Gauge-Invariant Localization of Infinitely Many Gravitational Energies from All Possible Auxiliary Structures

Abstract

The problem of finding a covariant expression for the distribution and conservation of gravitational energy-momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann's realization that there are infinitely many conserved gravitational energy-momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors of a given type, such as the Einstein pseudotensor, in_every_ coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau-Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether's theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many.

Added section on spinors without a tetrad, added section on errors inspired by supposed nonlocalization of energy, added references, minor amplifications and clarifications. Forthcoming in_General Relativity and Gravitation_

References in corpus (4)

• The Hamiltonian formulation of General Relativity: myths and reality
• Diffeomorphism Invariance in the Hamiltonian formulation of General Relativity
• Nonlinear Perturbations and Conservation Laws on Curved Backgrounds in GR and Other Metric Theories
• Noether's theorem, the stress-energy tensor and Hamiltonian constraints

Cited by in corpus (18)

• Change in Hamiltonian General Relativity from the Lack of a Time-like Killing Vector Field
• Gravitational Energy for GR and Poincare Gauge Theories: a Covariant Hamiltonian Approach
• On Geometric Objects, the Non-Existence of a Gravitational Stress-Energy Tensor, and the Uniqueness of the Einstein Field Equation
• Permanent Underdetermination from Approximate Empirical Equivalence in Field Theory: Massless and Massive Scalar Gravity, Neutrino, Electromagnetic, Yang-Mills and Gravitational Theories
• The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure
• On Gravitational Energy in Newtonian Theories
• Methodological Reflections on the MOND/Dark Matter Debate
• The Non-Relativistic Geometric Trinity of Gravity
• Universally Coupled Massive Gravity, III: dRGT-Maheshwari Pure Spin-$2$, Ogievetsky-Polubarinov and Arbitrary Mass Terms
• Einstein's Equations for Spin $2$ Mass $0$ from Noether's Converse Hilbertian Assertion
• Underdetermination in Classic and Modern Tests of General Relativity
• Einstein's Physical Strategy, Energy Conservation, Symmetries, and Stability: "but Grossmann & I believed that the conservation laws were not satisfied"
• How Dualists Should (Not) Respond to the Objection from Energy Conservation
• The Mass of the Gravitational Field
• Intrinsic vanishing of energy and momenta in a universe
• Progress and Gravity: Overcoming Divisions between General Relativity and Particle Physics and between Physics and HPS
• What Represents Space-time? And What Follows for Substantivalism \emph{vs.} Relationalism and Gravitational Energy?
• A Quasigroup Approach for Conservation Laws in Asymptotically Flat Spacetimes