Static energetics in gravity

Abstract

A stress-energy tensor, τ ab , for linear gravity in the physical spacetime, M, approximated by a flat background, M, and adapted to the harmonic gauge, was recently proposed by Butcher, MPH and ANL. By removing gauge constraints and imposing full metrical general relativity, we find a natural generalisation of τ ab to the pseudotensor of Einstein, E t ab . Møller's pseudotensor, M t ab , is an alternative to E t ab formulated using tetrads, and is thus naturally adapted to e.g. Einstein-Cartan gravity. Gauge theory gravity uses 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, M 4 . We obtain the pseudotensor of Møller for gauge theory gravity, M t(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 M T the pseudotensors in the spacetime algebra, M t(a) and E t(a), describe gravitational stress-energy as if the gravitational potential were a scalar (i.e. Klein-Gordon) field, ϕ, coupled to gravitational mass density, , on the Minkowski background M 4 . The old Newtonian formula ϕ = −M T /r successfully describes even strong fields in this picture. The Newtonian limit of this effect was previously observed in τ ab onM for linear gravity. We also draw fresh attention to the conserved mass of a static system, M T < M T . When compared with either or the density of the Komar mass in the Newtonian limit, M T produces a local virial theorem -such behaviour is usually associated with the proper mass of the system, M T > M T . We observe that the gravitational energy of Einstein and Møller add to M T on M 4 to give M T .We demonstrate the Klein-Gordon correspondence and mass functions using the 'Schwarzschild star' solution for an incompressible perfect fluid ball.

Non-zero torsion and late cosmology

“…[32,33]) and nonminimal couplings with topological invariants (see Refs. [34][35][36][37][38][39][40][41][42][43][44][45][46][47]). In these kinds of theories, torsion propagates in a vacuum, 1 and it is not necessary a spin tensor to have a nonvanishing torsion.…”

Current survey of Clifford geometric algebra applications

“…Static energetics for gravity is treated in Barker et al 160 The authors remove gauge constraints and impose full metrical general relativity for a stress energy for linear gravity in physical space time, approximated by a flat background, and find a natural generalization of the stress energy tensor to the pseudotensor of Einstein, with Møller's pseudotensor being an alternative (with tetrads adapted to Einstein–Cartan gravity). Gauge theory gravity (tetrads and spin connection as gauge fields on Minkowski space), a Poincaré gauge theory for spacetime algebra, uses GA for Einstein–Cartan gravity.…”

“…Gauge theory gravity (tetrads and spin connection as gauge fields on Minkowski space), a Poincaré gauge theory for spacetime algebra, uses GA for Einstein–Cartan gravity. The new work of Barker et al 160 uses a variational approach to obtain the Møller pseudotensor yielding a method for obtaining conserved currents. This is applied to static, spherical spacetimes, with a limit to Newton's $$$ -M/r $$$ potential formula, and conserved static system mass $$$ {M}_t $$$ (which produces a local virial theorem).…”