Abstract
Several authors have claimed that prediction is essentially impossible in the general theory of relativity, the case being particularly strong, it is said, when one fully considers the epistemic predicament of the observer. Each of these claims rests on the support of an underdetermination argument and a particular interpretation of the concept of prediction. I argue that these underdetermination arguments fail and depend on an implausible explication of prediction in the theory. The technical results adduced in these arguments can be related to certain epistemic issues, but can only be misleadingly or mistakenly characterized as related to prediction.
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Notes
By a relativistic spacetime I mean a smooth, connected four-dimensional manifold \(\mathscr {M}\) and a smooth, non-degenerate pseudo-Riemannian metric of Lorentz signature g defined on \(\mathscr {M}\).
The timelike (respectively, causal) past of a point q in a time-orientable spacetime \(\mathscr {M}\) is the set of all points p such that there exists a future-directed curve from p to q that is timelike (timelike or null). If such a curve exists, one writes \(p \ll q\) (\(p < q\)). The timelike (causal) past of a point q is usually denoted \(I^-(q)\) (\(J^-(q)\)). The timelike (causal) future of a point q, \(I^+(q)\) (\(J^+(q)\)) are defined analogously.
A closed spacelike surface S in \(\mathscr {M}\) is a three-dimensional subset of M where every smooth curve into the subset is a spacelike curve. A subset S is achronal if the intersection of \(I^+(S)\) and S is empty. The domain of dependence of a subset S, denoted D(S), is the set of all points p in \(\mathscr {M}\) such that every past or future inextendible causal curve through p intersects S.
Of course in a relativistic spacetime not all events which fall outside an observer’s past are future, since some events are spacelike related to the observer, thus one may as well allow events which are spacelike-related to the observer to count as predictable.
“Let it be assumed that the observer at q has full epistemic access to the influences in [the causal past of q]” (Hogarth 1993, p. 723).
“For present purposes take prediction to mean deterministic prediction from the laws of physics” (Earman 1995, p. 128).
Note that the condition does remove all reference to the observer, but simply for the reason that “observer” is not part of the primitive language of GTR. One may of course easily make a suitable definition. For example, “we can associate with each observer his space–time trajectory or cosmic world-line which is itself, necessarily, a future-directed timelike curve” (Malament 1977, p. 63).
For example: “Given a spacetime \(\mathscr {M}, g\) and a point q of \(\mathscr {M}\) such that [the domain of prediction of q] contains a point to the future of q, then \(\mathscr {M}, g\) is a closed universe, in the sense that it admits a compact spacelike surface” (Geroch 1977, p. 92). “The domain of prediction for each point in Minkowski space–time is empty. The same is true of many of the cosmological models of general relativity, e.g., the Robertson-Walker models which have been used to describe a universe beginning or ending with a big bang” (Earman 1986, p. 193).
Beisbart asserts that common intuitions record a difference: “In physics, we distinguish between dynamical theories and the initial conditions. The former are often supposed to hold necessarily, if they hold, whereas initial conditions are supposed to be contingent. As a consequence, induction may be on firmer grounds for laws than for initial conditions” (Beisbart 2009, p. 201). Maudlin (2007) too shares this intuition, but provides no compelling argument in support. There is, so far as I can see, nothing decisive in such metaphysical intuitions, especially given the epistemological parity of dynamical and non-dynamical conditions. Thus the burden is on those who wish to make the distinction to justify it.
Norton (2011) too recognizes that condition (ii) rests on “an excessively optimistic assessment of our observational abilities”.
“Using observationally determined initial data, completed for those which are determined not precisely enough or which correspond to observationally not accessible regions on the past light cone, one may construct essentially local cosmological models inside the past light cone...” (Dautcourt 1983, p. 153, emphasis added).
Examples of this contemporary understanding of prediction abound. Here is one: “The cosmological singularity (in all examples where its character is not known to be unstable) involves infinite curvature and infinite density. One’s abhorrence of such a theoretical prediction is particularly heightened by the correlative prediction that these infinities occurred at a finite proper time in the past,...” (Misner et al. 1973, p. 813).
To be sure this is an epistemology that was easier to maintain in pre-relativistic days. I am inclined in fact to interpret this as one of the central points of Geroch (1977).
“One may wonder if the definition of the domain of prediction given above accurately reflects the physical notion of ‘making predictions in general relativity.’ It is [my] position that it does not” (Manchak 2008, p. 319).
An inextendible spacetime is a spacetime for which all isometric embeddings into another spacetime are surjective. Every extendible spacetime has an (not necessarily unique) inextendible extension.
As written, this definition is too weak. There may be multiple isometric embeddings of q’s causal past into another spacetime, only one of which has the required isometric embedding of the union of q’s causal past and p’s causal past. In short, the observer may not only inhabit one of many observationally indistinguishable spacetimes, but may also suffer from self-locating uncertainty within a spacetime. Requiring that the first embedding is unique avoids this issue, as does requiring that the second embedding exists for each of the first.
As written, this definition is also too strong. The predicted event p’s causal past need not be shared in all observationally indistinguishable spacetimes, since the entire causal past of p is unnecessary to determine p. A suitable requirement would only have that the intersection of the future domain of dependence of the surface S that determines p and p’s causal past is shared in all spacetimes.
As Norton comments, “Our observable spacetime is four-dimensional and has a Lorentz signature metrical structure. We are allowed the inductive inference that this will persist in the unobserved part. More generally, we are allowed to infer inductively to the persistence of any local condition, such as the obtaining of the Einstein gravitational field equations, in both the observer’s and the [observationally indistinguishable] spacetimes” (Norton 2011, p. 170).
Manchak (2009b) agrees, and for this reason adopts the view that it may be possible to demonstrate hole-freeness in GTR by assuming suitably physically motivated conditions.
“But if a prediction depends crucially upon the precise data—if it undergoes a drastic change under even arbitrarily small perturbations of that data—then our prediction, while perhaps suggestive and useful, has little physical significance” (Geroch 1971a).
Geroch, at least, acknowledges that to follow this method is to operate in a “rather narrow framework” (Geroch 1977, p. 83).
As I observe in Sect. 2, there is no epistemologically significant difference between the way we come to learn about dynamical and non-dynamical constraints. Indeed, some dynamical laws—the EFE being a particularly salient example in this case—can be understood as incorporating dynamical and non-dynamical constraints, such as when GTR is treated as a constrained Hamiltonian system. Tradition and metaphysical intuitions may incline one toward inflating the distinction’s importance in interpretational contexts, but nobody has made it clear why one should. There are, in any case, reasonably good accounts of laws which do not presuppose that laws are necessarily dynamical. Cf. the relevant parts of Callender (2004, 2007).
Although I do not propose a positive definition of prediction in this paper, I think the two conditions I suggest in the paper, namely conditions (ii\(^\prime \)) and (iii\(^\prime \)), are important and relevant to having a clear conception of prediction in GTR. Nevertheless, I do not think they necessarily constitute an adequate definition of prediction.
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McCoy, C.D. Prediction in general relativity. Synthese 194, 491–509 (2017). https://doi.org/10.1007/s11229-015-0954-3
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DOI: https://doi.org/10.1007/s11229-015-0954-3