Source
Belot, “The Representation of Time and Change in Mechanics”, in Butterfield & Earman (eds.), Philosophy of Physics (Elsevier 2007), pp. 133–227; LIT-122, read in full in NOTE-118.
What was actually shown
The well-behaved case (§5). Given global existence, uniqueness and a dynamical time-translation group, the space of solutions S and the space of initial data I are symplectic. A slicing gives a one-parameter family of symplectic isomorphisms T_Σt : S → I that intertwine time translation on S with time evolution on I. That family lets initial data represent recurring states, and lets functions on S represent changeable quantities.
Damage that can be repaired (§6). Singular dynamics (the planar Kepler problem’s solution space is non-Hausdorff), gauge freedom (presymplectic spaces, isomorphic after reduction “when all goes well”) and time-dependence each break part of the picture, and each has a repair.
General relativity (§7.1–7.2). For vacuum GR with compact Cauchy surfaces the Hamiltonian obtained by the usual rule is h ≡ 0. Dynamical trajectories stay inside gauge orbits and are constant on the reduced space of initial data (p. 202). There is no ℝ-action implementing time on S, on the reduced S′ or on the reduced I′. Hence: “time is not represented in general relativity by a flow on a symplectic space and change is not represented by functions on a space of instantaneous or global states” (p. 209).
What it is not caused by (p. 210). The problem is derived in the covariant Lagrangian picture, so it is not an artefact of the 3+1 decomposition. In asymptotically flat GR the Poincaré group at infinity supplies a non-zero Hamiltonian and change is represented in the familiar way (pp. 204–206). Lack of a preferred slicing, slicing freedom and diffeomorphism invariance are therefore not sufficient. The source is geometry that is fully dynamical, with no background structure “at spatial infinity or elsewhere”.
Not an illusion (pp. 209–210). The reduced space of solutions contains worlds with change (Big Bang solutions) and without it (Einstein static). So the formalism fails to represent change; it does not show there is none.
What could have come out otherwise. The problem could have disappeared in the covariant picture, or appeared equally in asymptotically flat GR. It did neither.
What this does not say
- That geometric time solves it. Constant-mean-curvature time gives a time-dependent Hamiltonian for some spatial topologies (Example 54). Belot offers §7.3 “by way of further clarification of the problem of time, rather than as a suggested resolution” (p. 196).
- That every generally covariant theory escapes or suffers it. That an artificially covariant theory on a fixed background escapes is hedged (“I think”, p. 210), with a residual worry about the conformal class.
- That the reduced spaces are canonically isomorphic. That is “presumed” (p. 202), citing Fischer & Moncrief.
- That GR’s content is the set of all its Hamiltonianizations. That is a stated view, “entirely in the spirit” of GR (p. 220), not an argued result.
- Anything settled about quantum gravity. Distinct geometric times are not expected to give unitarily equivalent quantizations; the hopes are “long shots” (pp. 220–221).
- That the result covers all of GR. It is for vacuum, zero cosmological constant, globally hyperbolic, spatially compact solutions. The problem “only arises in those versions of general relativity most appropriate to the cosmological setting” (fn. 7, p. 137).
Connections
- The record’s reading recommends this as the work to cite against claims that general relativity shows time to be unreal (NOTE-118).
- Its treatment of gauge reduction as removing “physically otiose” variables is the move Le Bihan & Read consider for dualities (LIT-113; THEORY-687).