Abstract
A classic problem in general relativity, long studied by both physicists and philosophers of physics, concerns whether the geodesic principle may be derived from other principles of the theory, or must be posited independently. In a recent paper [Geroch & Weatherall, “The Motion of Small Bodies in Space-Time,” Comm. Math. Phys. (forthcoming)], Bob Geroch and I have introduced a new approach to this problem, based on a notion we call “tracking.” In the present paper, I situate the main results of that paper with respect to two other, related approaches, and then make some preliminary remarks on the interpretational significance of the new approach. My main suggestion is that “tracking” provides the resources for eliminating “point particles”—a problematic notion in general relativity—from the geodesic principle altogether.
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Notes
- 1.
For a recent review of the physics literature on this subject, see Poisson et al. (2011); for other recent work, see Asada et al. (2011), Gralla and Wald (2011), and the contributions to Puetzfeld et al. (2015). For the recent philosophical literature, which generally stems from a discussion of the geodesic principle by Brown (2005), see Malament (2012a), Tamir (2012), Sus (2014), Samaroo (2018), Lehmkuhl (2017b,a), and Weatherall (2011b, 2017, 2019).
- 2.
- 3.
Once again, see the references in note 1. One approach in particular that has been widely influential, but which I do not discuss at all, is the method of matched asymptotic expansion, as developed, for instance, by D’Eath (1975), Thorne and Hartle (1985), Mino et al. (1997), and Gralla and Wald (2011).
- 4.
There is a certain trade-off between, on the one hand, strength and simplicity and, on the other hand, information relevant in special cases, such as possible deviations from geodesic motion that might arise from finite body effects “on the way to the limit.” Compare, for instance, Thorne and Hartle (1985) or Gralla and Wald (2011), who describe, in the presence of additional (strong) assumptions, higher order “corrections” to geodesic motion for finite bodies, with the results to be described here, which might be understood to characterize (without these strong assumptions) the universal limiting, or order zero, behavior of small bodies. My perspective is that for foundational purposes, the more general and precise results are of greater value, though this is not necessarily true for other purposes, such as studying binary black holes. On the other hand, see footnotes 10 and 22 for ways in which the perspective taken here may bear fruitfully on widely accepted results from other approaches.
- 5.
- 6.
More precisely, we take test fields to be densities of weight 1; see Geroch and Weatherall (2018, Appendix A).
- 7.
We take derivatives by analogy with integration by parts. Fix a manifold M, a derivative operator ∇ on M, and a distribution X on M. Then ∇aX is that distribution whose action on a smooth test field α a is given by ∇aX{α a} = −X{∇aα a}. For background on distributions, including tensor distributions, see Geroch and Weatherall (2018, Appendix A), Grosser et al. (2001), or Steinbauer and Vickers (2006). The details of the theory of distributions do not particularly matter for the arguments that follow.
- 8.
That is, T ab vanishes on all test fields of the form ∇aα b.
- 9.
This is the distributional analog to the result proved in Malament (2012a).
- 10.
Although it is a side issue for present purposes, observe that this result points to a problem with certain approaches to treating the motion of rotating particles that represent “spin” by higher order distributions supported on a curve (Papapetrou, 1951; Souriau, 1974): such particles are incompatible with the energy condition. There is good physical reason for this. For ever smaller bodies to have large angular momentum (per unit mass), their rotational velocity must increase without bound—leading to superluminal velocities, which are incompatible with the energy condition.
- 11.
There are extensions to the theory of distributions—namely, the theory of Colombeau algebras (Colombeau, 2000)—that permit one to multiply distributions. But these have some undesirable properties, including that multiplication is not uniquely defined for distributions, and it does not reduce to pointwise multiplication for all (continuous) functions, conceived as distributions.
- 12.
Observe that we assume from the start that the curve is timelike; if one wants to conclude that the curve must be timelike, a stronger energy condition is required (Weatherall, 2012).
- 13.
- 14.
