The Gravitational Hamiltonian in the Presence of Non-Orthogonal Boundaries

TL;DR

The paper extends the gravitational Hamiltonian framework to non-orthogonal boundaries between spacelike hypersurfaces and a timelike boundary, introducing the non-orthogonality parameter $\eta$ and identifying a tilting term $H_t$ that encodes this dependence. By decomposing the action into canonical form and incorporating a background subtraction, the authors obtain a finite Hamiltonian with four pieces: $H_c$, $H_k$, $H_t$, and $H_m$, where $H_t$ vanishes upon proper background matching. Through Schwarzschild examples with both flat and tilted slicings, the work shows how the physical Hamiltonian depends on slicing and boundary matching, with tilting effects canceling in the background subtraction and momentum contributions becoming physically meaningful under boosts. The results underpin the interpretation of conserved charges as energies and momenta relative to a background spacetime, and highlight the role of boundary geometry in defining gravitational dynamics and thermodynamics in non-orthogonal settings.

Abstract

This paper generalizes earlier work on Hamiltonian boundary terms by omitting the requirement that the spacelike hypersurfaces $Σ_t$ intersect the timelike boundary $\cal B$ orthogonally. The expressions for the action and Hamiltonian are calculated and the required subtraction of a background contribution is discussed. The new features of a Hamiltonian formulation with non-orthogonal boundaries are then illustrated in two examples.

Figures (1)

• Figure 1: The spacetime manifold $\cal M$ and its submanifolds are shown, with one spacelike dimension suppressed. The unit normals $n^\mu$, and $u^\mu$ to the spacelike hypersurface $\Sigma_t$ and the timelike boundary $\cal B$ are shown at points on the two-surface $B_t$.