The general set of Noetherian energy-momentum tensors in linearized gravity: mathematical framework

Abstract

Energy-momentum tensors are foundational objects which are uniquely defined in standard physical field theories such as electrodynamics and Yang-Mills theory. In general relativity, and in particular linearized gravity where symmetries required for an energy-momentum tensor derived from Noether's first theorem are well defined, there exists a long standing non-uniqueness problem; numerous distinct energy-momentum expressions exist in the literature, and there is not consensus which, if any, is the unique expressions for the theory. Recently, the viability of the superpotential `improvement' method was shown to be insufficient for addressing the non-uniqueness problem of energy-momentum tensors in linearized gravity. In the present article, the mathematical framework for the general set of Noetherian energy-momentum tensors in linearized gravity is derived using Noether's first theorem, which consists of all possible energy-momentum tensors from the Noether current which yield the linearized Einstein field equations in the corresponding Euler-Lagrange equation of the Noether identity without introducing any `improvement' terms. This result has several advantages in addition to not requiring `improvements', such as the ability to impact the Lagrangian proportional piece of the energy-momentum tensor. Numerous common published gravitational energy-momentum expressions are then compared to these general results to assess which can be classified as Noetherian, and which cannot. Standard physical criteria such as symmetry and tracelessness are then used to prove that it is possible to directly obtain an energy-momentum tensor which is simultaneously symmetric and traceless from the Noether current; such an expression is derived from the general results. Consequences of these results and their relation to the aforementioned non-uniqueness problem are discussed.

References in corpus (25)

• Pseudotensors and quasilocal energy-momentum
• The Energy-Momentum Tensor for the Gravitational Field
• Currents and the Energy-Momentum Tensor in Classical Field Theory: A fresh look at an Old Problem
• From Gravitons to Gravity: Myths and Reality
• Quasilocal quantities for GR and other gravity theories
• Gravity from self-interaction redux
• Unimodular gravity and general relativity from graviton self-interactions
• The Energy-Momentum Tensor(s) in Classical Gauge Theories
• Canonical and gravitational stress-energy tensors
• Gravitational Energy for GR and Poincare Gauge Theories: a Covariant Hamiltonian Approach
• Bootstrapping gravity: a consistent approach to energy-momentum self-coupling
• Energy-Momentum Distribution: A Crucial Problem in General Relativity
• The Role of Energy and a New Approach to Gravitational Waves in General Relativity
• Post-Newtonian Celestial Dynamics in Cosmology: Field Equations
• The Energy-Momentum Problem in General Relativity
• Some remarks on the Einstein and Møller pseudotensors for static and spherically-symmetric configurations
• Energy Localization Invariance of Tidal Work in General Relativity
• Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent
• Symmetry properties of the metric energy-momentum tensor in classical field theories and gravity
• Energy-momentum tensors in linearized Einstein's theory and massive gravity: The question of uniqueness
• A connection between linearized Gauss-Bonnet gravity and classical electrodynamics
• Canonical Noether and the energy-momentum non-uniqueness problem in linearized gravity
• Improvement of a conserved current density versus adding a total derivative to a Lagrangian density
• Energy-momentum tensor and duality symmetry of linearized gravity in the Fierz formalism
• Two body dynamics in a quadratic modification of general relativity

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