Bootstrapping gravity: a consistent approach to energy-momentum self-coupling

TL;DR

This paper introduces a new graviton action that ensures consistent energy-momentum self-coupling, reproducing Einstein's equations order by order, and extends the formalism to include matter and cosmological constant effects.

Contribution

The authors propose a novel graviton action that guarantees consistent energy-momentum self-coupling and systematically derives Einstein's equations through perturbative expansion.

Findings

The new graviton action supplies the correct second-order term in Einstein's equations.

The formalism demonstrates a straightforward self-coupling procedure generating Einstein-Hilbert action.

Inclusion of matter and cosmological constant extends the applicability of the approach.

Abstract

It is generally believed that coupling the graviton (a classical Fierz-Pauli massless spin-2 field) to its own energy-momentum tensor successfully recreates the dynamics of the Einstein field equations order by order; however the validity of this idea has recently been brought into doubt [1]. Motivated by this, we present a graviton action for which energy-momentum self-coupling is indeed consistent with the Einstein field equations. The Hilbert energy-momentum tensor for this graviton is calculated explicitly and shown to supply the correct second-order term in the field equations; in contrast, the Fierz-Pauli action fails to supply the correct term. A formalism for perturbative expansions of metric-based gravitational theories is then developed, and these techniques employed to demonstrate that our graviton action is a starting point for a straightforward energy-momentum self-coupling procedure that, order by order, generates the Einstein-Hilbert action (up to a classically irrelevant surface term). The perturbative formalism is extended to include matter and a cosmological constant, and interactions between perturbations of a free matter field and the gravitational field are studied in a vacuum background. Finally, the effect of a non-vacuum background is examined, and the graviton is found to develop a non-vanishing ``mass-term'' in the action.