Transfer of energy in general relativity

Abstract
This paper studies time sequences of axially symmetric static configurations which can be continuously deformed into each other. We find the restriction which characterizes those time sequences which are physically allowed. This restriction can be interpreted as the law governing the non-radiative, or near field, transfer of gravitational energy, -m = ∯ P. dS and helps to clarify the concept of mass-energy in general relativity. We consider only the regions of space surrounding a source where, for quasi-static systems, the exact, static, empty space solutions of Weyl and Levi-Civita are good approximations at all times, and exact whenever the motion stops. Our results are then valid for arbitrarily strong fields. The restriction on the time sequences can be expressed as a restriction on the time dependence of the multipole moments Al, Bl (which are defined by the Weyl and Levi Civita solutions), becoming then -d/dt[ A0 + ½G ∑l = 0∞ (2l + 1) AlBl] = ½ G ∑l = 0∞ (AlBl - AlBl). The right-hand side of this expression corresponds exactly to the flux of energy across a surface, as given by Bondi’s Newtonian Poynting vector P = (1/8 πG) (ϕ∇ϕ-ϕ∇ϕ). It is natural to identify A0 + ½G ∑l = 0∞ (2l + 1)AlBl with the total mass-energy m enclosed by the surface.m can also be expressed as a surface intergal. Our restriction was derived from the vanishing of a surface integral in much the same manner as the equations of motion are derived in post-Newtonain approximations. However, it holds in arbitrarily strong fields, whereas the usual post-Newtonian methods use in lowest order a trivial solution (flat space) and can only be used in weak fields.

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