Gralla et al. (2009) extend a version of a curve-first approach to treat the Lorentz force law, and also derive leading order “self-force” corrections to it. But the relationship between their arguments and the sort of result envisaged here is the same as the relationship between the Gralla and Wald (2011) results and, say, the Geroch and Jang (1975) theorem, which is that they require much stronger assumptions. (Recall footnote 4.)
- 15.
For instance, in his classic textbook Wald (1984) describes the Geroch–Jang theorem as capturing the sense in which small bodies follow timelike geodesics, but then does not invoke this result to establish that light rays traverse null geodesics—appealing, instead, to a completely different construction.
- 16.
- 17.
By “generic” at a point p, I mean that x ab lies in the interior of the cone of tensors satisfying the dual energy condition at a p: that is, for any non-vanishing tensor T ab satisfying the dominant energy condition at p, T abx ab > 0.
- 18.
Observe the notational convention adopted here: previously we had used boldface for distributions; now we are using bold symbols to refer to the distributions associated with (determined by) smooth fields represented by the same, non-bolded, symbol.
- 19.
These results rephrase Theorem 3 and the subsequent discussion of Geroch and Weatherall (2018).
- 20.
The small modification involves the definition of a δ distribution supported on a null curve, which requires a choice of parameterization (since null curves cannot be parameterized by arc-length). It does not affect the conclusion that γ is a geodesic.
- 21.
Observe that this converse may be understood to capture a sense in which superluminal propagation is impossible, at least in a point-particle limit. One might take this result to be in tension with the arguments of Geroch (2011) and Weatherall (2014). But in fact, the tension is only apparent: this result assumes the dominant energy condition, while the discussions in those other papers do not (see also Earman (2014) for a discussion of the relation between the dominant energy condition and the notion of “superluminal propagation” discussed there). That said, the present result, in connection with those earlier papers, raises an interesting question. Can one generalize the notion of tracking considered here to hyperbolic systems whose solutions do not satisfy the dominant energy condition, and if so, do solutions always track their characteristics? I am grateful to an anonymous referee for raising the possible tension.
- 22.
Consider this result in connection with the remarks in footnote 10: as noted there, the dominant energy condition for distributions is incompatible with higher order distributions, and thus, with point particles carrying non-vanishing angular momentum. Here we see an even stronger result, which is that, in the small body limit, extended bodies all satisfying the dominant energy condition must have vanishing angular momentum (per unit mass).
- 23.
Note, too, that the subtleties regarding the status of the conservation condition discussed in Weatherall (2011b, 2019) arise here, too: in particular, although for sources to Maxwell’s equation, ∇bT ab = F a bJ b holds automatically, as a consequence of Maxwell’s equations (just as ∇bT ab = 0 holds for sources in Einstein’s equation), here we are considering test matter in Maxwell’s equations, since the background field F ab is fixed in advance. One could imagine considering a variation of this result, along the lines of the Ehlers–Geroch theorem, that allows electromagnetic backreaction, or even that allows both electromagnetic and gravitational backreaction. Though I do not know of any technical barriers to such results, formulating them is a delicate matter and we have not pursued it.
- 24.
We require that the spacetime be globally hyperbolic so that we are certain to have “enough” solutions to Maxwell’s equations; one could imagine relaxing this requirement.
- 25.
See Malament (2012b, §2.6) for a discussion of this point.
- 26.
The relevant arguments concerning Maxwell’s equations, and the other equations discussed below, are given in Geroch and Weatherall (2018, §4).
- 27.
In what follows, when I write of “total mass,” readers who are troubled by this notion should suppose we are in Minkowski spacetime, or a suitable asymptotically flat spacetime, where such notions make sense.
- 28.
One might respond that the Geroch–Jang and Ehlers–Geroch theorems do not explicitly refer to point particles, and so these, too, permit one to state the geodesic principle without reference to point particles. Fair enough. But from my perspective the main appeal of the current proposal is precisely that it is an assertion about field equations, and as we have seen, this is precisely what one cannot get from the Geroch–Jang and Ehlers–Geroch constructions. I am grateful to an anonymous referee for raising this objection.
- 29.
