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On Gravitational Energy in Newtonian Theories

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Abstract

There are well-known problems associated with the idea of (local) gravitational energy in general relativity. We offer a new perspective on those problems by comparison with Newtonian gravitation, and particularly geometrized Newtonian gravitation (i.e., Newton–Cartan theory). We show that there is a natural candidate for the energy density of a Newtonian gravitational field. But we observe that this quantity is gauge dependent, and that it cannot be defined in the geometrized (gauge-free) theory without introducing further structure. We then address a potential response by showing that there is an analogue to the Weyl tensor in geometrized Newtonian gravitation.

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Notes

  1. See, for instance, Misner et al. [32, pp. 466–468], for a classic argument that there cannot be a local notion of energy associated with gravitation in general relativity; see also Choquet-Brouhat [8], Goldberg [21], and Curiel [10] for other arguments and discussion.

  2. One of us (Weatherall) learned of this argument from David Malament, who attributes it to Bob Geroch; it is also described by Erik Curiel [10], who likewise heard of it from Malament.

  3. Here \(R^a{}_{bcd}\) is the Riemann tensor associated with \(g_{ab}\), \(R_{ab}=R^n{}_{abn}\) is the Ricci tensor, and \(R=R^a{}_a\) is the curvature scalar. For details and background, consult Misner et al. [32], Wald [41], or Malament [30]. (Our conventions follow Malament [30], and in particular, we work in geometric units where \(c=G=1\).)

  4. Of course, one could consider situations in which there is matter whose energy-momentum precisely cancel the gravitational energy-momentum. But one can likewise consider situations in which there is no such matter present, and then the argument follows.

  5. By “purely gravitational” phenomena, we mean only that these are phenomena that may exist in the absence of matter, or which may vary for fixed matter distribution; we do not mean to identify these phenomena with a “gravitational field” [28]. We are grateful to an anonymous referee for urging us to clarify this point.

  6. One can also define various notions of quasi-local energy-momentum, such as the Hawking energy [22] and the Geroch energy [20]. These generally make sense relative to some choice of 2-sphere embedded in a spacelike hypersurface. See Szabados [37] for a detailed discussion of the various options.

  7. See Lam [26] for a more detailed discussion of the role of background structures in defining gravitational energy in general relativity. Note that in some cases, one can argue that the extra structure necessary to define gravitational energy has a physical interpretation, related to (for instance) some class of observers. But it does not follow that there is some well-defined, observer-independent quantity that is being observed from various perspectives; to the contrary, what is captured by the available notions of gravitational energy is better understood as relative to the chosen structures. Some authors embrace this: Pitts [33], for instance, has argued that there are infinitely many gravitational energies in general relativity, and that one might as well take the collection of all of them to be the appropriate “covariant” characterization of gravitational energy in the theory.

  8. See also Szabados [37].

  9. Note the parallel here to previous arguments about the ways in which geometrized Newtonian gravitation can inform our understanding of the foundations of general relativity, concerning, for instance, the character of spacetime singularities [45]; the status of inertial motion [44, 50, 51]; the conventionality of geometry [52]; classical analogues of exact solutions of Einstein’s equation [15, 30]; and the significance of spacetime curvature more generally [6, 7, 19].

  10. The fields \(t_a\) and \(h^{ab}\) are not metrics in the strict sense of the term (i.e. neither is a symmetric nondegenerate tensor field of rank (0, 2)); however, they do determine metrical structure on (respectively) the class of spacelike hypersurfaces and on any given spacelike hypersurface. See [30, §4.1] for further discussion.

  11. For further details on classical spacetimes, see Malament [30, Ch. 4]. We assume that, in addition to M being \(\mathbb {R}^4\), the spacelike hypersurfaces determined by \(t_a\) are all diffeomorphic to \(\mathbb {R}^3\); and that \(t_a\) and \(h^{ab}\) are complete.

