Abstract
I discuss the debate between dynamical versus geometrical approaches to spacetime theories, in the context of both special and general relativity, arguing that (a) the debate takes a substantially different form in the two cases; (b) different versions of the geometrical approach—only some of which are viable—should be distinguished; (c) in general relativity, there is no difference between the most viable version of the geometrical approach and the dynamical approach. In addition, I demonstrate that what have previously been dubbed two ‘miracles’ of general relativity admit of no resolution from within general relativity, on either the dynamical or ‘qualified’ geometrical approaches, modulo some possible hints that the second ‘miracle’ may be resolved by appeal to recent results regarding the ‘geodesic principle’ in GR.
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Notes
- 1.
For further discussion regarding how this debate changes on moving from SR to GR, see Brown (2005), Brown and Read (2020), Read et al. (2018). In light of the fact that the advocate of the dynamical approach does not attempt to undertake an ontological reduction of the above-described kind in the context of GR, one might be inclined to conclude: ‘So much the worse for the dynamical approach in the context of GR, as a distinct view in the landscape’. Below, I will argue that there is something to this concern, for (I maintain) there is no difference in the GR context between the dynamical approach and the most defensible version of the geometrical approach.
- 2.
It is worth flagging that I will offer these two notions of explanation without claiming (or seeking) to give a full conceptual analysis of the notion of scientific explanation; in my view, the distinction between ‘qualified’ and ‘unqualified’ explanations is still a valuable and comprehensible one (providing, as I see it, at least some of the ‘explanatory concepts’ which Norton suggests may be necessary for ‘a full understanding of constructivism [i.e., the dynamical approach]’ (Norton 2008, p. 824)), even in the absence of such an analysis. (In this regard, cf. the methodology of Weatherall 2017, pp. 15–16.)
- 3.
By contrast, there is a sense in which advocates of the dynamical approach need not speak of ‘spacetime’ at all—cf. Brown and Read (2020, §3.1).
- 4.
It is worth noting that these two ‘miracles’ of GR may admit of resolution in a successor theory to GR, in a manner analogous to that in which the ‘miracle’ of the coincidence of gravitational and inertial masses in Newtonian mechanics was resolved on moving to GR. See Weatherall (2011a) for a detailed discussion of the explanation of the coincidence of gravitational and inertial masses in GR, and Read (2019) for how the two ‘miracles’ of GR may be resolved on moving to one particular successor theory—viz., perturbative string theory.
- 5.
In principle, we should not exclude other types of field on M—e.g. spinor fields; pseudotensors; tensor densities; etc. (For arguments for taking these latter two classes of object seriously, see Pitts 2006, 2010.) In this paper, however, I focus exclusively upon the case in which the Φi are tensor fields.
- 6.
Here, ∇a is the torsion-free derivative compatible with η ab, so that ∇aη bc = 0.
- 7.
The torsion-free derivative operator ∇a now compatible with g ab, so that ∇ag bc = 0.
- 8.
These are the Einstein field equations with vanishing cosmological constant Λ. For Λ ≠ 0, the field equations read G ab + Λg ab = 8πT ab.
- 9.
Recall that the stress-energy tensor is defined through \(T^{ab} := \frac {2}{\sqrt {g}}\frac {\delta S}{\delta g_{ab}}\), where g is the metric determinant, and S is the action to which the matter Lagrangian—here \(\mathcal {L}_{\mathrm {EM}}\)—is associated. T ab is defined from T ab via T ab := g acg bdT cd.
- 10.
Here, I switch to a coordinate-based description—hence the transition from Roman (abstract) to Greek indices.
- 11.
In this paper, I mean by ‘matter fields’, or ‘non-gravitational fields’, those for which there exists an associated stress-energy tensor, and by ‘gravitational fields’ those for which there exists no such stress-energy tensor—this distinction is in the spirit of Lehmkuhl (2011). In the context of GR, this means that the metric field is identified as a gravitational field, whereas all other fields typically of interest (e.g., Klein–Gordon fields, electromagnetic fields, etc.) are matter fields. Clearly, there exist subtle issues here regarding the possibility of defining a stress-energy tensor associated with the metric field in GR—see Curiel (2018) for a proof against this possibility, and e.g. Hoefer (2000), Lam (2011), Read (2018) for related discussion. Note also that this distinction between matter and gravitational fields may break down in the case of other spacetime theories—for example, in Newtonian gravitation theory (cf. Sect. 5.2.2), it is possible to define a stress-energy tensor associated with the potential φ, in spite of this field naturally being regarded as ‘gravitational’ (cf. Dewar and Weatherall 2018). Nevertheless, for my purposes, the above distinction will suffice.
- 12.
Here, again, I use a coordinate-based description. Note that I do not transform the fixed fields—cf. Pooley (2017, p. 115).
