[Paper Review] The Energy-Momentum Tensor for the Gravitational Field

S. Babak, L. P. Grishchuk|Jul 7, 1999

Cosmology and Gravitation Theories17 citations

TL;DR

This paper proposes a uniquely defined gravitational energy-momentum tensor in general relativity by treating gravity as a nonlinear tensor field in flat Minkowski spacetime. Using a field-theoretic formulation, the authors derive a tensor that is symmetric, conserved, free of second derivatives, and uniquely determined by six physical and mathematical criteria, offering a consistent framework for gravitational energy and momentum in curved spacetime.

ABSTRACT

The search for the gravitational energy-momentum tensor is often qualified as an attempt of looking for ``the right answer to the wrong question''. This position does not seem convincing to us. We think that we have found the right answer to the properly formulated question. We have further developed the field theoretical formulation of the general relativity which treats gravity as a non-linear tensor field in flat space-time. The Minkowski metric is a reflection of experimental facts, not a possible choice of the artificial ``prior geometry''. In this approach, we have arrived at the gravitational energy-momentum tensor which is: 1) derivable from the Lagrangian in a regular prescribed way, 2) tensor under arbitrary coordinate transformations, 3) symmetric in its components, 4) conserved due to the equations of motion derived from the same Lagrangian, 5) free of the second (highest) derivatives of the field variables, and 6) is unique up to trivial modifications not containing the field variables. There is nothing else, in addition to these 6 conditions, that one could demand from an energy-momentum object, acceptable both on physical and mathematical grounds. The derived gravitational energy-momentum tensor should be useful in practical applications.

Motivation & Objective

• To resolve long-standing ambiguities in defining gravitational energy-momentum by formulating gravity as a field on flat spacetime.
• To address the criticism that the search for a gravitational energy-momentum tensor is ill-posed by showing it is possible to derive a physically and mathematically consistent tensor.
• To ensure the derived tensor satisfies six stringent criteria: derivability from Lagrangian, tensor transformation, symmetry, conservation, absence of second derivatives, and uniqueness up to trivial terms.
• To provide a practical and consistent tool for energy-momentum analysis in gravitational systems, especially in radiating or dynamic spacetimes.

Proposed method

• Adopt a field-theoretic approach where gravity is treated as a nonlinear tensor field on a fixed Minkowski background, not as a dynamical metric.
• Use the variational principle on a Lagrangian density that depends on the field variables and the Minkowski metric, ensuring consistency with Noether's theorem.
• Derive the energy-momentum tensor via the standard variational derivative with respect to the metric, ensuring it transforms as a tensor under arbitrary coordinate changes.
• Enforce symmetry and conservation through explicit algebraic manipulation of the Lagrangian and its variation, eliminating non-tensorial or non-conserved components.
• Apply constraints and symmetries (e.g., Riemann tensor symmetries) to simplify the variational derivative and isolate the physical energy-momentum contribution.
• Verify that the resulting tensor is free of second derivatives and uniquely determined up to trivial terms not involving field variables.

Experimental results

Research questions

• RQ1 Can a consistent gravitational energy-momentum tensor be derived in general relativity without relying on the dynamical metric as the fundamental field?
• RQ2 What are the minimal physical and mathematical conditions that a gravitational energy-momentum tensor must satisfy to be considered acceptable?
• RQ3 Is it possible to construct a tensor that is symmetric, conserved, and free of second derivatives while being derivable from a Lagrangian in a flat spacetime formulation?
• RQ4 How does the proposed tensor compare to canonical and metrical energy-momentum tensors in standard formulations of GR?
• RQ5 Does the field-theoretic approach on Minkowski spacetime yield a unique and physically meaningful expression for gravitational energy and momentum?

Key findings

• The derived gravitational energy-momentum tensor is uniquely determined up to trivial modifications not involving field variables, satisfying all six required criteria.
• The tensor is symmetric, conserved due to the equations of motion, and transforms as a proper tensor under arbitrary coordinate transformations.
• The tensor is free of second derivatives of the field variables, ensuring it depends only on first-order derivatives and is well-behaved in field equations.
• The variational derivation from the Lagrangian yields a consistent and unambiguous expression, with the final form given by Eq. (B17) in the paper.
• The tensor is shown to be equivalent to the metrical energy-momentum tensor in the limit of the standard formulation, but derived via a more fundamental field-theoretic approach.
• The method successfully resolves the ambiguity in defining gravitational energy-momentum by proving that no further conditions are physically or mathematically necessary beyond the six stated criteria.