Abstract
We present a precise definition of a conserved quantity from an arbitrary covariantly conserved current available in a general curved space–time with Killing vectors. This definition enables us to define energy and momentum for matter by the volume integral. As a result we can compute charges of Schwarzschild and BTZ black holes by the volume integration of a delta function singularity. Employing the definition we also compute the total energy of a static compact star. It contains both the gravitational mass known as the Misner–Sharp mass in the Oppenheimer–Volkoff equation and the gravitational binding energy. We show that the gravitational binding energy has the negative contribution at maximum by 68% of the gravitational mass in the case of a constant density. We finally comment on a definition of generators associated with a vector field on a general curved manifold.
Highlights
Since Einstein submitted papers on general relativity,[1] classical or quantum field theory on a curved space–time has extensively been investigated
We have proposed a general definition of a conserved charge from any covariantly conserved current, which requires no specific asymptotic behaviors/approximations for the metric, or no subtraction of boundary contributions, as long as a Killing vector exists
Since the presented formula requires the matter energy–momentum tensor to define the mass, it is clear that black holes inevitably have nonzero matter energy–momentum tensor at singularity
Summary
Since Einstein submitted papers on general relativity,[1] classical or quantum field theory on a curved space–time has extensively been investigated. They are defined by a surface integral in the asymptotic region, by which the invariance under a class of general coordinate transformations preserving a boundary condition was achieved. This result has been further extended for a more general curved space–time with surface terms suitably incorporated.[9,10,11,12,13] A caveat in this extension is that boundary terms accompany with divergence even in the flat space–time, so that one needs to subtract it by comparing a reference frame or by adding local counterterms. Applying our definition to the energy of a compact star, we discover a correction to the mass formula obtained from the Oppenheimer–Volkoff equation, which represents a contribution from the gravitational interaction and becomes 68% of the mass at the maximum for a constant density
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