativity theory and loop quantum gravity. (shrink)

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)

It is argued that Minkowski space-time cannot serve as the deep structure within a ``constructive'' version of the special theory of relativity, contrary to widespread opinion in the philosophical community.

This chapter is concerned with the representation of time and change in classical (i.e., non-quantum) physical theories. One of the main goals of the chapter is to attempt to clarify the nature and scope of the so-called problem of time: a knot of technical and interpretative problems that appear to stand in the way of attempts to quantize general relativity, and which have their roots in the general covariance of that theory. The most natural approach to these questions is via (...) a consideration of more clear cases. So much of the chapter is given over to a discussion of the representation of time and change in other, better understood theories, starting with the most straightforward cases and proceeding through a consideration of cases that lead up to the features of general relativity that are responsible for the problem of time. (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.

I argue that the best spacetime setting for Newtonian gravitation (NG) is the curved spacetime setting associated with geometrized Newtonian gravitation (GNG). Appreciation of the ‘Newtonian equivalence principle’ leads us to conclude that the gravitational field in NG itself is a gauge quantity, and that the freely falling frames are naturally identified with inertial frames. In this context, the spacetime structure of NG is represented not by the flat neo-Newtonian connection usually made explicit in formulations, but by the sum of (...) the flat connection and the gravitational field. 1 Introduction2 Newtonian Gravity: The Orthodox Approach3 Newtonian Gravity: Additional Symmetries4 Cosmological Considerations5 A Newtonian Equivalence Principle: Inertial Frames in Newtonian Gravitation6 Theory Equivalence?7 Conclusion. (shrink)

In publications in 1914 and 1918, Einstein claimed that his new theory of gravity somehow relativizes the rotation of a body with respect to the distant stars and the acceleration of the traveler with respect to the stay-at-home in the twin paradox. What he showed was that phenomena seen as inertial effects in a space-time coordinate system in which the non-accelerating body is at rest can be seen as a combination of inertial and gravitational effects in a space-time coordinate system (...) in which the accelerating body is at rest. Two different relativity principles play a role in these accounts: the relativity of non-uniform motion, in the weak sense that the laws of physics are the same in the two space-time coordinate systems involved; what Einstein in 1920 called the relativity of the gravitational field, the notion that there is a unified inertio-gravitational field that splits differently into inertial and gravitational components in different coordinate systems. I provide a detailed reconstruction of Einstein's rather sketchy accounts of the twins and the bucket and examine the role of these two relativity principles. I argue that we can hold on to but that is either false or trivial. (shrink)

The cosmological constant is back. Several lines of evidence point to the conclusion that either there is a positive cosmological constant or else the universe is filled with a strange form of matter (“quintessence”) that mimics some of the effects of a positive lambda. This paper investigates the implications of the former possibility. Two senses in which the cosmological constant can be a constant are distinguished: the capital Λ sense in which lambda is a universal constant on a par with (...) the charge of the electron, and the lower case λ sense in which lambda is a humble constant of integration. The latter interpretation has been touted as the means to a solution to various problems in physics. These claims are critically examined with an eye to discerning the implications for philosophy of science and foundations of physics. (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)

Einstein regarded as one of the triumphs of his 1915 theory of gravity - the general theory of relativity - that it vindicated the action-reaction principle, while Newtonian mechanics as well as his 1905 special theory of relativity supposedly violated it. In this paper we examine why Einstein came to emphasise this position several years after the development of general relativity. Several key considerations are relevant to the story: the connection Einstein originally saw between Mach's analysis of inertia and both (...) the equivalence principle and the principle of general covariance, the waning of Mach's influence owing to de Sitter's 1917 results, and Einstein's detailed correspondence with Moritz Schlick in 1920. (shrink)

We investigate the interplay and connections between symmetry properties of equations, the interpretation of coordinates, the construction of observables, and the existence of physical relativity principles in spacetime theories. Using the refined notion of an event as a "point-coincidence" between scalar fields that completely characterise a spacetime model, we also propose a natural generalisation of the relational local observables that does not require the existence of four everywhere invertible scalar fields. The collection of all point-coincidences forms in generic situations a (...) four-dimensional manifold, that is naturally identified with the physical spacetime. (shrink)

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

This is a contribution to a book on quantum gravity and philosophy. I discuss nature and origin of the problem of quantum gravity. I examine the knowledge that may guide us in addressing this problem, and the reliability of such knowledge. In particular, I discuss the subtle modification of the notions of space and time engendered by general relativity, and how these might merge into quantum theory. I also present some reflections on methodological questions, and on some general issues in (...) philosophy of science which are are raised by, or a relevant for, the research on quantum gravity. (shrink)

James L. Anderson analyzed the novelty of Einstein's theory of gravity as its lack of "absolute objects." Michael Friedman's related work 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 the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson's proscription of "irrelevant" variables, 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 so naturally lack a timelike 4-velocity, so diffeomorphic equivalence to (1,0,0,0) is spoiled. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson's definition, GTR itself has an absolute object (as Robert Geroch has observed recently): a change of variables to a conformal metric density and a scalar density shows that the latter is absolute. (shrink)

Intuitively, a classical field theory is background-in- dependent 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 background-independent 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. (shrink)

John Earman's recent proposal that a substantive version of general covariance consists in the requirement that diffeomorphism invariance be a gauge symmetry is critically assessed. I argue that such a principle does not serve to differentiate general relativity from pre-relativistic theories. A model-theoretic characterization of two formulations of specially-relativistic theories is suggested. Diffeomorphisms are symmetries of only one such style of formulation and, I argue, Earman's proposal does not provide a reason to deny diffeomorphisms the status of gauge transformations relative (...) to this formulation. Carlo Rovelli's distinction between "passive" and "active" diffeomorphism invariance is also clarified. (shrink)

Background independence is generally considered to be ‘the mark of distinction’ of general relativity. However, there is still confusion over exactly what background independence is and how, if at all, it serves to distinguish general relativity from other theories. There is also some confusion over the philosophical implications of background independence, stemming in part from the definitional problems. In this paper I attempt to make some headway on both issues. In each case I argue that a proper account of the (...) observables of such theories goes a long way in clarifying matters. Further, I argue, against common claims to the contrary, that the fact that these observables are relational has no bearing on the debate between substantivalists and relationalists, though I do think it recommends a structuralist ontology, as I shall endeavour to explain. (shrink)

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)

An elementary notion of gauge equivalence is introduced that does not require any Lagrangian or Hamiltonian apparatus. It is shown that in the special case of theories, such as general relativity, whose symmetries can be identified with spacetime diffeomorphisms this elementary notion has many of the same features as the usual notion. In particular, it performs well in the presence of asymptotic boundary conditions.

We cast the flat space theory of a scalar field in generally covariant form by introducing an auxiliary field $\lambda$. The resulting theory is couched in terms of an action integral $S$, and all the fields (the scalar, the spacetime metric, and $\lambda$) are dynamical in the sense of being varied freely in $S$. Conservation of energy-momentum emerges as a formal consequence of diffeomorphism invariance, in close analogy with the situation in ordinary general relativity.