Deriving the Einstein Lagrangian from conservation of the symmetrized Belinfante tensor

TL;DR

This paper shows that the Einstein Lagrangian can be derived from the conservation of the symmetrized Belinfante energy-momentum tensor, clarifying its role in the spin-two derivation of general relativity.

Contribution

It demonstrates that conservation of the Belinfante tensor leads uniquely to the Einstein Lagrangian under certain conditions, linking energy-momentum conservation to the form of gravity.

Findings

Conservation of the total energy-momentum tensor is equivalent to Feynman's consistency condition.

The Einstein Lagrangian density is uniquely determined by conservation principles.

The symmetrized Belinfante tensor relates to the Papapetrou pseudotensor for the Einstein Lagrangian.

Abstract

We revisit the field-theoretic derivation of the Einstein Lagrangian for a symmetric rank-two tensor field on a Minkowski background. Instead of imposing Feynman's consistency condition directly, we ask whether it can be obtained from the conservation of a definite energy-momentum object. We take the field contribution to the total energy-momentum tensor to be the symmetrized Belinfante tensor associated with a Poincar\'{e}-invariant field Lagrangian. For a point particle coupled universally to the symmetric tensor field, we show that conservation of the total energy-momentum tensor is equivalent to Feynman's consistency condition. Assuming that the field Lagrangian is local, Lorentz invariant, and quadratic in first derivatives, this condition determines the Lagrangian, up to a total divergence, to be the Einstein Lagrangian density. This result clarifies the role of conservation of the symmetrized Belinfante energy-momentum tensor in the spin-two derivation of general relativity. For the Einstein Lagrangian, the resulting symmetrized Belinfante tensor is further shown to be related to the Papapetrou energy-momentum pseudotensor.