论文标题
高阶曲率局部重力的能量摩托明复合物
Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity
论文作者
论文摘要
总体而言，已经有许多建议定义重力能量密度，尤其是爱因斯坦，托尔曼，兰道和利希茨，帕帕普特鲁，莫勒和温伯格提出的建议。在这篇评论中，我们首先探索了$ n^{th} $ order fordational lagrangian $ l = l \ left（g_ {μν}，g_ {μn，i_ {1}}， g_ {μν，i_ {1} i_ {2}}，g_ {μν，i_ {1} i_ {2} i_ {2} i_ {3}}，\ cdots，g_ {μν，I_ {1} i_ {1} lagrangian as \ mbox {$ l_ {g} =（\ overline {r}+a_ {0} r^{2}+\ sum_ {k = 1}^{p} a_ {k} a_ {k} r \ box^{k} {k} {k} r）它的引力部分是通过使用Noether定理在无穷小刚性翻译下的重力作用不变性获得的。我们还表明，通常，这种张量不是协变量的对象，而是一个仿射对象，即伪张量。因此，如果我们将自己限制在一般相对论上，并且已经明确指出了伪tensor $τ^η_α$，则将成为爱因斯坦引入的伪tens子。使用了同样的方法来得出$ f \ left（r \ right）$ the palatini和公制方法中的$ f \ weft（r \ right）$重力。此外，在弱场近似中，计算了伪张量$τ^η_α$在度量扰动$ h $中的最低顺序。作为一种实际应用，通过在适当的计量的固定波数$ \ mathbf {k} $的方向上，由重力波沿方向携带的局部源$ \ mathbf {k} $发出的局部源$ \ hat {x} $在适当的指数下通过伪tensor的平均价值获得了适当的空间范围内的平均值，从而获得了固定波数$ \ mathbf {k} $。作为一种宇宙学应用，在平坦的弗里德曼 - 罗伯逊 - 罗伯逊 - 步行者时期，提出了$ f（r）$重力的重力和物质能量密度，均提出了palatini和公制形式主义。
In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy--momentum complex in an $n^{th}$ order gravitational Lagrangian $L=L\left(g_{μν}, g_{μν,i_{1}}, g_{μν,i_{1}i_{2}},g_{μν,i_{1}i_{2}i_{3}},\cdots, g_{μν,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ 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 $τ^η_α$ 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 $τ^η_α$ to lowest order in the metric perturbation $h$ was calculated. As a practical application, the power per unit solid angle $Ω$ 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--Lemaître--Robertson--Walker spacetime, the gravitational and matter energy density in $f(R)$ gravity both in Palatini and metric formalism was proposed.
