What Represents Space-time? And What Follows for Substantivalism \emph{vs.} Relationalism and Gravitational Energy?

TL;DR

This paper proposes a novel non-perturbative split in general relativity that models space-time with a constant background matrix, avoiding traditional issues with gravitational energy and substantivalism.

Contribution

It introduces a background matrix approach to represent space-time, circumventing the need for a metric-based substantivalist view and addressing gravitational energy debates.

Findings

The background matrix approach avoids extra gauge freedoms.

Traditional objections to pseudotensors are weakened and addressed.

The approach is compatible with strong fields, arbitrary coordinates, and topology.

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; particle physics egalitarianism encourages realism about gravitational energy. This essay proposes a third view, involving a particle physics-inspired non-perturbative split that characterizes space-time with a constant background _matrix_ (not a metric), avoiding the inference from gravitational energy to substantivalism: space-time is (M, eta), where eta=diag(-1,1,1,1) is a spatio-temporally constant numerical signature matrix (already used in GR with spinors). The gravitational potential, bearing any gravitational energy, is g_(munu)(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 ones, involves no extra gauge freedom and so lacks their obscurities and carpet lump-moving. After a discussion of Curiel's GR exceptionalist 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 of Noether's theorem (infinitely many energies) largely answers Schroedinger's false-negative coordinate dependence problem. Bauer's nonuniqueness (false positive) objection has several answers.