Localizing the energy and momentum of linear gravity

A framework is developed which quantifies the local exchange of energy and momentum between matter and the linearized 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 analyze 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 energy-momentum 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 energy-momentum content of gravitational plane-waves, the linearized Schwarzschild spacetime (extending to arbitrary static linear spacetimes) and the gravitational radiation outside two compact sources: a vibrating rod, and an equal-mass binary.

Figure 1

Schematic showing plane-wave regions sewing various source frames to the detector frame of $D$. $S_1$ and $S_2$ represent compact sources many wavelengths apart; if they have different velocities, then each will determine a separate source frame. In contrast, $S_3$ and $S_4$ are only separated by a small number of wavelengths; as there is no plane-wave region dividing the sources, they must share their source frame. It is natural to base this joint source frame on the velocity of the center of mass of the multicomponent system.

Figure 2

Plots of the energy-density of the dynamical gravitational field outside two monochromatic compact sources: a vibrating rod, and an equal-mass binary. Only half a period is shown, as ${\tau }_{\mu \nu }$ oscillates with twice the frequency of the source. Although the rod and the binary are much smaller than one wavelength, they have been magnified to illustrate the phase of their motion. The propagation of gravitational energy is more easily appreciated in the animated versions of these plots, available at www.mrao.cam.ac.uk/~lmb62/animations.

Figure 3

Comparison of the standard method for achieving transverse-traceless gauge in the vicinity of a source [13, chap. 4.4b] and the “persistent” method described here. In the two diagrams, $S$ represents an arbitrary source (a region with $T_{\mu \nu }\ne 0$) moving relative to $u^{\mu }$. The hypersurface $t=t_0$ used to define the gauge in the standard method is also shown, but plays no role in our method.