Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity

TL;DR

This paper reviews the energy-momentum complex in higher order curvature gravity theories, deriving pseudo-tensors in various frameworks, and applies these concepts to gravitational waves and cosmological models, highlighting the non-covariant nature of the energy-momentum pseudo-tensor.

Contribution

It extends the energy-momentum complex formulation to higher order curvature gravity theories, including $f(R)$ gravity, and explores their applications in gravitational wave emission and cosmology.

Findings

Derived energy-momentum pseudo-tensors for higher order gravity theories.

Calculated gravitational wave power emission using the pseudo-tensor.

Proposed energy densities in cosmological $f(R)$ models.

Abstract

In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, M{\o}ller, and Weinberg. In this review, we firstly explored the energy--momentum complex in an $n^{th}$ order gravitational Lagrangian $L=L\left(g_{\mu\nu}, g_{\mu\nu,i_{1}}, g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots, g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ and then in a gravitational Lagrangian as \mbox{$L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$}. Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether's theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor $\tau^{\eta}_{\alpha}$ becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy--momentum complex in $ f\left (R \right) $ gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor $\tau^{\eta}_{\alpha}$ to lowest order in the metric perturbation $h$ was calculated. As a practical application, the power per unit solid angle $\Omega$ emitted by a localized source carried by a gravitational wave in a direction $\hat{x}$ for a fixed wave number $\mathbf{k}$ under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann--Lema\^itre--Robertson--Walker spacetime, the gravitational and matter energy density in $f(R)$ gravity both in Palatini and metric formalism was proposed.