Diffeomorphism invariance is sometimes taken to be a criterion of background independence. This claim is commonly accompanied by a second, that the genuine physical magnitudes (the ``observables'') of background-independent theories and those of background-dependent (non-diffeomorphism-invariant) theories are essentially different in nature. I argue against both claims. Background-dependent theories can be formulated in a diffeomorphism-invariant manner. This suggests that the nature of the physical magnitudes of relevantly analogous theories (one background free, the other background dependent) is essentially the same. The temptation (...) to think otherwise stems from a misunderstanding of the meaning of spacetime coordinates in background-dependent theories. (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)

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)

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)

Metaphysics of Quantum Gravity The metaphysics of quantum gravity explores metaphysical issues related to research programs in theoretical physics clustered under the term quantum gravity. These research programs aim at the formulation of a theory that reconciles the theory of general relativity with quantum theory. The goal is not necessarily to come up with a … Continue reading Metaphysics of Quantum Gravity →.

Background independence begins life as an informal property that a physical theory might have, often glossed as 'doesn't posit a fixed spacetime background'. Interest in trying to offer a precise account of background independence has been sparked by the pronouncements of several theorists working on quantum gravity that background independence embodies in some sense an essential discovery of the General Theory of Relativity, and a feature we should strive to carry forward to future physical theories. This paper has two goals. (...) The first is to investigate what a world must be like in order to be truly described by a background independent theory given extant accounts of background independence. The second is to argue that there are no non-empirical reasons to be more confident in theories that satisfy extant accounts of background independence than in theories that don't. The paper concludes by drawing a general moral about a way in which focussing primarily on mathematical formulations of our physical theories can adversely affect debates in the metaphysics of physics. (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 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 : a change of variables to a conformal metric density and a scalar density shows that the latter is absolute. (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)

Much discussion was inspired by the publication of Harvey Brown's book \textit{Physical Relativity} and the so-called dynamical approach to Special Relativity there advocated. At the center of the debate there is the question about the nature of the relation between spacetime and laws or, more specifically, between spacetime symmetries and the symmetries of laws. Originally, the relation was mainly assumed to be explanatory and the dispute expressed in terms of the arrow of explanation – whether it goes from spacetime (symmetries) (...) to (symmetries of) laws or vice-versa. Not everybody agreed with a setting that involves leaving ontology out. In a recent turn, the relation has been claimed to be analytical or definitional. In this paper I intend to examine critically this claim and propose a way to generally understand the relation between spacetime symmetries and symmetries of laws as deriving from constitutive principles. (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)

Analysis of Emmy Noether's 1918 theorems provides an illuminating method for testing the consequences of coordinate generality, and for exploring what else must be added to this requirement in order to give general covariance its far-reaching physical significance. The discussion takes us through Noether's first and second theorems, and then a third related theorem due originally to F. Klein. Contact will also be made with the contributions of, principally, J.L. Anderson, A. Trautman, P.A.M. Dirac, R. Torretti and the father of (...) the whole business, A. Einstein. (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)

Recently the neglected issue of the causal structure in the flat space-time approach to Einstein's theory of gravity has been substantially resolved. Consistency requires that the flat metric's null cone be respected by the null cone of the effective curved metric. While consistency is not automatic, thoughtful use of the naive gauge freedom resolves the problem. After briefly recapitulating how consistent causality is achieved, we consider the flat Robertson–Walker Big Bang model. The Big Bang singularity in the spatially flat Robertson–Walker (...) cosmological model is banished to past infinity in Minkowski space-time. A modified notion of singularity is proposed to fit the special relativistic approach, so that the Big Bang becomes nonsingular. (shrink)

I examine a property of theories called "background-independence" that Einsteinian gravitation is thought to exemplify. This concept has figured in the work of Rovelli (2001, 2004), Smolin (2006), Giulini (2007), and Belot (2011), among others. I propose and evaluate a few candidates for background-independence, and I show that there is something chimaerical about the concept. I argue, however, that there is a proposal that clarifies the feature of Einsteinian gravitation that motivates the concept.

According to conventional wisdom, local gauge symmetry is not a symmetry of nature, but an artifact of how our theories represent nature. But a study of the so-called theta-vacuum appears to refute this view. The ground state of a quantized non-Abelian Yang-Mills gauge theory is characterized by a real-valued, dimensionless parameter theta—a fundamental new constant of nature. The structure of this vacuum state is often said to arise from a degeneracy of the vacuum of the corresponding classical theory, which degeneracy (...) allegedly arises from the fact that “large” (but not “small”) local gauge transformations connect physically distinct states of zero field energy. If that is right, then some local gauge transformations do generate empirical symmetries. In defending conventional wisdom against this challenge I hope to clarify the meaning of empirical symmetry while deepening our understanding of gauge transformations. I distinguish empirical from theoretical symmetries. Using Galileo’s ship and Faraday’s cube as illustrations, I say when an empirical symmetry is implied by a theoretical symmetry. I explain how the theta-vacuum arises, and how “large” gauge transformations differ from “small” ones. I then present two analogies from elementary quantum mechanics. By applying my analysis of the relation between empirical and theoretical symmetries, I show which analogy faithfully portrays the character of the vacuum state of a classical non-Abelian Yang-Mills gauge theory. The upshot is that “large” as well as “small” gauge transformations are purely formal symmetries of non-Abelian Yang-Mills gauge theories, whether classical or quantized. It is still worth distinguishing between these kinds of symmetries. An analysis of gauge within the constrained-Hamiltonian formalism yields the result that “large” gauge transformations should not be classified as gauge transformations; indeed, nor should “global” gauge transformations. In a theory in which boundary conditions are modeled dynamically, “global” gauge transformations may be associated with physical symmetries, corresponding to translations of these extra dynamical variables. Such translations are symmetries if and only if charge is conserved. But it is hard to argue that these symmetries are empirical, and in any case they do not correspond to any constant phase change in a quantum state. (shrink)

The Anderson-Friedman absolute objects project is reviewed. The Jones-Geroch dust 4-velocity counterexample is resolved by eliminating irrelevant structure. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields and using a modified spinor formalism. 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)

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)

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)

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)