Abstract
How does general relativity reduce, or explain the success of, special relativity? Answering this question, which Einstein took as a desideratum in the formulation of the former, is of acknowledged importance, yet there continues to be disagreement about how exactly it is best answered. I advocate here that part of the best answer involves showing that every relativistic spacetime has an approximate local Poincaré spacetime symmetry group, the spacetime symmetry group of Minkowski spacetime. This explains the application of Minkowski spacetime concepts that depend on, e.g., the conserved quantities that spacetime symmetries guarantee. I contrast this approach with another that instead invokes the strong equivalence principle, which focuses on the distinct notion of Poincaré invariance of dynamical equations. After showing with some examples that neither notion is necessary for the other, I use those examples to illuminate contrasting positions on the explanatory role of local approximate spacetime geometry, defending Torretti (1996) against criticisms by Brown and Pooley (2001). Finally, I acknowledge that establishing approximate local Poincaré spacetime symmetry is not a complete answer to the explanatory question with which I led, discussing in the concluding section further work that could lead to a complete answer. This includes specifying the circumstances under which matter fields in a general relativistic spacetime “behave” locally like those in Minkowski spacetime.
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Notes
- 1.
Rosaler (2019) declines from calling this sort of reduction-as-domain-subsumption an explanation, citing the deep controversies over accounts of explanation. I agree with him that the present sense of explanation is not well modeled, e.g., within the deductive-nomological or other standard philosophical accounts of scientific explanation, but so much the worse for those accounts.
- 2.
I also assume that M is connected, Hausdorff, and paracompact, and that the metric signature is (+ −−−). Throughout, roman sub- and superscripts denote abstract indices, while Greek and numerical ones denote components in a contextually specified basis. (See, e.g., Wald (1984, §2.4) for more on abstract index notation.) When an expression does not involve index contraction, I will often omit the indices to reduce notational clutter when no confusion should arise from doing so.
- 3.
I am implicitly using the identity map on M to compare the image of the pushforward with its argument. This is ultimately a convention: there is nothing mathematically or representationally privileged about the identity over any other diffeomorphism of the manifold, but choosing a different standard of comparison would yield an entirely representationally equivalent set of spacetime symmetries (Fletcher, 2020). This is all because diffeomorphisms are the isomorphisms in the category of smooth manifolds.
- 4.
Not all spacetime symmetries are such: consider so-called discrete symmetries such as reflections or time-reversal.
- 5.
This condition is equivalent with κ satisfying Killing’s equation, ∇(aκ b) = 0, where ∇ is the Levi-Civita covariant derivative operator compatible with the metric g. The Killing vector fields also form a Lie algebra, which will play a role in Sect. 3.
- 6.
That said, it does bring out a lacuna in the argument for the chronogeometric significance of the metric—the argument for why, according to the dynamical perspective on relativity theory (Brown, 1997; Brown and Pooley, 2001; Brown, 2005), the spacetime metric (perhaps only approximately) measures or surveys times and distances. The argument hinges on observing that “The symmetries of the dynamical laws governing non-gravitational fields in the appropriate local neighborhood …coincide with the symmetries of the dynamical metric field in this neighborhood” (Read et al., 2018, p. 19), with the latter understood as spacetime symmetries in the sense I have discussed (Read et al., 2018, p. 19n25). While the symmetry groups coincide, they act on different objects: the former acts on coordinates assignments to points and fields in a fixed region, while the latter acts on spacetime points and fields thereon. Why should this coincidence of two different types of objects deliver the interpretation of one?
- 7.
I am eliding some inconsequential technicalities regarding the relationships between local diffeomorphisms, local transformation groups (associated with a connected Lie group), and infinitesimal transformation groups (associated with a Lie algebra). For more on these, including references, see Hall (2004, Ch. 5.11).
- 8.
In this mode of presentation, I have assumed that the region in question is temporally and spatially orientable, for this is equivalent to the existence of a frame field. (See footnote 11 for definitions of these properties.) However, even if the region did not have those properties, one can always start with some smooth (inverse) Riemannian that can be decomposed locally into a frame field.
- 9.
Here I follow Read et al. (2018, p. 21) in construing one form to be simpler than another if it contains fewer terms. Although I am skeptical of the cogency of this notion—cf. my similar remarks about the hyperintensionality of minimal coupling in the concluding Sect. 5—insofar as it undergirds the definition and application of Poincaré invariance, which frames the question of its relation to Poincaré spacetime symmetry, I adopt it for those purposes without broader endorsement.
- 10.
- 11.
Recall that a spacetime is time-orientable when there exists a continuous classification of timelike vector fields on the spacetime into future- and past-directed; it is space-orientable when there exists a continuous classification of orthogonal spacelike vector field triads on the spacetime into left- and right-handed (Wald, 1984, p. 60).
- 12.
Astute readers may wonder just how the second claim is supposed to be a paraphrase of the first. This will play a role in the emerging dispute and its resolution, below.
- 13.
To make this last statement, Torretti (1996, p. 54) also invokes Einstein’s “Principle of Relativity,” that “The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or to the other of two Lorentz charts.”
- 14.
It is important to distinguish Minkowski spacetime, which, having no preferred origin, has at most the structure of an affine space, from the tangent space of a relativistic spacetime at a point—what Hall (2004, p. 147) calls Minkowski space—which does have a preferred origin and thus has the structure of a vector space. Charitably, then, by “isometric with Minkowski spacetime” Torretti means either “isometric with that of any point of Minkowski spacetime” or, what is essentially equivalent, “isometric with Minkowski space.”
- 15.