  12. Units have been chosen so as to set \(G = 1\).

  13. More generally, we will raise indices on objects defined with covariant indices by acting with the spatial metric \(h^{ab}\). Unlike in general relativity, however, we cannot in general lower indices once they have been raised, because \(h^{ab}\) is not invertible.

  14. The argument below is loosely based on Tong [38]; for philosophical discussion of Noether’s theorems, see Brading and Brown [5].

  15. Observe that the flat derivative operator \(\nabla \) on M, in the presence of the assumptions we have made above, makes M an affine space.

  16. Here we mean that if \(t_a\delta x^a \ne 0\), then \(|t_a\delta x^a| = \epsilon \); otherwise \(h^{ab}\sigma _a\sigma _b = \epsilon ^2\), where \(h^{ab}\sigma _b = \delta x^a\).

  17. See [34, §8] for further discussion of non-variational fields.

  18. There is another reason why one would not expect this current to be conserved in general: Poisson’s equation only constrains the gravitational potential on a time slice; its change over time is determined by the matter dynamics (and boundary conditions).

  19. See also the discussion in Synge [36].

  20. We observe, too, that Curiel [10] appears to take this quantity to be the natural candidate for the energy density of the gravitational field.

  21. Note, however, that it also has some unusual features: most important is that it does not satisfy the so-called mass condition, that \(T^{ab}t_at_b > 0\) whenever \(T^{ab}\ne \mathbf {0}\). It follows that the gravitational field has momentum, but no mass! Moreover, the 4-momentum, as determined by any observer, will always be spacelike, i.e., tangent to a spacelike hypersurface. (See [30, §4.1] for the definition of a spacelike hypersurface.)

  22. When we consider “generic matter fields” we are working in a classical setting—and assuming, in particular, that there are no “spin” degrees of freedom that might require non-symmetric \(T^{ab}\).

  23. A discussion of mass-momentum tensors in this formulation of Newtonian gravitation is given by Malament ([30], §4.1); see also Duval and Künzle [13] and Dixon [12].

  24. For a discussion of this idea, see Malament [30, § 4.2] and Weatherall [46]; see also Weatherall [49]. For recent discussions of how to understand Newtonian gravitation in light of this gauge structure, see Saunders [35], Saunders [24], Weatherall [47], Wallace [42], and Dewar [11].

  25. Curves that can be parameterized in this way are called timelike; \(\xi \) is then called the unit tangent vector.

  26. Here we are using the fact that the action of any derivative operator on M may be expressed using a fixed derivative operator and a smooth tensor field \(C^a{}_{bc}\). For further details, see Malament [30, §1.7].

  27. This relationship is analogous to way that the gauge transformation equation for the electromagnetic potential, \(A'_\mu = A_\mu - \nabla _\mu \lambda \), specifies how to interpret \(A'_\mu \) in terms of \(A_\mu \) and the gauge transformation field \(\lambda \).

  28. Observe that this requires taking the acceleration of a body relative to \(\nabla \) in one model to be the same physical quantity as the acceleration relative to \(\nabla '\) in the other model; by contrast, the translation given by Eq. (16b) identifies the acceleration relative to \(\nabla '\) with the acceleration relative to \((\nabla , C^a{}_{bc})\).

  29. For systematic treatments of Newton–Cartan theory, see Trautman [39] or Malament [30, Ch. 4].

  30. Of course, observing that the notion of gravitational energy already defined does not carry over to the geometrized theory does not ipso facto rule out the possibility of defining some other notion of gravitational energy density in geometrized Newtonian gravitation. But if such a quantity exists, it would apparently follow that it does not correspond to the natural candidate for gravitational energy density in the non-geometrized theory—and so, it would fail to satisfy a reasonable desideratum on any notion of gravitational energy in the geometrized theory.

  31. See Malament [30, §4.3] for a discussion of the significance of these conditions. Note that they hold of any derivative operator \(\tilde{\nabla }\) determined by a pair \((\nabla ,\varphi )\) as above.

