On the `simple' form of the gravitational action and the self-interacting graviton

TL;DR

This paper reformulates General Relativity as a self-interacting graviton theory using the $ ext{ΓΓ}$-form of the Lagrangian, introducing a covariant approach and clarifying the relation between different energy-momentum tensor definitions.

Contribution

It provides a covariant reformulation of Einstein's equations as a graviton field equation and explicitly constructs superpotentials linking various energy-momentum definitions.

Findings

Recast Einstein equations as a graviton field equation in standard field theory form.

Explicit superpotentials relate different energy-momentum tensor definitions.

Demonstrates the $ ext{ΓΓ}$-form's efficiency in generating graviton vertices.

Abstract

The so-called $\Gamma\Gamma$-form of the gravitational Lagrangian, long known to provide its most compact expression as well as the most efficient generation of the graviton vertices, is taken as the starting point for discussing General Relativity as a theory of the self-interacting graviton. A straightforward but general method of converting to a covariant formulation by the introduction of a reference metric is given. It is used to recast the Einstein field equation as the equation of motion of a spin-2 particle interacting with the canonical energy-momentum tensor symmetrized by the standard Belinfante method applicable to any field carrying nonzero spin. This represents the graviton field equation in a form complying with the precepts of standard field theory. It is then shown how representations based on other, at face value completely unrelated definitions of energy-momentum (pseudo)tensors are all related by the addition of appropriate superpotential terms. Specifically, the superpotentials are explicitly constructed which connect to: i) the common definition consisting simply of the nonlinear part of the Einstein tensor; ii) the Landau-Lifshitz definition.