We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge-Teitelboim approach. For the embedding theory we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case the independent definition of gravitational energy-momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory) we consider Noether procedure and the alternative method of energy-momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed.
Isometric Embeddings and Noncommutative Branes in Homogeneous Gravitational Waves
我们刻画了由一个常数Neveu-Schwarz三形式通量支持的六维Cahen-Wallach pp波中对称D膜上的世界体积理论。我们发现一类平坦的非对易欧氏D3-膜类似于膜在一个恒定的磁场，以及弯曲的非对易洛伦兹D3-膜类似于膜在一个电的背景。在前一种情况下，膜上的非对易场论是从第一性原理构造的，与世界体积中模糊球的动力学有关，并用于分析弦论的平坦空间极限。背景中所有其他对称膜上的世界体积理论都是局域场论。所有这些理论的物理起源都是通过时空中膜的等距嵌入和具有Neveu-Schwarz三形式通量的AdS 3 x S3的Penrose-Gueven极限之间的相互作用来描述的。非对称时空填充D-膜的非对易场论也被构造，给出了类似于Dolan-Nappi模型的空间变化但与时间无关的非对易性。
On the absence of the usual weak-field limit, and the impossibility of embedding some known solutions for isolated masses in cosmologies with f(R) dark energy
