Moving Observers, Non-orthogonal Boundaries, and Quasilocal Energies

TL;DR

This paper extends the Hamilton-Jacobi method for defining quasilocal energies by removing the orthogonality assumption of boundaries, allowing for more general observer motion and boundary configurations.

Contribution

It introduces a new formulation that considers non-orthogonal boundaries and focuses on the boundary foliation, enhancing the applicability of quasilocal energy definitions.

Findings

Derived a generalized expression for quasilocal energies with moving observers.

Calculated energies for observers moving relative to a Schwarzschild black hole.

Showed the advantages of non-orthogonal boundary treatment in theoretical and computational contexts.

Abstract

The popular Hamilton-Jacobi method first proposed by Brown and York for defining quasilocal quantities such as energy for spatially bound regions assumes that the spatial boundary is orthogonal to the foliation of the spacetime. Such a restriction is undesirable for both theoretical and computational reasons. We remove the orthogonality assumption and show that it is more natural to focus on the foliation of the spatial boundary rather than the foliation of the entire spatially bound region. Reference spacetimes which define additional terms in the action are discussed in detail. To demonstrate this new formulation, we calculate the quasilocal energies seen by observers who are moving with respect to a Schwarzschild black hole.