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Introduction to the mathematics of general relativity
About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.
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TL;DR: In this paper, the Runge-Lenz vector was used to derive the results for precession of particle orbits and bending of null rays in the case of spherically symmetric Schwarzschild geometry.
Abstract: The standard General Relativity results for precession of particle orbits and for bending of null rays are derived as special cases of perturbation of a quantity that is conserved in Newtonian physics, the Runge–Lenz vector. First, this method is applied to give a derivation of these General Relativity effects for the case of the spherically symmetric Schwarzschild geometry. Then the lowest order correction due to an angular momentum of the central body is considered. The results obtained are well known, but the method used is rather more efficient than that found in the standard texts, and it provides a good occasion to use the Runge–Lenz vector beyond its standard applications in Newtonian physics.
15 citations
TL;DR: In this article, a formulation of multipole moments generalizing that of Thorne is proposed for the stationary, vacuum region of spacetime surrounding a source of gravity, without assuming asymptotic flatness.
Abstract: A formulation of multipole moments generalizing that of Thorne is proposed for the stationary, vacuum region of spacetime surrounding a source of gravity, without assuming asymptotic flatness. In this formalism, such a region of spacetime is characterized by four sets of moments, the internal mass and current moments (those of the internal source) and the external mass and current moments (those of the external universe), which are read out from a de Donder coordinate expansion of the metric density. These moments uniquely determine the vacuum region of spacetime. The interactions between a gravitating body and an external gravitational field can be described in terms of these moments, in close analogy with Newtonian theory. A derivation, using the vacuum Einstein equation alone, is given of the laws of force and torque for an isolated body acted on by an external field. These laws generalize the results of Thorne and Hartle and of Zhang.
15 citations
TL;DR: In this article, the compatibility of the stress tensors corresponding to different formulas of the EM force density with special relativity was examined by examining the exerted electromagnetic force on boundary of an object in a few examples.
Abstract: By examination of the exerted electromagnetic (EM) force on boundary of an object in a few examples, we look into the compatibility of the stress tensors corresponding to different formulas of the EM force density with special relativity. Ampere-Lorentz's formula of the EM force density is physically justifiable in that the electric field and the magnetic flux density act on the densities of the total charges and the total currents, unlike Minkowski's formula which completely excludes the densities of the bounded charges and the bounded currents inside homogeneous media. Abraham's formula is fanciful and devoid of physical meaning. Einstein-Laub's formula seems to include the densities of the total charges and the total currents at first sight, but grouping the bounded charges and the bounded currents into pointlike dipoles erroneously results in the hidden momentum being omitted, hence the error in [Phys. Rev. Lett. 108, 193901 (2012)]. Naturally, the Ampere-Lorentz stress tensor accords with special relativity. The Minkowski sress tensor is also consistent with special relativity. It is worth noting that the mathematical expression of the Minkowski stress tensor can be quite different from the well-known form of this stress tensor in the literature. We show that the Einstein-Laub stress tensor is incompatible with special relativity, and therefore we rebut the Einstein-Laub force density. Since the Abraham momentum density of the EM fields is inherently corresponding to the Einstein-Laub force density [Phys. Rev. Lett. 111, 043602 (2013)], our rebuttal may also shed light on the controversy over the momentum of light.
15 citations
01 Apr 1936
15 citations
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TL;DR: In this article, it was shown that under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity, such that it is necessarily non-local.
Abstract: Based on an analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime.
14 citations