Abstract
It has been claimed, recently, that the fact that all the non-gravitational fields are locally Poincaré invariant and that these invariances coincide, in a certain regime, with the symmetries of the spacetime metric is miraculous in general relativity (GR). In this paper I show that, in the context of GR, it is possible to account for these so-called miracles of relativity. The way to do so involves integrating the realisation that the gravitational field equations (the Einstein field equation in GR) impose constraints on the behaviour of matter in a novel interpretation of the equivalence principle, which dictates the determination of local inertial frames through gravitational interaction. This proposed explanation of the miracles can also deal with the problematic cases for attempts at explaining them in the context of the standard geometrical perspective on relativity theory.
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Notes
Read et al. (2018, p. 20).
A similar claim can be found in Pitts (2019).
It must be noted that the form of the Einstein field equation does not determine the signature of the metric. As Brown stresses when discussing the chronogeometrical significance of the metric, inspired by Anderson’s formulation of the equivalence principle (see Brown 2005, section 9.2.3), according to the form of the equations the signature of the metric is indeterminate. At the end of the paper I will have more to say on the significance of this fact.
See the development of this argument in Read et al. (2018, section 4.1)
Claims like this one, pertaining to the original formulation of the DA, might be too crude: the chronogeometrical significance of the metric must presuppose, beyond the SEP, the existence of stable rods and clocks. According to this, the structure recovered through the equivalence principle is what Read (2019) call theoretical spacetime, distinct from the operational spacetime that would presuppose the existence of stable rod and clocks.
The question and its answer here refer to theoretical reasons; without any doubt, the miracles view admits empirical reasons for the validity of SEP.
Further clarification about the relation of minimal coupling and SEP seems necessary. As it is implicit in the presentation of the different versions of SEP, the authors of Read et al. (2018) do not think that any version of SEP is incompatible with non-minimal coupling; in fact, they show in that paper that it is possible to have non-minimally coupled equations in GR, a theory for which some version of SEP holds. Nonetheless, even for these equations it is true that they recover their special relativistic form in neighbourhoods of every point in which Riemann tensor terms can be ignored. This condition can be called approximate minimal coupling and it is this what I claim to be equivalent to the form of SEP that is taken as equivalent to the statement of the miracles. It is obvious that the same effect can be achieved by the condition of minimal coupling simpliciter, but this would be unnecessarily strong. These claims also assume that the matter fields do only couple to the metric and its derivatives, usually referred as universal coupling. Whether it is legitimate to use the term minimal coupling to what this version of the DA takes as primitive is mainly a question of terminology and not essential for the core argument of the paper. The essential question is, whatever the principle, whether there is one justification for it or it is taken as a brute fact. I thank an anonymous reviewer for pressing me for clarifying this point.
An anonymous reviewer objects my use of modal terms, like contingent, to refer the fact that there can be theories for which some metric field might not have chronogemetric significance, matter laws might not have all the same symmetries or these might not coincide with the symmetries of the metric. Even if I agree that issues about modality might obscure the discussion, I find this an economical way to synthesise this idea. Necessary and contingent in this paper should be understood in this way, as referring to putative existence of theories that do or do not have those particular features.
Read et al. (2018, Section 6)
In 4.2 I will discuss some subtleties involved in these arguments.
See Knox (2013, p. 348)
According to Read and Menon’s distinction (Read 2019) mentioned above this would be theoretical (not operational) spacetime.
This can be understood as an idealisation of the notion of force free-body (a body for which interactions are turned off) which incorporates, first, the possibility of universal interactions like gravity and also a field theoretical formulation. Taking this into account, one could say that the equivalence principle provides the determination of inertial structure as that detected by the matter dynamics when it does not interact with other matter.
Knox (2013, p. 353)
In Section 5, dedicated to the interpretation of the principle, I will expand on the connection between the motivating idea of the equivalence principle and GR’s implementation.
I will argue in Section 5 that this is a mistake: it is precisely the link to the original reference to free bodies what gives physical content to EP. This translates in the idea of inertial frames being determined through gravitational field that can be found, arguably, in some versions of the principle that we will meet below.
Knox introduces the notion of locally normal frames as a localisation of the notion of normal and orthonormal frames defined in SR (see Knox 2013, p. 349 for details). The aim is to have a characterisation for frames that can play the role of inertial frames in GR. The basic idea is that for locally normal frames on a geodesic one can find a neighbourhood where the connection coefficients vanish and the metric takes the Minkowskian form.
