Abstract
In General Relativity (GR), it has been claimed that inertia receives a dynamical explanation. This is in contrast to the situation in other theories, such as Special Relativity, because the geodesic principle of GR can be derived from Einstein’s field equations. The claim can be challenged in different ways, all of which question whether the status of inertia in GR is physically different from its status in previous spacetime theories. In this paper I state the original argument for the claim precisely, discuss the different objections to it and then propose a formulation that avoids the problems the original claim encounters. My conclusion is that one can say meaningfully that inertia is dynamically explained in GR. There are two senses in which the derivation of geodetic motion can be said to provide a (more) dynamical explanation of inertia in GR: it holds for any material test body that is a source of the gravitational field; and it is derivable without assuming inertial structures that are fixed independently of matter.
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Notes
Soon after the formulation of GR, it was noted that it is a consequence of the theory that the gravitational field equations constrain the dynamics of the matter fields. See, for instance, Weyl (1952).
For a brief review of the different approaches and difficulties associated with them see Geroch and Jang (1975).
I am aware that the use of the term source applied to the material part of EFE is very problematic. Strictly speaking, neither matter fields nor the stress-energy-momentum tensor are sources of the gravitational field in the same sense that charges can be said to be sources of the electromagnetic field. The dissimilarity is traceable to the non-linearity of EFE, which makes the same gravitational field also a source of itself. I use the term source here, for historical reasons, to mean, in general, non-gravitational fields (and in particular, material particles) that are partially responsible for the shape of the gravitational field.
For the Anderson–Friedman program to characterise the peculiarity of GR, see, for instance, Piits (2006).
For now, I am going to concentrate on relativistic theories, for which our previous Route 1 is available in the Lagrangian formulation (or something analogous if a Lagrangian is not available). The comparison between the GR and Newtonian gravity derivations is left for Sect. 5.
It might be useful to think of electromagnetic theory in this context. Maxwell’s equations are field equations for the electromagnetic field but do not constrain the motion of the sources (electrons, for instance) of the fields.
This will be further discussed in Sect. 6 under the label "‘universal coupling"’.
We must be careful here. Even if we thought that formal conditions (such as general covariance) could somehow be justifiably introduced, they would prove to be insufficient to play a substantive role in any attempted dynamical explanation.
It is interesting to introduce a consideration here regarding the possibility of non-minimal coupling. Even if non-minimal coupling does not, by itself, preclude the derivation of the response equations, the energy-momentum tensor for which they would hold would be a different one. This prompts a question about the physical interpretation of that tensor. In principle, in a special relativistic theory, the Hilbert energy-momentum tensor obtains its physical meaning from its connection to the symmetrical version of the canonical energy-momentum tensor. But such a connection is only available for Lagrangians with minimal coupling (see Leclerc 2006). So, allowing for Lagrangians with non-minimal coupling makes the response equations as general as in GR but it leaves the energy-momentum tensor without a natural interpretation.
In this section I will explore the implication of such an objection for the contrast between GR and SR; its effects on the comparison with Newtonian theory are considered in the next section.
Again, one could say that Lorentz covariance needs no further motivation than being taken as a property that matter fields happen to have. But in that case, the explanation loses generality by not being applicable to all matter fields.
Note that for other relativistic theories of gravity the previous scheme of analysis, namely comparing the status of the response equations, can be applied. If we do so, we will find that different justification of inertial structures plus a different scope of applicability of the response equations results from it; as a consequence of which inertia does not receive the same level of explanation in all of them.
Since the first version of the present paper, Weatherall (2011b) has published his own view with respect to the comparative status of inertial motion in geometrised Newtonian gravity and GR. There he argues that the weak energy condition needed in the Newtonian context is less demanding than SDEC for GR.
I stress here that Weatherall believes that this condition can be relaxed, in which case this is not the place where to look for a difference between GR and geometrised Newtonian theory. The main difference comes from what I say below.
