Static energetics in gravity

Abstract

A stress-energy tensor for linear gravity adapted to the harmonic gauge was recently proposed by Butcher, Hobson and Lasenby. By removing gauge constraints and imposing full metrical GR, we find a natural generalisation to the pseudotensor of Einstein. M{\o}ller's pseudotensor is an alternative to that of Einstein formulated using tetrads. We obtain the pseudotensor of M{\o}ller for gauge theory gravity (GTG) using a variational approach, identifying a potentially interesting recipe for constructing conserved currents in that theory. We show that in static, spherical spacetimes with a central gravitational mass M{\o}ller's pseudotensor describes gravitational stress-energy as if the gravitational potential were a scalar (i.e. Klein-Gordon) field coupled to a gravitational mass density on the Minkowski background. The old Newtonian formula successfully describes the potential of even strong fields in this picture. The Newtonian limit of this effect was previously observed in the tensor of Butcher; we recover a local virial theorem in this limit. We demonstrate using the `Schwarzschild star' solution for an incompressible perfect fluid ball.