See Jacobson and Mattingly (2001)
Read et al. (2018, p. 8)
See Bekenstein’s paper (2010) for the original formulation of the theory.
See Read (2019) for discussion of different versions of the GA.
This assuming that, as it seems to be assumed in this version of the DA, there is no other way of accounting for the spatio-temporal character of the metric in which the miracles are not presupposed. The Ehlers-Pirani-Schild construction (Ehlers et al. 1972) and the proposal developed in the rest of this paper can be taken as attempts at proving this posit wrong. Read(2018, footnote 55) acknowledges the first possibility, although he argues that it does not provide a complete explanation of the miracles.
Read et al. (2018, Section 8)
Claims like this one assume a general operational perspective under which the spatiotemporal character of the metric is acquired through the physical behaviour of rods and clocks made out of matter fields and a formalist concretion of the functionalist intuition (of which Knox 2013 is an example) to the effect that such behaviour of rods and clocks will be encoded on the symmetries of the equations of motion of matter fields. If something like this is assumed, the miracle claims would regard the spatiotemporal character of spacetime as a brute unexplained feature in the DA. For more on the relation between the operational perspective of the DA and its relations to Knox’ spacetime functionalism see Read (2019).
An anonymous reviewer points out that my use of the formula “gravity determining inertia” is highly questionable when applied to Einstein’s version of the principle, as he would not concede a fundamental status to any of the concepts involved. Even agreeing with this historical remark, and being aware of the differences between the two principles here presented, I still think that it is illustrative to introduce them here as examples of the explicit reference to gravity in the formulation of the principle, something that will be exploited in my use of the equivalence principle as part of the explanation of the miracles.
See Knox (2013). For a discussion on whether the metric field can be understood as a gravitational field in GR and a characterisation of geometrical and gravitational significance for mathematical objects, see Lehmkuhl (2008). According to Lehmkuhl’s analysis, it is possible to maintain that the metric has both gravitational and geometrical significance.
This is only a first expression of the principle. At this point, without further explanation, it can be seen as not substantially different from claims identifiable with some version of the GA (for instance, Read’s QGA presented in Read 2018) indicating that some field, the metric field, constrains the couplings of all the non-gravitational fields. This is right insofar as any explanation that intends to elude the miracles must refer to some common origin of the constraints, but it is not the full story. The rest of the story comes from making explicit the physical content of the principle in GR (and other theories) as implemented by the gravitational field equations. See below for further discussion.
Geodesic motion of test bodies can be derived from the conservation equation under certain conditions through different geodesic theorems (the Geroch-Jang being one of them). For test bodies made out of the conserved matter fields the results would establish that its worldtube contains a timelike geodesic. See Geroch and Jang (1975), Ehlers and Geroch (2004), Geroch and Weatherall (2017), Weatherall (2011c, 2012). More on the relevance of geodesic theorems for the explanation of the miracles below.
See Weatherall (2019) for a detailed derivation of this result.
Read (2018)
The qualification here is due to the fact that one could restrict further the symmetries of matter laws through the introduction of fixed fields that break local Poincaré invariance. This being so, the question relevant for the present approach is whether this is a problem for the claim of the miracles being explained in GR. The answer is that it is not so: a theory thus modified would not contain an explanation of the miracles but it would neither satisfy the version of the equivalence principle put forward in this paper. If we had a Lagrangian formulation for it, we would need a term in which the fixed field couples to the matter fields. This would violate the version of SEP that I present here as an explanation of the miracles, as it would not be true that just the gravitational field equations would determine the local inertial frames. One might think that the fixed field could perhaps be dynamically determined through equations that one might want to call gravitational; if this were the case, it would satisfy the principle as formulated here and allegedly explain the miracles, but the theory, if consistent, would not be GR anymore.
A Killing field λ with respect a metric gab is defined as a smooth vector field for which the Lie derivative of gab associated with it vanishes: £λgab = 0. The (local) flow maps determined by the Killing fields are isometries of the metric. See, for instance, Malament (2012b, p. 75)
In her 1918 seminal paper (Noether 2011), Noether proved two theorems and their converses. The general result connects the transformations of the dependent and independent variables that are invariances of an action integral with certain combinations of the Euler-Lagrange expressions. From this, she derives the two theorems: the first applies to finite groups of transformations (transformations that depend on constant parameters) and derive that certain combinations of the Euler-Lagrange expressions are divergences; assuming that such equations are satisfied, the divergences vanish and we obtain the conserved currents. The second, for transformations depending on arbitrary functions, provides further identities between the Euler-Lagrange expressions. Noether also proves the converse of these two theorems. The one that is of interest here is the converse of the first one, that would allow derivation of invariances of the action from the ten conserved currents that we have.
