Localizing the energy and momentum of linear gravity

Abstract

A framework is developed which quantifies the local exchange of energy and momentum between matter and the linearised gravitational field. We derive the unique gravitational energy-momentum tensor consistent with this description, and find that this tensor only exists in the harmonic gauge. Consequently, nearly all the gauge freedom of our framework is naturally and unavoidably removed. The gravitational energy-momentum tensor is then shown to have two exceptional properties: (a) it is gauge-invariant for gravitational plane-waves, (b) for arbitrary transverse-traceless fields, the energy-density is never negative, and the energy-flux is never spacelike. We analyse in detail the local gauge invariant energy-momentum transferred between the gravitational field and an infinitesimal point-source, and show that these invariants depend only on the transverse-traceless components of the field. As a result, we are led to a natural gauge-fixing program which at last renders the energymomentum of the linear gravitational field completely unambiguous, and additionally ensures that gravitational energy is never negative nor flows faster than light. Finally, we calculate the energymomentum content of gravitational plane-waves, the linearised Schwarzschild spacetime (extending to arbitrary static linear spacetimes) and the gravitational radiation outside two compact sources: a vibrating rod, and an equal-mass binary.

Trapping and Guiding Bodies by Gravitational Waves Endowed with Angular Momentum

“…The most natural source of gravitational radiation is a close binary system of compact massive objects. The distribution of the angular momentum of the gravitational radiation generated by such system in the linearized approximation can be computed using the framework developed in [17,18]. Consider a binary system moving on quasi-Keplerian orbits in the xy-plane.…”

“…Consider a binary system moving on quasi-Keplerian orbits in the xy-plane. We compute the radiative (time dependent) components of the metric generated by such system using the formula (73) in [17]. Next, we compute the energy-momentum tensor τ µν for this metric using the formula (23) from [17].…”

Energy-momentum tensors in linearized Einstein’s theory and massive gravity: The question of uniqueness

“…Recently, Butcher et al from the Cambridge Kavli Institute for Cosmology published a series of papers on "localized energetics of linear gravity" [9], [10], [11]. By examining the transfer of energy and momentum between local matter and the gravitational field within the linearized theory, they constructed a symmetric energymomentum tensor of linearized gravity which exhibits plausible physical properties and is quadratic in the first derivatives h ik,l ; however, the whole framework leads to the use of the harmonic gauge [9]. Later the same authors extended their work to the study of the localized angular momentum of linearized gravity [10].…”

“…e.g. the recent monograph [8]), (ii) to give relations to new developments due to the Cambridge group [9], [11] and (iii) to generalize our method of studying the uniqueness of energy-momentum tensors to the case of massive gravity. The massive gravity has been studied with an "oscillatory interest" for the past 70 years.…”

“…Within the linearized gravity we also investigate the role of the harmonic gauge condition and generalized gauge condition since we wish to analyze the uniqueness of the energy-momentum tensor presented in [9]. Taking into account the harmonic gauge condition ab initio, our procedure gives the five-parameter family of, generally nonsymmetric, conserved quantities.…”