The Behaviour of Rods and Clocks in General Relativity and the Meaning of the Metric Field

The notion that the metric field in general relativity can be understood as a property of space-time rests on a feature of the theory sometimes called universal coupling—the claim that rods and clocks “measure” the metric in a way that is independent of their constitution. It is pointed out that this feature is not strictly a consequence of the central dynamical tenets of the theory, and argued that the metric field would better be regarded as a (possibly emergent) field in space-time, rather than as the very fabric of space-time itself.

1. 1.

   For a discussion of the role of the clock hypothesis within special and general relativity, see Brown (2005, section 6.2.1).

2. 2.

   It is noteworthy that this famous Harvard experiment was not the first to test the red-shift hypothesis; it was preceded in 1960 by a similar experiment using the Mössbauer effect performed at Harwell in the UK (Cranshaw et al. 1960). The Harwell group also performed in the same year another red-shift test, again using the Mössbauer effect, but this time involving a source at the centre of a rotating wheel which contained a thin iron absorber ((Hay et al. 1960), see also Sherwin (1960)). For a detailed account of the history of the early redshift experiments, see Hentschel (1996).

   In 1960 it was clarified by Schild (1960) that red-shift tests are by nature insensitive to the precise form of Einstein’s field equations. In the same paper, he also noted that nonetheless the red-shift phenomenon itself leads naturally to the idea that gravitational fields are related to the curvature of space-time. It is worth emphasizing that this argument for curvature depends on an appeal to the inhomogeneity of the gravitational field and therefore to a series of red-shift experiments sufficiently separated on the surface of the Earth; a single example of the red-shift, such as in the Pound-Rebka experiment, in which the equipment is largely insensitive to tidal forces, is not enough. This point is sometimes overlooked, as in Carroll’s otherwise excellent (Carroll 2004, pp. 53–4). For more details, see Brown and Read (2016).

3. 3.

   If, for example, the source is at a higher temperature than the absorber, the shift is negative. For resonance absorption to occur in this case, the absorber must be given a small velocity away from the source so that the Doppler effect can compensate for the shift; see Sherwin (1960, p. 19).

4. 4.

   An earlier 1960 redshift test, also involving the Mössbauer effect, involved a source at the centre of a rotating wheel which contained a thin iron absorber ((Hay et al. 1960); see also Sherwin (1960)). Again a frequency shift due to relativistic time dilation—a case of clock retardation—was detected to an accuracy of a few percent; the radial acceleration in this experiment was of the order of 10⁴ g.

5. 5.

   See (Brown 2005, pp. 8 and 95).

6. 6.

   It is noted in (Brown 2005, p. 71, footnote 8) that the word “theorem” might be more happily translated from the original German as “statement”.

7. 7.

   In his 1949 discussion, Einstein clearly appreciates the difference between the two principles; see (Einstein 1969, p. 57).

8. 8.

   In 1940, Einstein wrote: “The content of the restricted relativity theory can accordingly be summarised in one sentence: all natural laws must be so conditioned that they are covariant with respect to Lorentz transformations.” (Einstein 1954, p. 329). It is worth recalling in this context the way Einstein described in his Autobiographical Notes the main contribution Minkowski made to relativity theory. It was not so much Minkowski’s ontological fusion of space and time into a single four-dimensional entity that Einstein praised, but his provision of a tensor calculus in which equations for the non-gravitational interactions are manifestly Lorentz covariant. For Einstein, Minkowski had done for relativity what Heaviside and others did for Maxwell theory when they introduced the three-vector formulation of electrodynamics (so that the physics is manifestly Euclidean covariant). Minkowski “showed that the Lorentz transformation …is nothing but a rotation of the coordinate system in the four-dimensional space” (Einstein 1969, p. 59), an insight which in fact Poincaré had anticipated.

9. 9.

   Suppose one considers the possibility of modeling a rigid rod by way of an infinite crystal composed of ions held together by electrostatic forces, rather in the spirit of Lorentz’s 1892 model of a system of charges held together in unstable equilibrium. Then the dynamical analysis seems to lead (as it did in the Lorentz case; see Brown (2001)) to a certain motion-induced deformation, rather than a strict longitudinal contraction: the conformal covariance of the equations has not been broken. This point was brought home to me in discussions some years ago with Adrian Sutton and his then 4th year undergraduate project students in the Department of Physics at Imperial College London: H. Anwar, V. Venkataraman, A. Wiener, C. Chan, B. Lok, C. Lin, and G. Abdul-Jabbar. This group has been studying a constructive approach to length contraction, similar to that of Bell (1976), but in which the effects of motion are calculated in a classical model of the attractive interatomic forces in an infinite ionic crystal. The question that has been thrown up, as I see it, is whether in such models it is possible to obtain strictly longitudinal length contraction without introducing quantum mechanics. (In the Bell atomic model, the conformal symmetry is broken by appeal to the Lorentz force law for the orbiting charge.) This question also applies to another electrostatic model of a rigid rod provided by Miller (2010), which was brought to my attention after discussions with the Imperial College group. Here, the way the author effectively breaks the conformal symmetry in the electrodynamics is not entirely consistent with the constructive nature of his approach.

10. 10.

    Note that the distinction here between strong and weak measurement is not relevant to the issue of the accuracy of the measurement.

11. 11.