As an anonymous reviewer points out, this derivation of the invariance properties of the action presupposes that the equations of motion for the matter fields are satisfied. The reviewer wonders whether this is a legitimate move; my impression is that it is so insofar as we are interested in deriving constraints for equations of motion that can be derived from an action for which the defined energy-momentum tensor is conserved.
It must be stressed here that the derivation that supports this claim presupposes the non-existence of fixed fields to which the matter fields couple. See my footnote 39,
One might object here that talk of gravitational interaction in the context of GR, or relativistic theories in general, is not sufficiently clear for the claim to have a substantive content. I think that it is enough, for the purpose of this paper, the identification of the equivalence principle with the claim that objects with gravitational significance (in the sense defined by Lehmkuhl 2008) are determined by equations that might impose the constraints that provide chronogeometrical meaning to some of these objects.
In Section 7 I will have more to say about this last question.
For a condensed presentation of different metatheoretical frameworks proposed to compare GR to other spacetime theories, see Lehmkuhl (2017).
Read et al. (2018, p. 22)
This claim is based in consistency reasons: the theory has been designed for the matter fields to couple to a physical metric that is defined from the metric and these other gravitational fields, meaning that the results from the two paths must be compatible for the theory to be consistent. Direct calculation should reflect this fact.
In order to prove the conservation condition of the energy-momentum tensor with respect to the derivative operator compatible with \(\tilde {g}_{ab}\), one must only note that the matter Lagrangian for this theory has the same form as the one in GR where gab is replaced by \({\tilde {g}}_{ab}\) (for an explicit proof see Weatherall 2019). To go from \(\widetilde {\nabla }_{b} \widetilde {T}^{ab}=0\) to the local Poincaré invariance of non-gravitational laws, one must apply the same derivation presented above for the case of GR (Section 5.2), taking into account that \({\tilde {g}}_{ab}\) is also a Lorentzian metric.
In the introduction of their paper, Brans and Dicke declare their intention of formulating a theory of gravitation that is more satisfactory form the standpoint of Mach’s principle than GR. By this they mean that Mach’s idea of the local inertial effects having their origin in the relative accelerated motion of distant masses would be better implemented in their theory.
See the previous discussion in 5.2.
Here I follow the presentation of the theory given in Weinstein (1996).
By this I mean not the specific explanation of the miracles through the constraints imposed by the Einstein field equation, only available in GR, but the general idea of explaining the symmetries of laws by re-interpreting some of the principles that define the theory. This is the strategy alluded to at the end of the paper.
Read (2018) considers the possibility of taking into account the constraints imposed by the Einstein field equation, which is part of my proposal, and, as I discuss in the text, dismisses it too quickly. The dismissal is based on Weatherall’s discussion of the geodesic theorem in Weatherall(2017, 2019)
Weatherall (2019, pp. 12–13)
One might challenge the claim that this provides any progress with respect to the statement of the miracles. Is it not an assumption that matter enters the Einstein field equation as source in the form of Tab? Is not this another way of stating the miracles? It is true that the miracles are contained in this interpretation of the equivalence principle and that it can be seen as a mere restatement of the miracles, but my claim is that the derivation from the equivalence principle allows a richer perspective. It is not a question of reducing the number of assumptions, but rather of placing them in a different way. The miracles claim that the coincidence of symmetries of non-gravitational fields is unexplainable. The perspective defended in this paper claims that this can be derived, in GR (although not in other theories) from the equivalence principle properly interpreted. From this point of view, the claim will only be valid for certain kinds of non-gravitational fields: those that can be described as non-interacting and acting as sources of the gravitational field (this, by the way, can be seen as giving content to a certain notion of matter). The hidden assumption in this perspective would then be, if one considers that GR is correct, that all matter meets such conditions. Nonetheless, one can say that this is all the matter that the gravitational interaction, as described by GR, sees. My claim is that this provides a richer look than merely claiming the miracles. Related to this is the fact that principles, beyond stating facts, can be generalised and used as templates for different theories.
There is a long tradition, starting with Kant, of understanding some physical principles as neither analytical nor empirical and, in this sense, as constitutive or transcendental. Although related to my proposal, this view on principles is not essential for the core discussion in this paper. Friedman (2001) and Disalle (2006) are standard references for this view in the context of relativity theory. See Sus (2016, 2019) for a more recent discussion.