- 13.
- 14.
One further observation about the distinction between special versus general relativistic theories as characterised above: Since the metric field η ab of SR is fixed identically in all KPMs, so too is the manifold M on which that field is defined. Not so for GR: since it is not definitional of a general relativistic theory that it contain a certain fixed field, there may exist models with distinct manifolds M.
- 15.
Other conditions which one may be interested in imposing upon the class of GR solutions in which one is interested are e.g. energy conditions, for such conditions are often understood to be tied to the restriction to ‘physically reasonable’ matter (for example, to conditions that energy cannot be negative). For a recent virtuoso study of energy conditions, see Curiel (2017).
- 16.
- 17.
- 18.
- 19.
One might reasonably pause over whether the ‘as if’ in the following passage is necessary, on Brown’s account.
- 20.
One might wonder whether satisfaction of the SEP should be regarded as being a necessary condition for the metric field to have its chronogeometric significance, or rather as being a sufficient condition, or rather something else. In Read et al. (2018, pp. 15–16), it is indeed claimed that the SEP constitutes a necessary condition for chronogeometricity; however, it is perhaps more conservative to state that, with auxiliary assumptions such as the existence of stable rods and clocks, it constitutes a jointly sufficient condition for chronogeometricity. In this way, one does not rule out other possible means of gaining operational access to the metric field—for example, by using test particles which traverse null and timelike geodesics to gain access to conformal and projective structure, from which (via Weyl’s theorem-type reasoning—cf. Ehlers et al. (1972), Weyl (1921), Weyl (1923), and with certain further additional assumptions) one can recover metric structure. There remains much further work to be done in order to understand fully these alternative means of gaining operational access to the metric field; cf. footnote 55 for some further discussion of Weyl’s theorem, and Butterfield (2007, §4) for related discussion.
- 21.
Similarly, one might argue that postulating that metric symmetries coincide with dynamical symmetries in SR is an important condition for the metric field η ab to have operational meaning in that case.
- 22.
- 23.
For the time being, my focus is on this mode of gaining operational access to the metric field—though I concede that there may be other means, as discussed in footnote 20 above, and in Sect. 6 below.
- 24.
The ‘locally’ qualification is of particular significance in GR, since the SEP ensures the local coincidence of metric and dynamical symmetries, in the neighbourhood of a given point p ∈M.
- 25.
There exist significant difficulties regarding attempts to tell such a story of ontological reduction in GR; an obvious illustration can be found in the existence of vacuum solutions in the theory. This said, the question of whether an ontological excision of the metric field in GR is possible remains of philosophical and conceptual interest—particularly to advocates of the dynamical approach, for whom this would afford a means of bringing their approach to GR into line with their approach to SR.
- 26.
See Read et al. (2018, §5), where the terminology of ‘miracles’ was introduced, for further discussion.
- 27.
There is some ambiguity regarding what is meant by a ‘theory’ here. To be clear, by ‘theory’ is meant here a theoretical framework such as that for SR or GR as presented in Sect. 2.3, rather than specific theories within those frameworks, such as KGS or KGG.
- 28.
The notation δ ab is chosen to emphasise the analogy with the Kronecker delta
; strictly, however, these are different objects, and should not be confused. - 29.
- 30.
- 31.
The first term is the Einstein–Hilbert action; F ab is the Faraday tensor associated with A a. In this paper, I take it that in GR (or, as here, the Jacobson–Mattingly theory) a vector ξ a at a point is timelike just in case g abξ aξ b < 0.
- 32.
Strictly, I will have to generalise the notion of a ‘metric symmetry’ in Sect. 5.2.2, to account for the examples given in that section. This, however, will be of no consequence.
- 33.
Cf. Brown and Read (2016, §5).
- 34.
It is worth making two related points here. (1): Technically, such coupling is not essential, for we might instead couple to e.g. a fixed Minkowski metric field η ab, or to a generic Lorentzian metric field which satisfies not the Einstein field equations, but some other set of dynamical equations. In the cases in which all dynamical laws feature coupling to g ab, however, this metric field may feature in explanations of the form of all these laws. (2): One need not make the assumption that all dynamical laws manifest certain (local) symmetries so explicitly—one might instead make assumptions of (e.g.) universal coupling of the metric field to matter fields in all dynamical equations for the latter; this may, then, entail the relevant facts about the symmetries of those laws. This, indeed, appears to be Maudlin’s stance, when he writes that ‘the fundamental requirement of a relativistic theory is that the physical laws should be specifiable using only the relativistic space-time geometry. For Special Relativity, this means in particular Minkowski space-time.’ Maudlin (2012, p. 117) The point here is that, on QGA, one may appeal to the metric field in giving certain generic explanations of the behaviour of matter fields in a certain restricted class of models of the theory—but the metric field itself does not account for those restrictions.
