Static energetics in gravity

A stress-energy tensor, τab, for linear gravity in the physical spacetime, M, approximated by a flat background, M̌, and adapted to the harmonic gauge, was recently proposed by Butcher, Hobson, and Lasenby. By removing gauge constraints and imposing full metrical general relativity, we find a natural generalization of τab to the pseudotensor of Einstein, Etab. Møller’s pseudotensor, Mtab, is an alternative to Etab formulated using tetrads and is thus naturally adapted to, e.g., Einstein-Cartan gravity. Gauge theory gravity uses the geometric algebra to reproduce Einstein-Cartan gravity and is a Poincaré gauge theory for the spacetime algebra: the tetrad and spin connection appear as gauge fields on Minkowski space, M4. We obtain the pseudotensor of Møller for gauge theory gravity, Mt(a), using a variational approach, also identifying a potentially interesting recipe for constructing conserved currents in that theory. We show that in static, spherical spacetimes containing a gravitational mass MT, the pseudotensors in the spacetime algebra, Mt(a) and Et(a), describe gravitational stress-energy as if the gravitational potential were a scalar (i.e., Klein-Gordon) field, φ, coupled to gravitational mass density, ϱ, on the Minkowski background M4. The old Newtonian formula φ = −MT/r successfully describes even strong fields in this picture. The Newtonian limit of this effect was previously observed in τab on M̌ for linear gravity. We also draw fresh attention to the conserved mass of a static system, MTMT. We observe that the gravitational energy of Einstein and Møller was added to MT on M4 to give MT. We demonstrate the Klein-Gordon correspondence and mass functions using the “Schwarzschild star” solution for an incompressible perfect fluid ball.