I borrow this term from Brown and Pooley (2001) to refer to any version of the principle that imposes certain features to the non-gravitational field equations without stating that this can be derived from the gravitational field equations.
References
Acuña, P. (2016). Minkowski spacetime and Lorentz invariance: The cart and the horse or two sides of a single coin? Studies in History and Philosophy of Modern Physics, 55, 1–12.
Balashov, I., & Janssen, M. (2003). Presentism and relativity. British Journal for the Philosophy of Science, 54, 327–346.
Bekenstein, J.D. (2010). Modified gravity as an alternative to dark matter. arXiv:1001.3876, [astro-ph.CO].
Brading, K. (2001). Symmetries, conservation laws, and Noether’s variational problem. D.Phil. Thesis. University of Oxford.
Brans, C., & Dicke, R.H. (1961). Mach’s principle and a relativistic theory of gravitation. Physical Review, 124(3), 925.
Brown, H.R. (2005). Physical relativity: space-time structure from a dynamical perspective. Oxford: Oxford University Press.
Brown, H.R., & Pooley, O. (2001). The origin of the spacetime metric: Bell’s ‘Lorentzian Pedagogy’, and its significance in general relativity. In C. Callender, & N. Huggett. (Eds.), Physics meets philosophy at the planck scale (pp. 256–276). Cambridge: Cambridge University Press.
Brown, H.R., & Pooley, O. (2006). Minkowski space-time: a glorious non-entity. In D. Dieks (Ed.), The ontology of spacetime (pp. 67–89). Amsterdam: Elsevier.
Brown, H., & Read, J. (2020). The dynamical approach to spacetime theories. In E. Knox, & A. Wilson (Eds.),The Routledge companion to philosophy of physics. Forthcoming. London: Routledge.
Disalle, R. (2006). Understanding spacetime: the philosophical development of physics from Newton to Einstein. Cambridge: Cambridge University Press.
Doughty, N. (1990). Lagrangian interaction: an introduction to relativistic symmetry in electrodynamics and gravitation. Westview Press.
Ehlers, J., & Geroch, R. (2004). Equation of motion of small bodies in relativity. Annals of Physics, 309, 232–236.
Einstein, A. (1918). Prinzipielles zur allgemainen Relativitätstheorie. Annalen der Physik. In M. Janssen, R. Schulmann, J. Illy, C. Lehner, & D.K. Buchwald (Eds.), The collected papers of Albert Einstein (pp. 37–44). Princeton: Princeton University Press.
Ehlers, J., Pirani, F.A.E., & Schild, A. (1972). Republication of: the geometry of free fall and light propagation. General Relativity and Gravitation, 1587-1609(2012), 44. https://doi.org/10.1007/s10714-012-1353-4.
Fletcher, S.C. (2020). Approximate Local Poincaré Spacetime Symmetry in General Relativity. In C. Beisbart, T. Sauer, & C. Wüthrich (Eds.), Thinking about space and time: 100 years of applying and interpreting general relativity, Einstein Studies series (Vol. 15). Basel: Birkähuser. Forthcoming.
Friedman, M. (2001). Dynamics of reason. Stanford: Csli Publications.
Frisch, M. (2011). Principle or constructive relativity. Studies in History and Philosophy of Modern Physics, 42, 176–183.
Geroch, R., & Jang, P.S. (1975). Motion of a body in general relativity. Journal of Mathematical Physics, 16(1), 65.
Geroch, R., & Weatherall, J.O. (2017). The motion of small bodies in space-time. arXiv:1707.04222 [gr-qc].
Jacobson, T., & Mattingly, D. (2001). Gravity with a dynamical preferred frame. Physical Review D, 64, 024028.
Janssen, M. (2002). COI Stories: Explanation and evidence in the history of science. Perspectives on Science, 10, 457–522.
Janssen, M. (2009). Drawing the line between kinematics and dynamics in special relativity. Studies in History and Philosophy of Modern Physics, 40, 25–52.
Knox, E. (2013). Effective spacetime geometry. Studies in History and Philosophy of Modern Physics., 44, 346–356. https://doi.org/10.1016/j.shpsb.2013.04.002.
Lehmkuhl, D. (2008). Is spacetime a gravitational field? In D. Dieks (Ed.), The ontology of Spacetime II. (pp. 83–110). Amsterdam: Elsevier.
