arXiv 0911.23402009-11-16gr-qc

与Friedmann-Robertson-Walker度规相关的爱因斯坦能量

Einstein energy associated with the Friedmann -Robertson -Walker metric

Abhas Mitra

备注：: By mistake a marginally different earlier version was loaded, now the journal version is uploaded
发表信息：: Gen.Rel.Grav.42:443-469,2010
DOI：: 10.1007/s10714-009-0863-1

摘要：

遵循爱因斯坦对拉格朗日密度和引力场能量密度的定义（Einstein，A.，Ann. Phys. Lpz.，49，806（1916）; Einstein，A.，物理Z.，19，115（1918）; Pauli，W.，{ \it相对论}，B. I. Publications，Mumbai，1963，Trans. by G.场），托尔曼推导出任意系统的总物质加引力场能量（$P_0$）的一般公式（托尔曼，R.C.，物理修订版，35（8），875（1930）; Tolman，R.C.，{ \it Relativity，Thermodynamics & Cosmology}，Clarendon Press，Oxford，1962））; Xulu，S.S.，arXiv：hep-th/0308070（2003））。对于一个静态孤立系，在准笛卡尔坐标系中，这个公式导致了众所周知的结果$P_0 = \int \sqrt{-g} (T_0^0 - T_1^1 -T_2^2 -T_3^3) ~d^3 x$，其中$g$是度规张量的行列式，而$T^a_b$是{ \em物质}的能量动量张量。虽然在文献中，这被称为“托尔曼质量”，但必须认识到，这本质上是“爱因斯坦质量”，因为这里的伪张量是由于爱因斯坦。事实上，Landau-Lifshitz在根本不使用任何伪张量的情况下获得了静态孤立系的“惯性质量“的相同表达式，这表明了爱因斯坦质量的物理意义和正确性（Landau，L.D.，和Lifshitz，E.M.，{ \it经典场论}，Pergamon出版社，牛津，第2版，1962年）！我们第一次应用这个一般公式找到一个表达式$P_0$的Friedmann-Robertson-Walker（FRW）度规通过使用相同的准笛卡尔基。当我们分析这个新结果时，在物理上，我们提出了一个没有宇宙学常数的空间平坦模型。最后，可以看出，$P_0$的守恒仅在静态极限下才成立。

英文摘要：

Following Einstein's definition of Lagrangian density and gravitational field energy density (Einstein, A., Ann. Phys. Lpz., 49, 806 (1916); Einstein, A., Phys. Z., 19, 115 (1918); Pauli, W., {\it Theory of Relativity}, B.I. Publications, Mumbai, 1963, Trans. by G. Field), Tolman derived a general formula for the total matter plus gravitational field energy ($P_0$) of an arbitrary system (Tolman, R.C., Phys. Rev., 35(8), 875 (1930); Tolman, R.C., {\it Relativity, Thermodynamics & Cosmology}, Clarendon Press, Oxford, 1962)); Xulu, S.S., arXiv:hep-th/0308070 (2003)). For a static isolated system, in quasi-Cartesian coordinates, this formula leads to the well known result $P_0 = \int \sqrt{-g} (T_0^0 - T_1^1 -T_2^2 -T_3^3) ~d^3 x$, where $g$ is the determinant of the metric tensor and $T^a_b$ is the energy momentum tensor of the {\em matter}. Though in the literature, this is known as "Tolman Mass", it must be realized that this is essentially "Einstein Mass" because the underlying pseudo-tensor here is due to Einstein. In fact, Landau -Lifshitz obtained the same expression for the "inertial mass" of a static isolated system without using any pseudo-tensor at all and which points to physical significance and correctness of Einstein Mass (Landau, L.D., and Lifshitz, E.M., {\it The Classical Theory of Fields}, Pergamon Press, Oxford, 2th ed., 1962)! For the first time we apply this general formula to find an expression for $P_0$ for the Friedmann- Robertson -Walker (FRW) metric by using the same quasi-Cartesian basis. As we analyze this new result, physically, a spatially flat model having no cosmological constant is suggested. Eventually, it is seen that conservation of $P_0$ is honoured only in the a static limit.

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