Transferring energy in general relativity

Abstract

A problem in general relativity is, how the gravitational field can transfer energy and momentum between different distant places. The issue is that matter stress tensor is locally conserved, with no explicit interaction with the free gravitational field, which is represented by the Weyl tensor. In this paper we explicitly construct an interaction tensor for free gravity and matter, that depicts the interplay between the energy momentum tensor of free gravity, which is taken to be the symmetric two index square root of Bell-Robinson tensor, and matter. This is examined both in the case of Coulomb-like Petrov type D spacetimes and radiation like Petrov type N spacetimes, where a unique square root exists. The first case generalises the Tweedledum and Tweedledee thought experiment regarding gravitational induction in Newtonian gravity to general relativistic scenarios, and the second gives a proposal for how gravitational radiation can transfer energy and momentum between distant objects separated by a vacuum.

Gravitational lensing beyond geometric optics: I. Formalism and observables

“…where the latter group of terms project out any dependence on χ a in the expansion (115) for e ab . This is qualitatively similar to the electromagnetic scalar Φ 1 as given by (62). Next, δΨ 2 appears at the same order as δΨ 3 but is significantly simpler: In terms of the shear (66),…”

“…is such a root. This has the same tensorial structure as the geometric-optics stress-energy tensors associated with scalar and electromagnetic fields, as has been noted previously for Bel-Robinson tensors associated with type-N curvature tensors [60,61,62]. However, the coefficient here does not support the analogy: Reasonable stress-energy tensors should scale like the square of the field amplitude, not the amplitude itself.…”

“…Even in spacetimes where shearfree rays may exist, they are special. Summarizing (59), (62), and (65), the generic frequency scalings associated with the electromagnetic Newman-Penrose scalars are…”

“…Noting that the Bel-Robinson tensor has units of (length) −4 , which differs from the (length) −2 associated with an ordinary stress-energy tensor, there has been some uncertainty regarding its physical interpretation (independently of any particular approximation). One possibility which has been proposed is that a rank-2 square root of the Bel-Robinson tensor may serve as a kind of gravitational stress-energy tensor [60]; this has been used to propose a notion of gravitational entropy [61] and also to discuss interactions between material and gravitational fields [62]. While square roots do not exist for all geometries, there are no difficulties at high frequencies; inspection of (131) shows that…”

An investigation on gravitational entropy of cosmological models

“…Based on this work, Clifton, Ellis and Tavakol [16] proposed that the symmetric 2-index square root t ab , of the BR tensor should act as the effective energy momentum tensor for free gravitational field. Subsequently Goswami and Ellis [21] constructed a tensor describing the interaction between free gravity and matter, which is taken to be the symmetric two index square root of the BR tensor.…”