Abstract
I describe how Einstein constructed General Relativity, discussing the warping of spacetime, gravitational time dilation and the distortions of time near singularities. Contrary to Einstein’s aims, GR does not evidence a complete physical equivalence of reference frames, with time simply relative to the observer’s coordinate system. I argue that the principle of local becoming is embodied in the geodesic principle of GR, which guarantees the same connection of time with inertia as was ensconced in Newton’s physics and preserved in SR.
Relativistic clocks are hodometers of timelike worldlines.
—Roberto Torretti, Relativity and Geometry.*
*Torretti (1996, 96). In this quotation I have silently corrected Torretti’s spelling of ‘hodometer’ as ‘hodometre’. A hodometer is a device consisting of a wheel attached to a dial (an analogue odometer). Another name for it is a waywiser; the term Harvey Brown uses in his (2005), and an illustration of the device is shown on the cover of his book.
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- 1.
This is a very small effect. If the mountain is Ben Nevis at about 1500 m, and the deepest mine is about the same depth, a rough calculation gives the clock at the top of the mountain running faster by a factor of about 1 + 3 × 10−12! That is, it would gain by a few microseconds per year over the one in the mine.
- 2.
“Query 1. Do not Bodies act upon Light at a distance, and by their action bend its rays; and is not this action (caeteris paribus) strongest at the least distance?” (Newton 1730, Bk III, 339) .
- 3.
In defence of this assumption Soldner wrote: “That light rays have all the absolute [basic] properties of matter one can see from the phenomenon of aberration which is possible only because light rays are truly material. And furthermore, one cannot think of a thing which exists and works on our senses that would not have the property of matter.” (Jaki 1978, 948).
- 4.
- 5.
- 6.
There is, admittedly, a slight hitch to this reasoning, as the time coordinates of the observer at rest and the accelerating observer are not the same. The rotating clock does not return to the same point in spacetime after one revolution. It would trace a spiral in spacetime. Nevertheless, the point stands that space would be warped by gravitational fields, according to the EP. See Janssen (2014, 181) for a discussion.
- 7.
Actually, as we’ll see, there are several other components to the overall red shift of the light from distant galaxies. One is due to the fact that each of the galaxies is moving apart from all the others with the overall expansion of space: this is the famous Hubble red shift, our best indicator of the expansion of the universe. The latter is due to the expansion of space itself over cosmic distances, and so comes into play when assessing the redshift of light from distant galaxies. But it is a very small effect over distances such as that of the Earth from the Sun.
- 8.
The discrepancy is due to the fact that for the original prediction Einstein applied the EP in a flat background space. As we shall see, the EP properly only applies locally, in arbitrarily small regions. The curved metric of GR knits these local regions together in such a way as to double the amount of the curvature.
- 9.
It may appear odd that in the above derivation Einstein uses the classical Doppler formula and not the relativistic Doppler formula that he had derived in his original SR paper (Einstein [1905] 1923, 56): ν1 = ν0 γ(1 − v/c). But as Einstein had already shown in his 1907 paper, by expanding γ in a power series, the formula can be shown to be correct “to a first approximation” in which terms of order v2/c2 are neglected.
- 10.
Here we should make a couple of caveats: first, as Stephen Hawking showed, because of quantum effects, black holes can in fact evaporate by (very slowly) leaking radiation; second, as has been discovered very recently by Joseph Polchinski and colleagues (Almheiri et al. 2013) , quantum effects should result in the production of a seething maelstrom of particles at the event horizon, whereas the Equivalence Principle predicts that the free fall of a body through the event horizon should be an inertial motion . This clash of predictions is known as the Firewall Paradox, and may not be resolved until we are closer to a theory of Quantum Gravity that supersedes GR and Quantum Theory while accounting for their successes.
- 11.
Richard Muller argues on this basis that “actually, there are no black holes” (Muller 2016, 86), reasoning that “it takes infinite time to form a black hole, measured in our time coordinate” (87). Meanwhile, the fall takes only ten minutes in the rest frame of matter that is accelerating towards the centre of the black hole, while from the standpoint of this frame “at eleven minutes, the time outside has gone to infinity and beyond” (83). Granted; but Muller fails to acknowledge that this entails that there is no such thing as the “universe at an instant of time” which he presupposes in his concept of Now as “the leading edge of time” (293), or of which he supposes one could have a “God-like complete knowledge” (121).