  32. Here, we take a “frame” to be identified with a congruence of timelike curves. A derivative operator determines a class of frames: namely, the frames such that every curve in the congruence is a geodesic of the derivative operator. Conversely, given such a class of frames, a derivative operator is uniquely determined (since a derivative operator is uniquely determined by its class of geodesics).

  33. Observe that in fact, by supposing that \(\varphi \) approaches zero as one approaches spatial infinity, we are not only choosing a privileged gauge, but also specifying boundary conditions for Poisson’s equation, which fixes a homogeneous solution. (Recall that the Dirichlet problem—finding homogeneous solutions to Poisson’s equation for fixed boundary conditions—has unique solutions for sufficiently well-behaved boundaries and boundary conditions [23, Ch. 4].)

  34. We will not develop this idea here; the details are given by Malament [30, §4.5].

  35. We do not wish to imply, however, that the situations are identical. To the contrary, as we have described, in geometrized Newtonian gravitation, one arrives at this canonical gravitational energy via a canonical degeometrization, whereas in general relativity you arrive at it via asymptotic Killing fields that may be defined when a spacetime is asymptotically flat. Thus one may think of the canonical gravitational energy one gets in the presence of asymptotic flatness as running through a canonical class of frames; in the relativistic case, meanwhile, the asymptotic Killing fields allows one to define a quantity that is frame-independent. This difference may not be as significant as it looks however, since asymptotic flatness in Newton–Cartan theory could be expected to yield an analogous notion of “asymptotic Killing field”, corresponding to vector fields whose induced one-parameter families of diffeomorphisms preserve the (curved) derivative operator at infinity. But we will defer pursuing this idea to future work. (Another difference, of course, is that one gets an energy density field in the Newton–Cartan case, but only an integrated global energy in general relativity.) We are grateful to an anonymous referee for encouraging us to emphasize this point.

  36. There are other arguments available to strengthen this point: for instance, there is a sense in which energy can be transfered from one system to another via a classical gravitational field: see Bondi and McCrea [4], Cooperstock and Booth [9], and Synge [36] for discussion.

  37. This is not to say that Einstein’s equation does not constrain the Weyl curvature: in particular, the Bianchi identities determine a relationship between the divergence of the Weyl tensor and the derivative of the Ricci tensor, which in turn is determined by Einstein’s equation.

  38. Once again (see fn. 5), we do not mean to say that the Weyl tensor is a candidate for a “gravitational field” in general relativity; rather, we mean that it captures those degrees of freedom in the theory that are not fixed by a matter distribution. One might use the expression “purely geometric” instead of “purely gravitational”.

  39. Wallace [43] also defines a (distinct, but closely related) Newtonian Weyl tensor and likewise associates it with homogeneous solutions to the Poisson equation; it encodes precisely the same data as the one we consider, and in this sense, it might be viewed as an equivalent proposal. But he does not work in the geometrized formulation, nor does he emphasize or develop the analogy to the relativistic Weyl tensor. See also Ellis and Dunsby [17] and Ellis [16] for discussions of the Weyl tensor in Newtonian theories.

  40. See Malament [30, §4.2]. The vector field \(\eta ^a\) satisfying these properties exists globally under the topological assumptions made in Sect. 2 above; more generally, one may always define such a vector field locally.

  41. Again, this inference relies on our topological assumptions; more generally, it holds only locally.

  42. To see this, suppose we have some such derivative operator \(\nabla \) and suppose \(\varphi \) is such that \(\nabla _n\nabla ^n\varphi = 0\). Now consider any other derivative operator \(\tilde{\nabla }\) compatible with \(t_a\) and \(h^{ab}\). Then \(\nabla = (\tilde{\nabla },2t_{(b}\kappa _{c)}{}^a)\), where \(\kappa _{ab}\) is some anti-symmetric tensor [30, §4.1]. It follows that \(0=\nabla _n\nabla ^n\varphi = \nabla _n\tilde{\nabla }{}^n\varphi = \tilde{\nabla }_n\tilde{\nabla }{}^n\varphi + C^n{}_{nm}\tilde{\nabla }{}^m\varphi = \tilde{\nabla }_n\tilde{\nabla }{}^n\varphi \), as desired.

