arXiv 1702.011622017-02-03gr-qchep-thmath-phmath.MP

高阶引力理论的引力能动量赝张量

The gravitational energy-momentum pseudo-tensor of Higher-Order Theories of Gravity

S. Capozziello, M. Capriolo, M. Transirico

备注：: 23 pages, to be published in Annalen der Physik
DOI：: 10.1002/andp.201600376

摘要：

我们推导出引力能动量张量$τ^η_α$的一般拉格朗日量的任何顺序$L=L\left(g_{μν}, g_{μν,i_{1}}, g_{μν,i_{1}i_{2}},g_{μν,i_{1}i_{2}i_{3}},\cdots, g_{μν,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$，特别是对于一个拉格朗日量，如$L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$。我们证明了这个张量在一般情况下不是协变的，而只是仿射的，那么它就是伪张量。此外，伪张量$τ^η_α$在弱场极限中计算，直到$h^{2}$阶的第一非零项，其中$h$是度量扰动。在适当的时空域上的伪张量的平均值。最后，我们计算了在适当规范下，当波数$\mathbf{k}$一定时，引力波在方向$\hat{x}$上携带的单位立体角功率$Ω$。这些结果是有用的，在寻找更多的模式的引力辐射超出了标准的广义相对论的两个模式，并处理非局部引力理论的条款涉及$\Box R$。该方法的总目标是在朗道伪张量的同一标准下处理任意阶的理论。

英文摘要：

We derive the gravitational energy momentum tensor $τ^η_α$ for a general Lagrangian of any order $L=L\left(g_{μν}, g_{μν,i_{1}}, g_{μν,i_{1}i_{2}},g_{μν,i_{1}i_{2}i_{3}},\cdots, g_{μν,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ and in particular for a Lagrangian such as $L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$. We prove that this tensor, in general, is not covariant but only affine, then it is a pseudo-tensor. Furthermore, the pseudo-tensor $τ^η_α$ is calculated in the weak field limit up to a first non-vanishing term of order $h^{2}$ where $h$ is the metric perturbation. The average value of the pseudo-tensor over a suitable spacetime domain is obtained. Finally we calculate the power per unit solid angle $Ω$ carried by a gravitational wave in a direction $\hat{x}$ for a fixed wave number $\mathbf{k}$ under a suitable gauge. These results are useful in view of searching for further modes of gravitational radiation beyond the standard two modes of General Relativity and to deal with nonlocal theories of gravity where terms involving $\Box R$ are present. The general aim of the approach is to deal with theories of any order under the same standard of Landau pseudo-tensor.

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摘要：

英文摘要：

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