The Gravitational Wave Stress-Energy (pseudo)-Tensor in Modified Gravity

TL;DR

This paper compares four methods for deriving the gravitational wave stress-energy pseudo-tensor in various gravity theories, demonstrating their consistency in energy loss calculations and exploring their properties in modified gravity contexts.

Contribution

It introduces and compares four different methods for deriving the gravitational wave stress-energy pseudo-tensor in modified gravity theories, including Einstein-e6ther, and applies them to compute energy and momentum fluxes.

Findings

All methods produce the same energy loss rate.

The Noether method yields a non-symmetric, gauge-dependent stress-energy tensor.

The stress-energy tensor in Einstein-e6ther theory is non-Lorentz-invariant.

Abstract

The recent detections of gravitational waves by the advanced LIGO and Virgo detectors open up new tests of modified gravity theories in the strong-field and dynamical, extreme gravity regime. Such tests rely sensitively on the phase evolution of the gravitational waves, which is controlled by the energy-momentum carried by such waves out of the system. We here study four different methods for finding the gravitational wave stress-energy pseudo-tensor in gravity theories with any combination of scalar, vector, or tensor degrees of freedom. These methods rely on the second variation of the action under short-wavelength averaging, the second perturbation of the field equations in the short-wavelength approximation, the construction of an energy complex leading to a Landau-Lifshitz tensor, and the use of Noether's theorem in field theories about a flat background. We apply these methods in General Relativity, scalar-tensor theories and Einstein-\AE{}ther theory to find the gravitational wave stress-energy pseudo-tensor and calculate the rate at which energy and linear momentum is carried away from the system. The stress-energy tensor and the rate of linear momentum loss in Einstein-\AE{}ther theory are presented here for the first time. We find that all methods yield the same rate of energy loss, although the stress-energy pseudo-tensor can be functionally different. We also find that the Noether method yields a stress-energy tensor that is not symmetric or gauge-invariant, and symmetrization via the Belinfante procedure does not fix these problems because this procedure relies on Lorentz invariance, which is spontaneously broken in Einstein-\AE{}ther theory. The methods and results found here will be useful for the calculation of predictions in modified gravity theories that can then be contrasted with observations.