Virial identities in relativistic gravity: 1D effective actions and the role of boundary terms

TL;DR

This paper explores the derivation of virial identities in relativistic gravity, emphasizing the importance of boundary terms like Gibbons-Hawking-York, and introduces a gauge choice that simplifies their computation in spherically symmetric scenarios.

Contribution

It provides a pedagogical derivation of virial identities from 1D effective actions and highlights the crucial role of boundary terms in relativistic gravity.

Findings

Boundary terms are essential for complete virial identities in gravity.

A specific gauge choice simplifies virial identity calculations in spherically symmetric cases.

Virial identities can be derived from matter actions alone under certain gauge conditions.

Abstract

Virial (aka scaling) identities are integral identities that are useful for a variety of purposes in non-linear field theories, including establishing no-go theorems for solitonic and black hole solutions, as well as for checking the accuracy of numerical solutions. In this paper, we provide a pedagogical rationale for the derivation of such integral identities, starting from the standard variational treatment of particle mechanics. In the framework of one-dimensional (1D) effective actions, the treatment presented here yields a set of useful formulas for computing virial identities in any field theory. Then, we propose that a complete treatment of virial identities in relativistic gravity must take into account the appropriate boundary term. For General Relativity this is the Gibbons-Hawking-York boundary term. We test and confirm this proposal with concrete examples. Our analysis here is restricted to spherically symmetric configurations, which yield 1D effective actions (leaving higher-D effective actions and in particular the axially symmetric case to a companion paper). In this case, we show that there is a particular "gauge" choice, $i.e.$ a choice of coordinates and parameterizing metric functions, that simplifies the computation of virial identities in General Relativity, making both the Einstein-Hilbert action and the Gibbons-Hawking-York boundary term non-contributing. Under this choice, the virial identity results exclusively from the matter action. For generic "gauge" choices, however, this is not the case.