Pitts, J. Brian (2022) What Represents Space-time? And What Follows for Substantivalism vs. Relationalism and Gravitational Energy? [Preprint]
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Abstract
The questions of what represents space-time in GR, the status of gravitational energy, the substantivalist-relationalist issue, and the (non-)exceptional status of gravity are interrelated. If space-time has energy-momentum, then space-time is substantival. Two extant ways to avoid the substantivalist conclusion deny that the energy-bearing metric is part of space-time or deny that gravitational energy exists. Feynman linked doubts about gravitational energy to GR-exceptionalism, as do Curiel and Duerr; particle physics egalitarianism encourages realism about gravitational energy.
In that spirit, this essay proposes a third possible view about space-time, one involving a particle physics-inspired non-perturbative split that characterizes space-time with a constant background _matrix_ (not a metric tensor), a sort of vacuum value, thus avoiding the inference from gravitational _energy to substantivalism. On this proposal, space-time is < M, eta>, where eta=diag(-1,1,1,1) is a spatio-temporally constant numerical signature matrix, a matrix already used in GR with spinors. The gravitational potential, to which any gravitational energy can be ascribed, is g_{\mu\nu}(x)- eta (up to field redefinitions), an _affine_ geometric object with a tensorial Lie derivative and a vanishing covariant derivative. This non-perturbative split permits strong fields, arbitrary coordinates, and arbitrary topology, and hence is pure GR by almost any standard. This razor-thin background, unlike more familiar backgrounds (e.g., Rosen's flat metric tensor field, Rosenfeld and Moeller's orthonormal tetrad, and Sorkin's background connection), involves no extra gauge freedom and so lacks their obscurities and carpet lump-moving.
After a discussion of Curiel's GR exceptionalist denial of the localizability of gravitational energy and his rejection of energy conservation, the two traditional objections to pseudotensors, coordinate dependence and nonuniqueness, are explored. Both objections are inconclusive and getting weaker. A literal interpretation involving infinitely many energies corresponding by Noether's first theorem to the infinite symmetries of the _action_ (or laws) largely answers Schroedinger's false-negative coordinate dependence problem. Bauer's false-positive objection has multiple answers. Non-uniqueness might be handled by Nester et al.'s finding physical meaning in multiplicity in relation to boundary conditions, by an optimal candidate, or by Bergmann's identifying the non-uniqueness and coordinate dependence ambiguities as one.
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