  43. Here we no longer make the topological assumptions of Sect. 2.

  44. Note that we do not assume that \(\nabla \) satisfies the geometrized Poisson equation for any \(\rho \). Rather, we assume that whatever \(\nabla \) is, it is compatible with metrics that are in turn compatible with at least one derivative operator that could satisfy the geometrized Poisson equation, for some \(\rho \) or other: it follows from this that \(\nabla \) is spatially flat. See [30, § 4.1] .

  45. We require condition 5 because we are trying to capture the “part” of the Riemann tensor that is not fixed by a matter distribution, in the sense that the Riemann tensor could be written as a sum of a candidate Weyl tensor and some quantity related to the Ricci tensor, as one can do in general relativity. Observe, too, that the relativistic Weyl tensor satisfies this condition.

  46. One can easily establish that all five conditions above are satisfied. Indeed, we claim that this tensor is the unique one satisfying all five. A sketch of the proof is that all terms proportional to \(R=R_{ab}h^{ab}\), \(t_{[c}t_{d]}\), or \(R_{[cd]}\) vanish identically, as do terms of the form \(t_bt_{[c}R_{d]n}h^{na}\). This list exhausts the candidates to appear in an expression satisfying the desiderata above.

  47. It is interesting to consider the relationship between these “gravitational waves” and the classical limit of gravitational plane waves discussed by Ehlers [15]. Briefly, the solutions we consider are homogeneous solutions to Poisson’s equation at a time; we do not place any constraints on how these vary over time. Ehlers [15], meanwhile, considers classical gravitational fields that are varying over time, so that at each time one has some solution to the homogeneous Poisson equation, but these smoothly vary according to a periodic function analogous to the time-dependence of a gravitational plane wave in general relativity. In this sense, one can take the solutions that Ehlers considers to be a special case of the ones we are considering.

  48. Wallace [43] emphasizes this point in discussing Newtonian cosmology. Note, though, that thinking of homogeneous solutions to Poisson’s equation as the analogue to Weyl degrees of freedom in general relativity leads to a caveat to his analysis: although in the initial value formulation of general relativity, one does not need to specify boundary conditions at all times to get unique evolution, one does need to specify essentially independent initial data for the Weyl tensor everywhere on a Cauchy surface (i.e., specifying the Weyl tensor “at infinity” at a time is not sufficient). Wallace is correct insofar as there is surely a metaphysical—and physical—difference between a situation where data at one time determines the physics everywhere, and one where boundary conditions must be specified at all times. But there is a certain epistemological parity, nonetheless: in both cases, one must make strong assumptions about regions of space that one has no access to.

  49. For a discussion of this limit—and specifically, the sense in which geometrized Newtonian gravitation may be understood as the classical limit of general relativity—see Ehlers [14, 15], Künzl [25], Malament [29], and Fletcher [18].

  50. Pace Synge [36], who suggests that work is an inappropriate notion in a field theory.

  51. For details, see for instance Trautman [40] or Weatherall [48].

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Acknowledgements

This paper is partially based upon work supported by the National Science Foundation under Grant No. 1331126. We are grateful to David Malament for helpful conversations about the material presented here, including suggesting the form of Eq. (23); to Erik Curiel for discussions about conformal invariance in the context of geometrized Newtonian gravitation; and to David Wallace for discussions of these and related ideas. The manuscript was improved by comments from James Read and two anonymous referees. Weatherall is grateful for feedback from an audience at the 18th UK/European Foundations of Physics Conference in London, UK.

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  1. Munich Center for Mathematical Philosophy, LMU Munich, Munich, Germany

    Neil Dewar

  2. Department of Logic and Philosophy of Science, University of California, Irvine, USA

    James Owen Weatherall

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Dewar, N., Weatherall, J.O. On Gravitational Energy in Newtonian Theories. Found Phys 48, 558–578 (2018). https://doi.org/10.1007/s10701-018-0151-6

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