- 12.
Lee Smolin has argued that a global time function is definable in Shape Dynamics, in which only the local shapes of Riemannian 3-geometries are dynamical (see Barbour 2011 for an introduction). This allows the slicing of spacetime into three-dimensional spaces of constant mean curvature, the so-called CMC foliation, where the spaces evolve in a “dynamically determined preferred global time” (Unger and Smolin 2015, 420). But as Smolin concedes, “the effects of the preferred global time, if it exists, are not detectible in experiments at less than the scale of the whole universe” (491–92). Such a time tracks the cosmological expansion of space, but not the local becoming of events in it.
- 13.
Cf. Roberto Torretti: “Gravity does not therefore impair the time-keeping functioning of natural clocks; it shapes the several strands of time measured by differently placed clocks” (Torretti 1999, 291). Cf. also George Ellis: “The relative flow of time along different world lines may be different: that is the phenomenon of time dilation, caused by the varying gravitational potentials represented by the metric tensor [78]. But this does not mean it is not well defined along each world line.” (Ellis 2012, 12). All this should be compared with Roberto Mangabeira Unger, who claims that “None of the classical empirical tests of general relativity … have any direct or proximate relation to time” and writes dismissively of “so-called time dilation” (Unger and Smolin 2015, 191) .
- 14.
That is true of the conventionalist reading of GR given by Henri Poincaré and Moritz Schlick, of the attempts by neo-Kantians like Ernst Cassirer and Hans Reichenbach to assimilate it to a revised neo-Kantianism, and of the objective idealism that Hermann Weyl and Arthur Eddington saw to be its main lesson. See Ryckman (2014) for a highly informative account of these philosophical interpretations of general relativity.
- 15.
In a 1907 yearbook article on relativity Einstein began: “So far we have applied the principle of relativity … only to non-accelerated systems. Is it conceivable that the principle of relativity should apply to systems that are accelerated with respect to each other?” (Isaacson 2007, 148). Similarly, in 1919 he asks rhetorically: “Should the independence of physical laws of the state of motion of the coordinate system be restricted to the uniform translation of coordinate systems in respect to each other?” (Einstein 1919; in 1954, 230)
- 16.
“The special theory, on which the general theory rests, applies to all physical phenomena with the exception of gravitation; the general theory provides the law of gravitation and its relations to the other forces of nature.” (Einstein 1919; in 1954, 228–29)
- 17.
This feature of relativity theory is usually called the “clock hypothesis”. As I have argued in my (2010), however, it is not a separate hypothesis: in SR it is a consequence of the definition of proper time , and in GR it follows from the Equivalence Principle.
- 18.
See also the lucid discussion in Penrose’s (2016, 52–59), where he details the later fortunes of Weyl’s theory as the foundation of the so-called “gauge theories” of strong and weak interactions in the standard model of particle physics.
- 19.
A point to note in passing here is that, despite the great success of his using the energy-momentum tensor in formulating GR, later in life Einstein became quite sceptical about representing matter by such tensors. As Torretti reports (Torretti 1999, 295–6), he came to regard them as idealized models rather than fundamental physical entities, “as purely temporary and more or less phenomenological devices for representing the structure of matter” (Einstein and Infeld 1938, 209).
- 20.
- 21.
Cf. Torretti (1996, 135), to whose explanations I am indebted here.
- 22.
Leibniz first articulates this view in an unpublished manuscript of 1676 (see Arthur 2013). There he notes the greater simplicity of the Copernican hypothesis in its dispensing with the imaginary epicycles and eccentric circles of the Ptolemaic system, the potential changes in the apparent diameters of the fixed stars and changes of situation of the Earth relative to the fixed stars, and observations of oscillations of hanging lamps, and of tides “impinging only on eastern and western shores” (A VI 3, 105). These things, he concludes, “can be explained more distinctly by the supposed motion of the Earth and its being reduced to a simple cause” (111).
- 23.
For a concise account of the bucket thought experiment (as well as other thought experiments of Newton and Leibniz), see (Arthur 2017).
- 24.
For a discussion of Huygens’s views on motion, see Stein’s (1977), esp. pp. 8–9.