Weatherall regards this difference between GR and the Newtonian case as irrelevant. His first reason for this view is that in GR, even assuming that the response equations hold, one needs an extra assumption regarding the nature of test bodies. I cannot see the strength of this concern if one understands test matter as a limiting case of ordinary matter. His second reason is that the conservation condition is better seen as a meta-principle both in Newtonian theory and relativistic theories of matter. I agree with him with respect to it playing such a role historically; but this by itself does not undermine my main argument: even if Einstein took energy conservation as a requisite for his sought-after theory, it is physically meaningful that uniquely in GR such a condition is a consequence of the gravitational field equations.
The situation in geometrised Newtonian gravitation can be thought of as analogous to that in electromagnetism where the field equations (Maxwell’s equations) do not restrict the motion of charged bodies and one must add the Lorentz force law.
In principle it is arguable that seeing this as a positive feature of the explanation is somehow arbitrary. Nonetheless, it seems difficult to imagine how a condition that fixes spacetime structure in a way that is independent of matter can have an impact on the derivation of the response equations without diminishing its generality.
See Norton (1985).
The idea is that while GR and other theories meet Will’s notion of EEP, GR also contains a kind of explanation of it. This might be thought of as being a consequence of EFE and seen as a fuller implementation of the equivalence between inertia and gravity. I leave the precise version of such a principle to be derived in future work elsewhere.
References
Bekenstein, J. D. (2004). Relativistic gravitation theory for the modified Newtonian dynamics paradigm. Physical Review D, 70, 083509.
Brading, K., & Brown, H. R., (2003). Symmetries and Noether’s theorems. In K. A. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 89–109). Cambridge: Cambridge University Press.
Brans, C., & Dicke, R. H. (1961). Mach’s principle and a relativistic theory of gravitation. Physical Review, 124(3), 925.
Brown, H. (2005). Physical relativity: Space–time structure from a dynamical perspective. Oxford: Clarendon Press.
Carroll, S. (2004). Spacetime and geometry. An introduction to general relativity. Reading: Addison Wesley.
Geroch, R., & Jang, J. S. (1975). Motion of a body in general relativity. Journal of Mathematical Physics, 16, 65–67.
Leclerc, M. (2006). Canonical and gravitational stress-energy tensors. International Journal of Modern Physics D, 15, 959.
Lee, D. L., Lightman, A. P., & Ni, W-.T. (1974). Conservation laws and variational principles in metric theories of gravity. Physical Review D, 10(6), 1685.
Malament, D., (2012). A remark about the ‘geodesic principle’ in general relativity. In M. Frappier, D. Brown & R. DiSalle (Eds.), Analysis and interpretation in the exact sciences: essays in honor of William Demopoulos. Berlin: Springer.
Norton, J. D. (1985). What was Einstein’s principle of equivalence. Studies in History and Philosophy of Science, 16(3), 203.
Pitts, J. B. (2006). Absolute objects and counterexamples: Jones–Geroch dust, Torretti constant curvature, tetrad-spinor, and scalar density. Studies in the History and Philosophy of Modern Physics, 37(2), 347–371.
Pooley, O. (2013) Substantivalist and relationalist approaches to spacetime. In R. Batterman (Ed.), The Oxford Handbook of Philosophy of Physics. Oxford University Press.
Rosen, N. (1980). General relativity with a background metric. Foundations of Physics, 10, 673–704.
Weatherall, J. O. (2011a). The motion of a body in Newtonian theories. Journal of Mathematical Physics, 52(3), 032502.
Weatherall, J. O. (2011b). 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.
Weyl, H. (1952). Space–time–matter. New York: Dover.
Will, C. D. (1981). Theory and experiment in gravitational physics. Cambridge: Cambridge University Press.
Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.
Zlosnik, T. G., Ferreira, P. G., & Starkman, G. D. (2006). The vector-tensor nature of Bekenstein’s relativistic theory of modified gravity. Physical Review D, 74, 044037.
Acknowledgments
My special thanks go to Carl Hoefer for his help, and patience, which led to major improvements in the substance and form of this paper. I am also grateful to Toffa Evans and Laura Felline who read, commented and suggested useful changes on different drafts of the paper. Needless to say, the remaining errors are only mine.
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Sus, A. On the Explanation of Inertia. J Gen Philos Sci 45, 293–315 (2014). https://doi.org/10.1007/s10838-014-9246-8
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DOI: https://doi.org/10.1007/s10838-014-9246-8