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Fresh look at boundary terms in Einstein-Hilbert gravity via an initial value variational principle
Phys. Rev. D 113, 124050 – Published 15 June, 2026
DOI: https://doi.org/10.1103/7psx-3kkg
Abstract
A key tenet of general relativity is the dynamical nature of spacetime, ideally represented as an initial value problem. Here we explore the variational formulation of classical Einstein-Hilbert gravity as an 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 is 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 action principle for gravity as an initial value problem is rendered well-posed in the variational sense, 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 from the action, bypassing the governing equations.
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