Pitts, J. Brian (2022) What Represents Spacetime? And What Follows for Substantivalism vs. Relationalism and Gravitational Energy? [Preprint]

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Abstract
The questions of what represents spacetime in GR, the status of gravitational energy, the substantivalistrelationalist issue, and the (non)exceptional status of gravity are interrelated. If spacetime has energymomentum, then spacetime is substantival. Two extant ways to avoid the substantivalist conclusion deny that the energybearing metric is part of spacetime or deny that gravitational energy exists. Feynman linked doubts about gravitational energy to GRexceptionalism, as do Curiel and Duerr; particle physics egalitarianism encourages realism about gravitational energy.
In that spirit, this essay proposes a third possible view about spacetime, one involving a particle physicsinspired nonperturbative split that characterizes spacetime 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, spacetime is < M, eta>, where eta=diag(1,1,1,1) is a spatiotemporally 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 nonperturbative split permits strong fields, arbitrary coordinates, and arbitrary topology, and hence is pure GR by almost any standard. This razorthin 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 lumpmoving.
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 falsenegative coordinate dependence problem. Bauer's falsepositive objection has multiple answers. Nonuniqueness 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 nonuniqueness and coordinate dependence ambiguities as one.
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