The values for the gravitational energy-momentum density, given by the famous classical pseudotensors: Einstein, Papapetrou, Landau-Lifshitz, Bergmann-Thompson, Goldberg, Møller, and Weinberg, in the small region limit are found to lowest non-vanishing order in normal coordinates. All except Møller's have the zeroth order material limit required by the equivalence principle. However for small vacuum regions we find that {\it none} of these classical holonomic pseudotensors satisfies the criterion of being proportional to the Bel-Robinson tensor. Generalizing an earlier work which had identified one case, we found another independent linear combination satisfying this requirement--and hence a one parameter set of linear combinations of the classical pseudotensors with this desirable property.
In 1985 Thorne and Hartle used the Landau-Lifshitz pseudotensor to demonstrate the tidal heating and angular momentum flux for a black hole. Later in 2004, Poisson used the gravitational perturbation method to study a black hole and obtained the same result. Poisson proposed a new idea, that the mass quardupole moment and current quadrupole moment can be written as the rate of change of the tidal gravitational field. Inspired by these two papers, we use the method of Thorne and Hartle to study other classical pseudotensors: Einstein, Bergmann-Thomson, Papapetrou and Weinberg. Moreover, we also…
Describing the gravitational energy and momentum, the Bel-Robinson tensor is the best tensor. However, the classical pseudotensors can also manage the the same job. As Deser mentioned in 1999, a certain linear combination between Einstein pseudotensor and Landau-Lifshitz pseudoetnsor give a pure Bel-Robinson tensor. Here we used the same idea but adapted the harmonic gauge, we found that all the classical pseudotensor alone cannot give a multiple of the Bel-Robinson tensor. But under a modification, all of them contribute the same energy-momentum and stress as the Bel-Robinson tensor does.
The current paper is devoted to the investigation of the general form of the energy-momentum pseudotensor (pEMT) and the corresponding superpotential for the wide class of theories. The only requirement for such a theory is the general covariance of the action without any restrictions on the order of derivatives of the independent variables in it or their transformation laws. As a result of the generalized Noether procedure, we obtain a recurrent chain of the equations, which allows one to express canonical pEMT as a divergence of the superpotential. The explicit expression for this superpoten…
We derive the gravitational energy-momentum pseudotensor $ τ^σ_ {\phantom σ λ} $ in metric $ f\left (R \right) $ gravity and in teleparallel $ f\left (T\right) $ gravity. In the first case, $R$ is the Ricci curvature scalar for a torsionless Levi-Civita connection; in the second case, $T$ is the curvature-free torsion scalar derived by tetrads and Weitzenböck connection. For both classes of theories the continuity equations are obtained in presence of matter. $ f \left (R \right) $ and $ f \left (T \right) $ are non-equivalent but differ for a quantity $ ω\left (T, B \right) $ containing the t…
In his study of tidal stabilization of fully relativistic neutron stars Thorne showed that the fully relativistic expression for tidal heating is the same as in non-relativistic Newtonian theory. Furthermore, Thorne also noted that this tidal heating must be independent of how one localizes gravitational energy and is unambiguously given by that expression. Favata calculated the tidal heating for a number of classical gravitational pseudotensors including that of M$ø$ller, and obtained the result that all of them produced the same (Newtonian) value. After a re-examination of the calculation us…
Energy and angular momentum of charged rotating black holes
我们表明，爱因斯坦，托尔曼，朗道和Lifshitz，Papapetrou和温伯格（ETLLPW）的伪张量给出相同的能量分布，线动量和角动量，任何克尔-Schild度规。这一结果推广了Gürses和Gürsey以前的工作，该工作只涉及爱因斯坦、Landau和Lifshitz的伪张量。我们计算这些分布的Kerr-Newman和Bonnor-Vaidya度量，并找到合理的结果。所有计算都是在Kerr-Schild笛卡尔坐标系中进行的，没有任何近似。对于Reissner-Nordström度规，这些定义给出了与彭罗斯准定域质量相同的结果。对于克尔黑洞，所有的能量都被限制在它的内部，而对于克尔-纽曼黑洞，正如预期的那样，能量由它的内部和外部共享。Kerr-Newman黑洞的总能量和角动量分别为$M$和$ M a$（$M$是质量参数，$a$是旋转参数）。Bonnor-Vaidya度规的能量分布与Tod得到的彭罗斯准局域质量相同。
We use the Møller and Landau-Lifshitz energy-momentum definitions in General Relativity (GR) to evaluate the energy-momentum distribution of the phantom black hole space-time. The phantom black hole model was applied to the supermassive black hole at the Galactic Centre. We obtain that in both pseudotensorial prescriptions the energy distribution depends on the mass $M$ of the black hole, the phantom constant $p$ and the radial coordinate $r$. Further, all the calculated momenta are found to be zero. The limiting cases $r\rightarrow 0$, $r\rightarrow \infty $ and $r\rightarrow -\infty $ have a…
The specification of energy for gravitating systems has been an unsettled issue since Einstein proposed his pseudotensor. It is now understood that energy-momentum is \emph{quasi-local} (associated with a closed 2-surface). Here we consider quasi-local proposals (including pseudotensors) in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) there are many possible expressions, (ii) they depend on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. The Hamiltonian perspective helped us to make a remarkable…
The Bel-Robinson tensor $B$ and the tensor $V$ have the same quasilocal energy-momentum in a small sphere. Using a pseudotensor approach to evaluate the energy-momentum in a half-cylinder, we find that $B$ and $V$ have different values, not proportional to the "Bel-Robinson energy-momentum". Furthermore, even if we arrange things so that we do get the same "Bel-Robinson energy-momentum" value, the angular momentum gives different values using $B$ and $V$ in a half cylinder. In addition, we find that $B$ and $V$ have a different number of independent components. The fully trace free property of…
In this article, the energy density of plane gravitational wave is studied by using Einstein and Møller's prescription of energy-momentum pseudotensors. The linearly polarized plan gravitational wave solution of Einstein field equation, which has been defined by Bondi et al., is represented by four kinds of different coodrinates. The energy distribution of gravitational wave solution in Einstein and Møller's prescription are obtained. Particularly the energy component is zero in null coordinates.
