A fresh look at boundary terms in Einstein-Hilbert gravity via an initial value variational principle

TL;DR

This paper reformulates Einstein-Hilbert gravity as an initial value problem using a Schwinger-Keldysh-Galley action, emphasizing boundary terms and providing a well-posed variational principle without extra boundary conditions.

Contribution

It introduces a boundary-inclusive variational formulation of gravity via the SKG action, avoiding additional boundary terms and enabling direct metric determination.

Findings

The SKG action naturally decomposes into bulk and boundary terms.

The variational principle is well-posed without extra boundary conditions.

The approach offers a new way to solve for spacetime metrics directly.

Abstract

A key tenet of general relativity is the dynamical nature of space-time, ideally represented as an initial value problem. Here we explore the variational formulation of classical Einstein-Hilbert gravity as initial value problem by constructing its Schwinger-Keldysh-Galley (SKG) action, including a careful treatment of boundary terms. The construction is based on a doubling of degrees of freedom and independent of a foliation. The action naturally decomposes into a bulk term furnishing Einstein's equations and a boundary term, which is related to conserved quantities, such as the Komar mass. We find that since only trivial connecting conditions must be specified on boundaries, the variational action principle for gravity as an initial value problem is rendered well-posed without the need to add additional boundary terms. The SKG approach to gravity offers a novel and complementary avenue to solve for the metric of spacetime directly from the action, bypassing the governing equations.