Actually, Brown (2005, p. 170) states that minimal coupling involves the non-appearance of terms depending on spacetime curvature in the dynamical equation, but Read et al. (2018, §3.2) rightly point out that this is in general not true, even accounting for the ambiguities of the application of the minimal coupling prescription as I have described it.
- 16.
Cf. the demand of Sonego and Faraoni (1993, p. 1185) for a real scalar field satisfying the homogeneous screened Poisson equation: “We require that the physical properties of wave propagation—rather than the form of the wave equation—should reduce locally to those valid in flat spacetime. More precisely, we require that the physical features of the solutions be locally the same in both cases.”
References
Brown HR (1997) On the role of special relativity in general relativity. International Studies in the Philosophy of Science 11(1):67–81
Brown HR (2005) Physical Relativity: Space-time structure from a dynamical perspective. Clarendon Press, Oxford
Brown HR, Pooley O (2001) The origin of the spacetime metric: Bell’s ‘Lorentzian pedagogy’ and its significance in general relativity. In: Callender C, Huggett N (eds) Physics meets philosophy at the Planck scale: Contemporary theories in quantum gravity, Cambridge University Press, Cambridge, pp 256–272
Carroll S, Lim E (2004) Lorentz-violating vector fields slow the universe down. Physical Review D 70:123525
Ehlers J (1986) On limit relations between, and approximative explanations of, physical theories. In: Barcan Marcus R, Dorn GJW, Weingartner P (eds) Logic, Methodology and Philosophy of Science VII, Elsevier, Amsterdam, pp 387–403
Einstein A (1923) The foundation of the general theory of relativity. In: The Principle of Relativity, Methuen, pp 109–164, trans. Perrett W, Jeffery GB
Einstein A (1956) The Meaning of Relativity, 5th edn. Princeton University Press, Princeton, trans. Adams EP, Strauss EG, Bargmann S
Feintzeig BH (2017) On theory construction in physics: Continuity from classical to quantum. Erkenntnis 82(6):1195–1210
Fletcher SC (2014) Similarity and spacetime: Studies in intertheoretic reduction and physical significance. PhD thesis, University of California, Irvine
Fletcher SC (2016) Similarity, topology, and physical significance in relativity theory. The British Journal for the Philosophy of Science 67(2):365–389
Fletcher SC (2018) Approximate symmetry of spacetime structure, unpublished manuscript
Fletcher SC (2020) On representational capacities, with an application to general relativity. Foundations of Physics 50(4):228–249
Fletcher SC, Manchak J, Schneider MD, Weatherall JO (2018) Would two dimensions be world enough for spacetime? Studies in History and Philosophy of Modern Physics 63:100–113
Hall GS (2004) Symmetry and Curvature Structure in General Relativity. World Scientific, Singapore
Hoffmann-Kolss V (2015) On a sufficient condition for hyperintensionality. Philosophical Quarterly 65(260):336–354
Jacobson T, Speranza AJ (2015) Variations on an aetherial theme. Physical Review D 92:044030
Janssen M (2014) “No success like failure…”: Einstein’s quest for general relativity, 1907–1920. In: Janssen M, Lehner C (eds) The Cambridge Companion to Einstein, Cambridge University Press, Cambridge, pp 167–227
Malament DB (2007) Classical relativity theory. In: Butterfield J, Earman J (eds) Philosophy of Physics, Handbook of the Philosophy of Science, vol A, Elsevier, Amsterdam, pp 229–274
Nickles T (1973) Two concepts of intertheoretic reduction. The Journal of Philosophy 70(7):181–201
Nolan D (2014) Hyperintensional metaphysics. Philosophical Studies 171(1):149–160
Norton JD (1984) How Einstein found his field equations: 1912–1915. Historical Studies in the Physical Sciences 14(2):253–316
Norton JD (1993) General covariance and the foundations of general relativity: eight decades of dispute. Reports on Progress in Physics 56:791–858
Pauli W (1958) Theory of Relativity. Pergamon, trans. Field G
Read J, Brown HR, Lehmkuhl D (2018) Two miracles of general relativity. Studies in History and Philosophy of Modern Physics 64:14–25
Renn J, Sauer T (1998) Heuristics and mathematical representation in Einstein’s search for a gravitational field equation. In: Goenner H, Renn J, Ritter J, Sauer T (eds) The Expanding Worlds of General Relativity, Einstein Studies, vol. 7, Birkhäuser, Boston, pp 87–125
Rosaler J (2019) Reduction as an a posteriori relation. The British Journal for the Philosophy of Science 70(1):269–299
Schrödinger E (1950) Space-time Structure. Cambridge University Press, Cambridge
Sonego S, Faraoni V (1993) Coupling to the curvature for a scalar field from the equivalence principle. Classical and Quantum Gravity 10(6):1185–1187
Torretti R (1996) Relativity and Geometry, Dover edn. Dover, Mineola, NY
Torretti R (1999) The Philosophy of Physics. Cambridge University Press, Cambridge
Wald RM (1984) General Relativity. University of Chicago Press, Chicago
Acknowledgements
Thanks to Dennis Lehmkuhl, one of whose questions to me during the September, 2017 “Thinking About Space and Time” conference in Bern spurred the idea of this paper. Thanks also to James Read, Chris Wüthrich, and an anonymous referee for their comments, and especially to Jim Weatherall for suggesting a way to formalize a part of the proof of the main theorem and catching a serious error in a previous version thereof. Part of the research leading to this essay was written with the support from a Marie Curie Fellowship (PIIF-GA-2013-628533).
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Fletcher, S.C. (2020). Approximate Local Poincaré Spacetime Symmetry in General Relativity. In: Beisbart, C., Sauer, T., Wüthrich, C. (eds) Thinking About Space and Time. Einstein Studies, vol 15. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-47782-0_12
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