Moving observers, nonorthogonal boundaries, and quasilocal energy

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 timelike 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 timelike boundary rather than the foliation of the entire four dimensional 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.