There is an interesting question lurking in the background here, which is: can we always unambiguously identify “source terms” in a differential equation? In standard cases in physics, it is generally clear what counts as a source. But I will not attempt to give an analysis of this concept here, and will proceed on the assumption that it is sufficiently clear for current purposes.
- 30.
This claim is well-known and widely discussed in the physics literature; see, for instance, Wald (1984, Appendix E) for an argument. For further discussion in a foundational context, with particular emphasis on the relationship between this claim and the geodesic principle in general relativity and other theories, see Weatherall (2019).
- 31.
For a discussion of the status of energy conditions in general relativity, see Curiel (2017).
- 32.
In effect, this is what is shown in Weatherall (2012). Note, however, that the strengthened dominant energy condition considered there, which is necessary for the Geroch–Jang theorem as stated, would not be natural in the current context. The reason is that distributions do not take values at points, and so requiring that they have certain behavior at points where they are non-vanishing is awkward to express. At best one would have to recast the condition in terms of the support of the distribution.
- 33.
Observe that this failure to satisfy the energy conditions is not obviously related to the fact that Dirac fields have “intrinsic” angular momentum (though it is related to the fact that they are spinors). (Recall fns. 10 and 22.)
- 34.
One might worry that this last observation is a problem for the proposed formulation of the geodesic principle in terms of tracking. But I do not think there is a real concern. Source-free matter that tracks non-geodesic curves is every bit as much a problem for other formulations of the geodesic principle as the present one—and at least on the proposed formulation, the tension between such matter and geodesic motion is immediately manifest.
- 35.
References
Asada, H., Futamase, T., Hogan, P. A., 2011. Equations of motion in general relativity. Oxford University Press, Oxford, UK.
Brown, H. R., 2005. Physical Relativity. Oxford University Press, New York.
Colombeau, J. F., 2000. New generalized functions and multiplication of distributions. Vol. 84. Elsevier.
Curiel, E., 2017. A primer on energy conditions. In: Lehmkuhl, D., Schiemann, G., Scholz, E. (Eds.), Towards a Theory of Spacetime Theories. Birkhäuser, Boston, MA, pp. 43–104.
D’Eath, P. D., 1975. Interaction of two black holes in the slow-motion limit. Physical Review D 12 (8), 2183.
Earman, J., 2014. No superluminal propagation for classical relativistic and relativistic quantum fields. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 48, 102–108.
Ehlers, J., Geroch, R., 2004. Equation of motion of small bodies in relativity. Annals of Physics 309, 232–236.
Einstein, A., Grommer, J., 1927. Allgemeine Relativitätstheorie und Bewegungsgesetz. Verlag der Akademie der Wissenschaften, Berlin.
Geroch, R., 1996. Partial differential equations of physics, arXiv:gr-qc/9602055.
Geroch, R., 2011. Faster than light? In: Plaue, M., Rendall, A., Scherfner, M. (Eds.), Advances in Lorentzian Geometry. American Mathematical Society, Providence, RI, pp. 59–80.
Geroch, R., Jang, P. S., 1975. Motion of a body in general relativity. Journal of Mathematical Physics 16 (1), 65.
Geroch, R., Traschen, J., 1987. Strings and other distributional sources in general relativity. Physical Review D 36 (4), 1017.
Geroch, R., Weatherall, J. O., 2018. The motion of small bodies in space-time. Communications in Mathematical Physics (forthcoming). https://doi.org/10.1007/s00220-018-3268-8.
Gralla, S. E., Harte, A. I., Wald, R. M., 2009. Rigorous derivation of electromagnetic self-force. Physical Review D 80 (2), 024031.
Gralla, S. E., Wald, R. M., 2011. A rigorous derivation of gravitational self-force. Classical and Quantum Gravity 28 (15), 159501.
Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R., 2001. Geometric theory of generalized functions with applications to general relativity. Vol. 537. Springer Science & Business Media, Dordrecht.
Lehmkuhl, D., 2017a. General relativity as a hybrid theory: The genesis of Einstein’s work on the problem of motion. Studies in History and Philosophy of Modern Physics (forthcoming).https://doi.org/10.1016/j.shpsb.2017.09.006.