- 35.
I am grateful to Oliver Pooley for impressing this point upon me.
- 36.
Scare quotes are included on ‘metric’ here, for strictly neither t ab nor h ab satisfies the metric non-degeneracy condition—cf. Malament (2012, §4.1).
- 37.
For details regarding Leibnizian, Galilean, and Newtonian structures, see Earman (1989, ch. 2).
- 38.
The exact mathematical forms of these groups are not relevant for our purposes—see Pooley (2013, §3.1) for details.
- 39.
A Galilean structure is traditionally considered to be the ‘most appropriate’ spacetime setting for NGT, for in this case structure symmetries and dynamical symmetries (are claimed to) coincide, thereby satisfying Earman’s ‘adequacy conditions’ on spacetime theories (see Earman 1989, §3.4). For recent philosophical discussion calling into question whether this orthodoxy is correct, see Dewar (2018), Knox (2014), Saunders (2013), Teh (2018), Wallace (2017), Weatherall (2016a, 2018); I do not discuss further such matters in this paper.
- 40.
Here,
is the Riemann tensor associated with the derivative operator ∇a defined in the Galilean structure. - 41.
Setting aside the issues indicated in footnote 39.
- 42.
Indeed, I here include scare quotes on the word ‘theory’, as there are good grounds to question whether such a ‘theory’ is really coherent, since it does not have sufficient structure in its KPMs to be able to write down the dynamical equations used to fix its DPMs—cf. Stein (1977, p. 6). (Belot puts the point pithily, when he accuses those working with such theories of ‘arrant knavery’ Belot 2000, p. 571; for further related discussion, cf. Dewar 2018, pp. 268–269.)
- 43.
On this possibility, cf. Pooley’s discussion at Pooley (2013, p. 94).
- 44.
At least on QGA—it is questionable whether this theory is coherent on the dynamical approach, according to which (as discussed above) metric/structure symmetries in theories with fixed metric/structure (such as both SR and NGT) just are dynamical symmetries. Cf. Brown and Read (2020, §3.1).
- 45.
Though in this case the theory is coherent, in a way that arguably NGT set in a Leibnizian structure is not—cf. footnote 42.
- 46.
Arguably, Maudlin falls into this camp, for he both (a) speaks of restricting dynamical equations in SR to those which couple universally to η ab, thereby placing him in QGA (cf. footnote 34); and (b) argues that, in any model of SR, there exists a clock which satisfies the clock hypothesis, and thereby (by definition) correctly reads off intervals along its worldline as given by the metric field (cf. Maudlin 2012, ch. 5). There are good reasons to doubt (b)—cf. Menon et al. (2018), discussed further below.
- 47.
For the full details, see Pitts (2019).
- 48.
Indices in this passage have been altered for consistency with the present paper; there is no change in content.
- 49.
Again, I am grateful to Oliver Pooley for impressing this point upon me. In this regard, cf. Pooley (2013, p. 63), where Pooley writes, ‘What, then, is at stake between the metric-reifying relationalist and the traditional substantivalist? Both parties accept the existence of a substantival entity, whose structural properties are characterised mathematically by a pseudo-Riemannian metric field and whose connection to the behaviour of material rods and clocks depends on, inter alia, the truth of the strong equivalence principle. It is hard to resist the suspicion that this corner of the debate is becoming merely terminological.’
- 50.
Here, I use the notation of Weatherall (2017, p. 6).
- 51.
For Brown’s own discussion of the geodesic principle, see Brown (2005, §9.3). With Brown’s central contention—that geodesic motion of small bodies in GR is a consequence of the Einstein field equations, and is therefore automatic in GR, in a way that it is not in antecedent theories (‘It is no longer a miracle.’ Brown (2005, p. 163))—Weatherall is in disagreement, for (a) geodesic motion is, in fact, independent of the Einstein field equations; (b) similar results can be derived in other theories, e.g. NGT, and Newton–Cartan theory. (For the details of Newton–Cartan theory, in which the gravitational potential φ of NCT is absorbed into a (curved) derivative operator, see Malament 2012, ch. 4.) For Weatherall’s work on the geodesic principle, see Weatherall (2017, 2011c,b, 2012, 2017); I am in agreement with him on these matters. Also worthy of mention in this regard are remarks in a similar vein to (a) made by Pooley (2013, p. 543); and an earlier paper of Malament (2012), in which it is pointed out (pace Brown) that geodesic motion in GR follows only on the assumption of the strengthened dominant energy condition.
- 52.
Here, Weatherall’s notation has been amended slightly: I use ‘∇a’ rather than ‘∇’.
- 53.
