On the energy of gravitational waves

Abstract

The energy of gravitational waves is a fundamental problem in gravity theory. The existing descriptions for the energy of gravitational waves, such as the well-known Isaacson energy-momentum tensor, suffer from several defects. Due to the equivalence principle, the gravitational energy-momentum can only be defined quasilocally, being associated with a closed spacelike 2-surface bounding a region. We propose a new approach to derive the energy of gravitational waves directly from the quasilocal gravitational energy. Such an approach is natural and consistent with the quasilocality of gravitational energy-momentum.

Appendices

Appendix A: Canonical form of Weyl tensor

Table 1 Canonical form of Weyl tensor for different Petrov types

Appendix B:Tetrad transformations

The Lorentz transformation with six parameters acting on a null tetrad $\{m^{a},\bar{m}^{a},l^{a},k^{a}\}$ can be classified to three types:

(i) $l'=l, \quad k'=k+a\bar{m}+\bar{a}m+a\bar{a}l, \quad m'=m+al$

(ii) $k'=k, \quad l'=l+b\bar{m}+\bar{b}m+b\bar{b}k, \quad m'=m+bk$

(iii) $k'=Ak, \quad l'=A^{-1}l, \quad m'=\mathrm {e}^{\mathrm {i}\theta }m$

where a and b are complex parameters, $A>0$ and $\theta$ are real parameters.