    A possible objection to this reasoning might go as follows. The generally covariant formulation of any specially relativistic dynamics (such as Maxwellian electrodynamics), or more generally any dynamical theory within an absolute space-time background, flat or curved, leads to equations of motion in which the absolute structure appears in the equations. Such structure appears to be causally relevant; indeed a violation of the action-reaction principle seems to obtain. Space-time structure acts on matter, but not the other way round. However, demanding general covariance in the context of special relativity is like demanding that (“pure gauge”) electromagnetic vector and scalar potentials appear in the Schrödinger equation for a free particle. Just as it would be odd to say that such potentials are physically acting on the particle, arguably the “action” of space-time structure in special relativity is merely an artifact of the generally covariant formulation, which is ill-suited to the theory. (For further discussion of the purported violation of the action-reaction principle in special relativity theory, see Brown and Pooley (2004) and Brown (2005, section 8.3.1). For a treatment of Einstein’s appeal to the principle in extolling the virtues of general relativity, see Brown and Lehmkuhl (2016).

12. 12.

    It is curious how infrequently this issue is raised. A rare case was Dieks, writing in 1987: “…it should be emphasized that the general theory of relativity is a fundamental physical theory …it can safely be said that constructs like macroscopic measuring rods and clocks cannot figure as essential elements in such a fundamental theory. …[T]he behaviour of macroscopic bodies like rods and clocks should be explained on the basis of their microscopic constitution” (Dieks 1987, p. 15); see also (Dieks 1984).) However, the nature of this explanation as suggested by Dieks differs from what follows. I note that Fletcher (2013) provides a theorem showing that, for any timelike curve in any spacetime, there is a light clock that measures the length of the curve as given by the metric to arbitrary accuracy. The proof of course assumes that light propagates along null geodesics, which is a consequence of the Einstein equivalence principle (see below).

13. 13.

    For a recent review of TeVeS, see Skordis (2009). It has been argued (Zlosnik et al. 2006) that TeVeS is not a true bimetric theory. First, it can be shown to be equivalent to a (mathematically more complicated) Tensor-Vector theory involving just the single metric \(\tilde {g}_{\mu \nu }\) in the total action. More significantly, these authors claim that tensor gravity waves propagate along the same light cone as electromagnetic ones. But this claim conflicts with the analysis of TeVeS and its generalizations by Skordis (2006, 2008, 2009).

14. 14.

    A gravitational theory that violates local Lorentz covariance is due to Jacobson et al. (2001). It contains a time-like unit vector field which serves to pick out a preferred frame. I take it the appearance of this field in the equations governing the matter fields is ruled out by the second component of the Einstein equivalence principle.

15. 15.

    A different analysis leading to the same conclusion was provided by James L. Anderson (1999). Anderson also argued that the (approximately) metrically-related behaviour of clocks can be derived from the dynamical assumptions of GR, in the same way that the motion of a free test particle can be derived. He claimed that this becomes particularly clear in the approximation scheme developed to address the problem of motion in GR due to Einstein, Infeld and Hoffmann in 1939 and 1940. I repeat Anderson’s concluding remarks:

    In this paper I have argued that a metric interpretation is not needed in general relativity and that the purposes for which it was originally introduced, i.e., temporal and spatial measurements and the determination of geodesic paths, can be all be derived from the field equations of this theory by means of the EIH [Einstein, Infeld and Hoffman] approximation scheme. As a consequence, the only ab initio space-time concept that is required is that of the blank space-time manifold. In this view what general relativity really succeeded in doing was to eliminate geometry from physics. The gravitational field is, again in this view, just another field on the space-time manifold. It is however a very special field since it is needed in order to formulate the field equations for, what other fields are present and hence couples universally with all other fields.

16. 16.

    See the reconstruction of GR due to Barbour et al. (2002) based on a dynamical 3-geometry approach and inspired by Mach’s relational reasoning.

17. 17.

    The emergent approach discussed by Barceló et al. (2001) relies on classical considerations related to the so-called “analog models” of GR to motivate the existence of the Lorentzian metric field, and effective theories arising out of the one-loop approximation to quantum field theory to generate the dynamics (and in particular the familiar Hilbert-Einstein term in the effective action) in the spirit of Sakharov’s 1968 notion of induced gravity. Such an approach clearly calls into question the appropriateness of quantizing gravity. For more recent developments along similar lines, but now based on a potentially deep connection between the field equations for the metric and the thermodynamics of horizons, see Padmanabhan (2007, 2008).

18. 18.

    The above-mentioned work of Barbour et al. (2002) was originally thought to provide a derivation of the Einstein equivalence principle. However, careful further analysis by Edward Anderson has cast doubt on this claim; for details see Anderson (2007). It should be mentioned that the validity of the Einstein equivalence principle seems to be genuinely mysterious in the case of the case of the emergent gravity approach; see Barceló et al. (2001, section 4).

19. 19.

    I am grateful to Dennis Lehmkuhl for recently bringing this letter to my attention. It is reproduced in the preface by John Stachel in (Earman et al. 1977, p. ix). An enlightening account of Einstein’s misgivings about the geometric nature of general relativity is given by Lehmkuhl (2014). Finally, a fuller treatment of most of the main arguments made in sections 3 and 4 of the present paper is given by Brown (2005).

I wish to thank David Rowe, Tilman Sauer and Scott Walter for the kind invitation to contribute to this volume. I am grateful to Dennis Lehmkuhl, David Rowe and Scott Walter for comments that led to improvements in the paper. I thank Norbert Straumann for bringing to my attention the 1987 work by Anton Eisele, and for further correspondence. I also benefited from discussions with Eleanor Knox, Constantinos Skordis and George Svetlichny, as well as Adrian Sutton and his project students (see note 9).

Authors and Affiliations

1. Faculty of Philosophy, University of Oxford, Radcliffe Observatory Quarter 555, Woodstock Road, Oxford, OX2 6GG, UK

   Harvey R. Brown

Corresponding author

Correspondence to Harvey R. Brown .

Editors and Affiliations

1. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

   David E. Rowe

2. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

   Tilman Sauer

3. Centre François Viète, Université de Nantes, Nantes Cedex, France

   Scott A. Walter

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