| dc.description.abstract | We present two complementary approaches for determining the reference for the co-
variant Hamiltonian boundary term quasi-local energy and test them on spherically
symmetric spacetimes. On the one hand, we extremize the energy in two ways, which
we call energy-extremization programs A and B. Both programs produce reasonable
results that allow us to discuss energies measured by diRerent observers. We show
that the energies produced by program A can be positive, zero, or even negative,
while in program B they are always non-negative. On the other hand, we match the
orthonormal frames of the dynamic and the reference spacetimes right on the two-
sphere boundary. If we further require that the reference displacement vector to be
the timelike Killing vector, the result is the same as program A. If, instead, we require
that the Lie derivatives of the two-area along the displacement vector in both the dy-
namic and reference spacetimes are the same, the result is the same as program B,
which satis¯es the usual criteria. In particular, the energies are non-negative and van-
ish only for Minkowski (or anti-de Sitter) spacetime. So by studying the spherically
symmetric spacetimes, both static and dynamic, we learn that the references deter-
mined by our energy extremization programs are those which isometrically match
the dynamic spacetimes on the boundary. And the energies determined by isometric
matching approach are actually the extremum measured by the associated observers.
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