Conservation of Energy-Momentum in Teleparallel Gravity
在一篇著名的论文arXiv：gr-qc/0003100 V.C.德安德拉德湖C. T. Guillen和J.G.佩雷拉定义了一个保守的规范电流，但他们说："这是，我们相信最远的一个可以去的方向张量定义的能量和动量的引力场。局部洛伦兹协方差的缺乏可以被认为是广义相对论中引力能-动量密度的赝张量特征的远程平行表现。''.我们相信，他们在给出这样一个张量定义之前，仅仅停留了不到一英寸，而且，由此产生的能量-动量张量具有零迹，可以使之对称，并且与物质场的能量-动量张量一起，它服从一个远程平行流形的自然守恒方程。特别是在最后一节中，关于不需要任何宇宙学常数而解释宇宙膨胀加速的可能性，得到了一些重要的结论。
The energy of gravitating systems has been an issue since Einstein proposed general relativity: considered to be ill defined, having no proper local density. Energy-momentum is now regarded as \emph{quasi-local} (associated with a closed 2-surface). We consider the pseudotensor and quasi-local proposals in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) many expressions, (ii) each depends on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. Our remarkable discovery is that with a 4D isometric Minkows…
The gravitational energy-momentum pseudo-tensor in $f(Q)$ non-metric gravity
Università di Napoli “Federico II”（那不勒斯费德里科二世大学）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli（意大利国家核物理研究所（INFN），那不勒斯分部）；Scuola Superiore Meridionale（南方高等学院）；Università di Salerno（萨莱诺大学）；INFN, Gruppo Collegato di Salerno（意大利国家核物理研究所（INFN），萨莱诺协作组）
Angular momentum at null infinity has a supertranslation ambiguity from the lack of a preferred Poincaré group and a similar ambiguity when the center-of-mass position changes as linear momentum is radiated. Recently, we noted there is an additional one-parameter ambiguity in the possible definitions of angular momentum and center-of-mass charge. We argue that this one-parameter ambiguity can be resolved by considering the generalized BMS charges that are constructed from local 2-sphere-covariant tensors near null infinity; these supertranslation-covariant charges differ from several expressio…
We present a new formula for the angular momentum $J^{μν}$ carried away by gravitational radiation in classical scattering. This formula, combined with the known expression for the radiated linear momentum $P^μ$, completes the set of radiated Poincare charges due to scattering. We parametrize $P^μ$ and $J^{μν}$ by non-perturbative form factors and derive exact relations using the Poincare algebra. There is a contribution to $J^{μν}$ due to static (zero-frequency) modes, which can be derived from Weinberg's soft theorem. Using tools from scattering amplitudes and effective field theory, we calc…
We present a formalism for computing classically measurable quantities directly from on-shell quantum scattering amplitudes. We discuss the ingredients needed for obtaining the classical result, and show how to set up the calculation to derive the result efficiently. We do this without specializing to a specific theory. We study in detail two examples in electrodynamics: the momentum transfer in spinless scattering to next-to-leading order, and the momentum radiated to leading order.
We prove a theorem on scalar-valued functions of tensors, where ``scalar'' refers to absolute scalars as well as relative scalars of weight $w$. The present work thereby generalizes an identity referred to earlier by Rosenfeld in his publication ``On the energy-momentum tensor''. The theorem provides a $(1,1)$-tensor identity which can be regarded as the tensor analogue of the identity following from Euler's theorem on homogeneous functions. The remarkably simple identity is independent of any internal symmetries of the constituent tensors, providing a powerful tool for deriving relations betw…