Lehmkuhl, D., 2017b. Literal versus careful interpretations of scientific theories: The vacuum approach to the problem of motion in general relativity. Philosophy of Science 84 (5), 1202–1214.
Malament, D., 2012a. A remark about the “geodesic principle” in general relativity. In: Frappier, M., Brown, D. H., DiSalle, R. (Eds.), Analysis and Interpretation in the Exact Sciences: Essays in Honour of William Demopoulos. Springer, New York, pp. 245–252.
Malament, D. B., 2012b. Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. University of Chicago Press, Chicago.
Mathisson, M., 1931. Die mechanik des materieteilchens in der allgemeinen relativitätstheorie”. Zeitschrift für Physik 67, 826–844.
Mino, Y., Sasaki, M., Tanaka, T., 1997. Gravitational radiation reaction to a particle motion. Physical Review D 55 (6), 3457.
Papapetrou, A., 1951. Spinning test-particles in general relativity. i. Proc. R. Soc. Lond. A 209 (1097), 248–258.
Poisson, E., Pound, A., Vega, I., 2011. The motion of point particles in curved spacetime. Living Reviews in Relativity 14 (7).
Puetzfeld, D., Lämmerzahl, C., Schutz, B. (Eds.), 2015. Equations of motion in relativistic gravity. Springer, Heidelberg, Germany.
Samaroo, R., 2018. There is no conspiracy of inertia. British Journal for the Philosophy of Science. 69 (4), 957–982.
Souriau, J.-M., 1974. Modèle de particule à spin dans le champ électromagnétique et gravitationnel. Annales de l’Institut Henri Poincaré Sec. A 20, 315.
Steinbauer, R., Vickers, J. A., 2006. The use of generalized functions and distributions in general relativity. Classical and Quantum Gravity 23 (10), R91.
Sternberg, S., Guillemin, V., 1984. Symplectic Techniques in Physics. Cambridge University Press, Cambridge.
Sus, A., 2014. On the explanation of inertia. Journal for General Philosophy of Science 45 (2), 293–315.
Tamir, M., 2012. Proving the principle: Taking geodesic dynamics too seriously in Einstein’s theory. Studies in History and Philosophy of Modern Physics 43 (2), 137–154.
Thorne, K. S., Hartle, J. B., 1985. Laws of motion and precession for black holes and other bodies. Physical Review D 31 (8), 1815.
Wald, R. M., 1984. General Relativity. University of Chicago Press, Chicago.
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.
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., 2014. Against dogma: On superluminal propagation in classical electromagnetism. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 48, 109–123.
Weatherall, J. O., 2017. Inertial motion, explanation, and the foundations of classical spacetime theories. In: Lehmkuhl, D., Schiemann, G., Scholz, E. (Eds.), Towards a Theory of Spacetime Theories. Birkhäuser, Boston, MA, pp. 13–42.
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.
Weyl, H., 1922. Space–Time–Matter. Methuen & Co., London, UK, reprinted in 1952 by Dover Publications.
Wong, W. W.-Y., 2011. Regular hyperbolicity, dominant energy condition and causality for Lagrangian theories of maps. Classical and Quantum Gravity 28 (21), 215008.
Acknowledgements
I am grateful to Harvey Brown, Bob Geroch, Dennis Lehmkuhl, David Malament, and Bob Wald for many helpful conversations related to this material. I am also grateful to audiences at the Eighth Quadrennial International Fellows Conference (2016) in Lund, Sweden; New Directions in Philosophy of Physics (2017) in Tarquinia, Italy; the Pacific Institute of Theoretical Physics at the University of British Columbia; the workshop “Thinking about Space and Time” at the University of Bern; the Max Planck Institute for Gravitational Physics in Potsdam, Germany; the Black Hole Initiative at Harvard University; the Einstein Papers Project at Caltech; and Utrecht University, and especially to Lars Andersson, Joshua Goldberg, Jim Hartle, Bob Wald, and Shing-Tung Yau for probing questions, comments, and objections.
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Weatherall, J.O. (2020). Geometry and Motion 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_10
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