- 25.
In a way, Einstein’s reasoning is the obverse of Huygens’s: where for Huygens gravity was a difference in centrifugal force, Einstein is making apparent centrifugal force an effect of gravity.
- 26.
There is also an analogous effect known as “frame-dragging”, a warping of spacetime outside a massive shell produced by its rotation.
- 27.
It is in this sense—around the local compass of inertia—that the universe itself can be said to rotate, as in Gödel’s rotating universe solution to the EFE.
- 28.
“These Lorentz frames at various events on the hypersurface do not mesh to form a global inertial frame, but their surfaces of simultaneity do mesh to form the spacelike hypersurface itself” (Misner et al. 1973, 714) .
- 29.
I can think of two further objections to Ellis’s “uneven blanket” conception of the present as a surface of equal proper time elapsed. One stems from the Twin Paradox discussed in Chaps. 5 and 6: for Astrid, less of her proper time will have elapsed since leaving Terence than will have for him, yet at their reunion they are certainly present to one another. Second, the worldlines of bodies that have penetrated the Schwarzschild radius of a black hole will not terminate on any such surface of constant proper time.
- 30.
Leibniz had effectively recognized this in his posthumously published Protogea. See (Arthur 2014) for a discussion.
- 31.
- 32.
It is worth remembering, even so, just how very small the constant is. Current estimates put it at 1.11 × 10−52 m−2!
- 33.
In fact Hermann Weyl had predicted the linear relationship later empirically confirmed by Hubble in his (1923), and it was H. P. Robertson’s endorsement of Lemaître’s prediction of the expansion of the universe on this basis in his (1928) that was confirmed by Hubble in his (1929) and to Hubble’s (1923).
- 34.
Robertson (1933, 64). Although Friedmann had assumed it was pressureless, this assumption was dropped by Lemaître, Robertson and Walker.
- 35.
Gödel also remarked on the lack of precision in the notion of a “mean motion of matter”: “What may be called the ‘true mean motion’ is obtained by taking regions so large that a further increase in their size does not any longer change essentially the value obtained. In our world this is the case for regions including many galactic systems” (Gödel 1949, 559, n. 7). He adds that this approximation could perhaps be improved, but would still involve “introducing more or less arbitrary elements (such as, e.g., the size of the regions or the weight function to be used in the computation of the mean motion of matter)” (560, n. 9).
- 36.
“Problems with the notion of a global cosmic time may arise if a privileged set of world lines becomes difficult to identify, e.g. in the very early universe above the electroweak (Higgs) phase transition or in a (complicated) inhomogeneous universe. A more serious problem for time (which is a problem even for a local definition of time) arises if a point is reached in the backward extrapolation where the world lines themselves can no longer be identified. In particular, this appears to be the case if some point is contemplated, e.g. at the onset of inflation, where all constituents of the universe are of a quantum nature, leading to what can be called the quantum problem of time.” (Rugh and Zinkernagel 2017, 379).
- 37.
There is also the defence that the existence of a preferred foliation in a particular solution does not in itself constitute a violation of Lorentz invariance or covariance. As Ellis points out, “These are symmetries of the general theory, not of its solutions. Interesting solutions break the symmetries of the theory” (Ellis 2012, 10). The same point is made by Kent Peacock (2018, 120).
- 38.
Smolin posits that “all theories of subsystems of the universe should be relativistically invariant so that the effects of the preferred global time, if it exists, are not detectible in experiments at less than the scale of the whole universe” (Unger and Smolin 2015, 491–2) .
- 39.
Here Gödel seems to be conveniently forgetting that in GR the very structure of spacetime is contingent, since it depends on the distribution of matter, as has been argued by Dorato (1995, 283). So the existence or not of CTCs is necessarily contingent.
- 40.
- 41.
For such a thorough philosophical discussion see Arntzenius and Maudlin (2013).
- 42.
For example, Robert Geroch begins his analysis by defining the identity of an event as follows: “We regard two events as being ‘the same’ if they coincide, that is, if they occur ‘at the same place at the same time.’” (Geroch 1978, 4).
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Arthur, R.T.W. (2019). Time in General Relativity. In: The Reality of Time Flow. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-030-15948-1_7
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