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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 Lagrangian method to generate conserved currents in field theories starting from the so-called Poincare-Cartan form is reviewed in this article, with examples of application to general relativity.
Abstract: The general 'Lagrangian' method to generate conserved currents in field theories starting from the so-called Poincare-Cartan form is reviewed, with examples of application to general relativity.
17 citations
TL;DR: In this paper, the authors formally discuss the post-Minkowskian limit of gravity without adopting conformal transformations but developing all the calculations in the original Jordan frame, and show that such an approach gives rise, in general, together with the standard massless graviton, to massive scalar modes whose masses are directly related to the analytic parameters of the theory.
Abstract: We formally discuss the post-Minkowskian limit of
$f(R)$-gravity without adopting conformal transformations but developing all the calculations in the original Jordan frame. It is shown that such an approach gives rise, in general, together with the standard massless graviton, to massive scalar modes whose masses are directly related to the analytic parameters of the theory. In this sense, the presence of massless gravitons only is a peculiar feature of General Relativity. This fact is never stressed enough and could have dramatic consequences in detection of gravitational waves. Finally the role of curvature stress-energy tensor of $f(R)$-gravity is discussed showing that it generalizes the so called Landau-Lifshitz tensor of General Relativity. The further degrees of freedom, giving rise to the massive modes, are directly related to the structure of such a tensor.
17 citations
TL;DR: In this article, the equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space are considered.
Abstract: The equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space are considered. A nonholonomic distribution defined by the differential form ω = A 0 dx 0 + A 1 dx 1 + A 2 dx 2 + A 3 dx 3 + dx 4 on a five-dimensional smooth Lorentzian manifold is studied. By means of the Pontryagin maximum principle, it is proved that the equations of horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general theory of relativity. Thus, a model of the Kaluza-Klein theory is built by means of the sub-Lorentzian geometry. Finally, the geodesic sphere, which appears in a constant magnetic field, is studied, as well as its singular points.
17 citations
TL;DR: In this article, the authors discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment.
Abstract: We briefly discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely, the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment. Furthermore, in his analysis of the foundations of Newtonian mechanics, Mach provided an operational definition for inertial mass and pointed out that time and space are conceptually distinct from their operational definitions by means of masses. Mach recognized that this circumstance is due to the lack of any a priori connection between the inertial mass of a body and its Newtonian state in space and time. One possible way to improve upon this situation in classical physics is to associate mass with an extra dimension. Indeed, Einstein's theory of gravitation can be locally embedded in a Ricci-flat 5D manifold such that the 4D energy-momentum tensor appears to originate from the existence of the extra dimension. An outline of such a 5D Machian extension of Einstein's general relativity is presented.
17 citations
TL;DR: In this article, the authors discuss the absence of stationary source-free solutions and the positive energy problem in general relativity at the linearized level in terms of the Bel-Robinson tensor, and raise the possibility that there may exist stationary solutions to the full Einstein equations in five dimensions.
Abstract: The absence of stationary source-free solutions and the positive energy problem in general relativity are discussed, at the linearized level, in terms of the Bel-Robinson tensor. The possibility is raised that there may exist stationary solutions to the full Einstein equations in five dimensions.
16 citations