Lehmkuhl, D. (2017). Introduction. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spacetime theories (pp. 1–12). Boston, MA: Birkhassuser.
Malament, D. (2012a). A remark about the “geodesic principle” in general relativity. In M. Frappier, D. Brown, & R. DiSalle (Eds.), Analysis and interpretation in the exact sciences. The Western Ontario Series in Philosophy of Science (Vol. 78). Dordrecht: Springer.
Malament, D. (2012b). Topics in the foundations of general relativity and newtonian gravitation theory. Chicago: University of Chicago Press.
Maudlin, T. (2012). Philosophy of physics Volume I: space and time. Princeton: Princeton University Press.
Myrvold, W. (2017). How could relativity be anything other than physical? Studies in History and Philosophy of Modern Physics. https://doi.org/10.1016/j.shpsb.2017.05.007.
Noether, E. (2011). Invariante Variationsprobleme, Nachrichten Von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, 1918, pp. 235–257. (English translation: Invariant Variational Problems in Kosmann-Schwarzbach, Y. (2011). The Noether Theorems). Springer.
Pauli, W. (1921). Relativitatstheorie, Encycl. Mathematischen Wistentchaften, vol. 5, part 2 (Translated [1958] as Theory of Relativity).
Pitts, B. (2019). Space-time Constructivismvs vs. Modal provincialism: Or, How Special Relativistic Theories Needn’t Show Minkowski Chrono- geometry. Studies in History and Philosophy of Modern Physics, 191–198.
Pooley, O. (2017). Background independence, diffeomorphism invariance, and the meaning of coordinates. In D. Lehmkuhl, G. Schiemann, & E. Scholz, (Eds.), Towards a theory of spacetime theories. Springer.
Read, J. (2019). On miracles and spacetime. Studies in History and Philosophy of Modern Physics, 65, 103–111. https://doi.org/10.1016/j.shpsb.2018.10.002.
Read, J. (2018). Explanation, geometry and conspiracy in relativity theory.
Read, J., Brown, H., & Lehmkuhl, D. (2018). Two miracles of general relativiy. Studies in History and Philosophy of Modern Physics, 64, 14–25. https://doi.org/10.1016/j.shpsb.2018.03.001.
Read, J. (2019). The limitations of inertial frame spacetime functionalism. Synthese. https://doi.org/10.1007/s11229-019-02299-2https://doi.org/10.1007/s11229-019-02299-2.
Sus, A. (2014). On the explanation of inertia. Journal for General Philosophy of Science, 45, 293–315.
Sus, A. (2016). CategorÌas, intuiciones y espacio-tiempo kantiano. Revista de Humanidades de ValparaÌso, 8, 223–249.
Sus, A. (2019). Explanation, analyticity and constitutive principles in spacetime theories. Studies in History and Philosophy of Modern Physics, 65, 15–24. https://doi.org/10.1016/j.shpsb.2018.08.002.
Sus, A. (2020). How to be a realist about Minkowski spacetime without believing in magical explanations-Theoria. An International Journal for Theory History and Foundations of Science, 35(2), 175–195. https://doi.org/10.1387/theoria.21065.
Weatherall, J.O. (2011c). On the status of the geodesic principle in Newtonian and relativistic physics. Studies in the History and Philosophy of Modern Physics, 42(4), 276–281.
Weatherall, J.O. (2012). A brief remark on energy conditions and the Geroch-Jang theorem. Foundations of Physics, 42(2), 209–214.
Weatherall, J.O. (2017). Inertial motion, explanation, and the foundations of classical spacetime theories. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spacetime theories (pp. 13–42). Boston, MA: Birkhassuser.
Weatherall, J.O. (2019). Conservation, inertia, and spacetime geometry. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 67, 144–159.
Weinstein, S. (1996). Strange couplings and Space-Time structure. Philosophy of Science, 63 (Proceedings), S63–S70.
Acknowledgments
I am grateful to two anonymous referees for very helpful comments and suggestions. Research for this article has been supported by the following projects: “Laws, explanation and realism in physical and biomedical sciences” (FFI2016-76799-P) and “Limits of quantum physics - formalism, interpretation, visualization and aesthetics” (FFI2016-77266-P), Ministerio de Economía y Competitividad (Spain).
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Sus, A. Relativity without miracles. Euro Jnl Phil Sci 11, 3 (2021). https://doi.org/10.1007/s13194-020-00311-y
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DOI: https://doi.org/10.1007/s13194-020-00311-y