The Weyl Curvature Tensor, Cotton-York Tensor and Gravitational Waves: A covariant consideration

TL;DR

This paper advocates for using the Cotton-York tensor in the 1+3 covariant formalism to better capture gravitational wave contributions, highlighting its comprehensive encoding of electric, magnetic, and shear effects.

Contribution

It demonstrates that the Cotton-York tensor encodes both electric and magnetic parts of the Weyl tensor and shear, proposing it as a natural choice for analyzing gravitational waves in the 1+3 formalism.

Findings

Cotton-York tensor encodes electric, magnetic, and shear contributions.

Using only Weyl or shear tensors results in information loss.

Cotton-York tensor could be useful in metric approaches as well.

Abstract

1+3 covariant approach to cosmological perturbation theory often employs the electric part ($E_{ab}$), the magnetic part ($H_{ab}$) of the Weyl tensor or the shear tensor ($\sigma_{ab}$) in a phenomenological description of gravitational waves. The Cotton-York tensor is rarely mentioned in connection with gravitational waves in this approach. This tensor acts as a source for the magnetic part of the Weyl tensor which should not be neglected in studies of gravitational waves in the 1+3 formalism. The tensor is only mentioned in connection with studies of 'silent model' but even there the connection with gravitational waves is not exhaustively explored. In this study, we demonstrate that the Cotton-York tensor encodes contributions from both electric, magnetic part of the Weyl tensor and in directly from the shear tensor. In our opinion, this makes the Cotton-York tensor arguably the natural choice for linear gravitational waves in the 1+3 covariant formalism. The tensor is cumbersome to work with but that should negate its usefulness. It is conceivable that the tensor would equally be useful in the metric approach, although we have not demonstrated this in the current study. We contend that the use of only one of the Weyl tensor or the shear tensor, although phenomenologically correct, leads to loss of information. Such information is vital particularly when examining the contribution of gravitational waves to the anisotropy of an almost -Friedmann-Lamitre-Robertson-Walker (FLRW) universe. The recourse to this loss is the use Cotton-York tensor.