In addition to the satisfaction of the strengthened dominant energy condition—again, see the Geroch-Jang theorem as stated above.
- 54.
In the words of Thorne et al., ‘Schiff’s conjecture states that any complete and self-consistent gravitation theory that obeys [the weak equivalence principle] must also, unavoidably, obey [the strong equivalence principle]’ (emphasis in original) (Thorne et al., 1973, p. 3575). In turn, the weak equivalence principle is defined as follows: ‘If an uncharged test body is placed at an initial event in spacetime, and is given an initial velocity there, then its subsequent worldline will be independent of its internal structure and composition’ (emphasis in original) (Thorne et al., 1973, p. 3571); the strong equivalence principle is defined as: ‘(i) [The weak equivalence principle] is valid, and (ii) the outcome of any local test experiment—gravitational or nongravitational—is independent of where and when in the universe it is performed, and independent of the velocity of the (freely falling) apparatus’ (Thorne et al., 1973, p. 3572). For the original presentation of Schiff’s conjecture, see Schiff (1960, p. 343); for ensuing discussion and attempted proofs of restricted versions of the conjecture, see Coley (1982), Lightman and Lee (1973), Ni (1977), Thorne et al. (1973). Clearly, the version of Schiff’s conjecture under consideration in this paper is different to that above—the gap to be bridged here is between the geodesic motions of small bodies, and the symmetries of matter fields tout court.
- 55.
Geroch and Weatherall demonstrate in Geroch and Weatherall (2018) that source-free Maxwell fields ‘track’ null geodesics—a new result. Since the geodesic theorems demonstrate that massive matter moves on timelike geodesics, this gives access to both conformal and projective structure, respectively. One might think, therefore, that one may appeal to the Ehlers–Pirani–Schild result Ehlers et al. (1972) (itself a generalisation of Weyl’s theorem—cf. Weyl 1921), that (subject to extra constraints) conformal and projective structure fixes metric structure, to strengthen the connection between these geodesic theorems and geometry. While such results do indeed yield a further sense in which local geometry may be inferred from geodesic motions, they continue to leave unbridged the gap between the geodesic motions of small bodies, and the local dynamics of matter tout court. That is, Schiff’s conjecture remains unproven, in general.
- 56.
In more detail, recall from footnote 44 that, on the dynamical approach, metric/structure symmetries in theories with fixed metric/structure just are dynamical symmetries—so how could it be the case that there exists a theory in which such symmetries do not coincide?
- 57.
I concede that it is somewhat strained to seek to read Weatherall as an advocate of the dynamical approach; a reading on which he endorses something like QGA is more natural. Nevertheless, it is at least worth noting that advocation of the dynamical approach is consistent with Weatherall’s writings. (Moreover—and interestingly—Weatherall has questioned in personal communication whether fixed metric structure, such as the Minkowski metric field of SR, should be regarded as being ontologically autonomous—in which case, his views are arguably closer to the dynamical approach than one might initially think. Whether, however, it is best to read Weatherall as endorsing the dynamical approach versus e.g. the version of the geometrical approach due to Janssen (2009), Balashov and Janssen (2003), Janssen (2002), in which the ontological autonomy of the metric field in e.g. SR is denied, remains unclear absent further work. Since the issues here are subtle, and it would take significant work to do justice to Janssen, these matters will have to wait for a future piece.)
- 58.
This coupling will ensure that a necessary condition on the metric field’s having chronogeometric significance is satisfied—cf. Sect. 3.
- 59.
Of course, it is also worth remaining conscious of the differences between Knox and Weatherall—for example, Weatherall makes no explicit commitment to inertial structure as the sine qua non of spacetime.
- 60.
‘Structure’ construed here in the sense of Sect. 5.2.2.
; strictly, however, these are different objects, and should not be confused.
is the Riemann tensor associated with the derivative operator ∇a defined in the Galilean structure.References
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Acknowledgements
I am grateful to Erik Curiel, Patrick Dürr, Niels Linnemann, Tushar Menon, Oliver Pooley, and Jim Weatherall for invaluable comments on earlier drafts of this paper; to Harvey Brown, Jeremy Butterfield, Dennis Lehmkuhl, and Brian Pitts for fruitful discussions on the dynamical approach; and to Jim Weatherall (again) for asking a pertinent question which motivated my writing this paper. I am also indebted to audiences in Bern, Bonn, Cambridge, Exeter, Kraków, and London for useful feedback. My D.Phil. studies were supported by an AHRC grant at the University of Oxford; a senior scholarship at Hertford College, Oxford; and a fellowship with the Space and Time After Quantum Gravity project at the University at Illinois at Chicago and the University of Geneva—the latter funded by the John Templeton Foundation.
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Read, J. (2020). Explanation, Geometry, and Conspiracy in Relativity Theory. 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_9
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