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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: The structure of the Poisson-brackets algebra of constraints of general relativity is reexamined using the recently introduced spinorial variables to provide a point of departure for a nonperturbative quantization scheme for general relativity.
Abstract: The structure of the Poisson-brackets algebra of constraints of general relativity is reexamined using the recently introduced spinorial variables. Three different combinations of constraints are analyzed and their relative merits are discussed. In each case we construct the corresponding expression of the Becchi-Rouet-Stora-Tyutin charge. These expressions provide a point of departure for a nonperturbative quantization scheme for general relativity.
22 citations
22 citations
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TL;DR: In this article, a description of the static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity is given, and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in general relativity.
Abstract: Using various gauges of the radial coordinate we give a description of the static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist innitely many such solutions to the Einstein equations which are physically dieren t and only some of them describe the gravitational eld of a single massive point particle. In particular, we show that the widespread Hilbert’s form of Schwarzschild solution does not solve the Einstein equations with a massive point particle’s stress-energy tensor. Novel normal coordinates for the eld and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravitational mass defect of a point particle and determine the dependence of the solutions on this mass defect. In addition we give invariant characteristics of the physically and geometrically dieren t classes of spherically symmetric static space-times created by one point mass.
22 citations
TL;DR: In this paper, the effects of non-commutativity on spin-2 amplitudes are taken into account by keeping the standard form of the Einstein tensor on the left hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side.
Abstract: Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in non-commutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy–momentum tensor as a source on the right-hand side. The present paper, relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, considers from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes is shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited. Approximate formulae for the particle emission rate are also obtained within this framework.
22 citations
TL;DR: In this article, the authors reformulate the general theory of relativity in the language of Riemann-Cartan geometry, and they show that the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field.
Abstract: We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.
22 citations