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.

Teukolsky equation and Penrose wave equation

Abstract: It is shown that Teukolsky's equation can be derived from a second-order wave equation for the Riemann tensor. The derivation is done in a way that emphasizes the modern tensor-analysis content of the Newman-Penrose formalism.

New Approach to General Relativity

Abstract: It is pointed out that if one did not know the Einstein--HamiltonJacobi equation one might hope to derive it straight off from plausible first principles, without ever going through the formulation of the Einstein field equations themselves. (auth)

A new gravitational energy tensor

Abstract: The Bel-Robinson tensor is the most used gravitational energy tensor; however, it has the dimensions of energy squared. How to construct tensors with the dimensions of energy by using Lancoz tensors is shown here. The resulting tensors have a large number of arbitrary parameters, frequently have spacelike currents, and frequently do not reduce to familiar pseudo-energy tensors in the weak field limit. Two particular examples of interest are one with well-behaved currents and one which reduces to an energy pseudo-tensor in the weak field limit.

Newton’s third law in the framework of special relativity

Abstract: Newton’s third law states that any action is countered by a reaction of equal magnitude but opposite direction. The total force in a system not affected by external forces is thus zero. However, according to the principles of relativity a signal cannot propagate at speeds exceeding the speed of light. Hence the action cannot be generated at the same time with the reaction due to the relativity of simultaneity, thus the total force cannot be null at a given time. The following analysis provides for a better understanding of the ways natural laws would behave within the framework of Special Relativity, and on how this understanding may be used for practical purposes. It should be emphasized that although momentum can be created in the material part of the system as described in the following work momentum cannot be created in the physical system, hence for any momentum that is acquired by matter an opposite momentum is attributed to the electromagnetic field.

The mathematical foundations of general relativity revisited

Abstract: The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide. The paper is therefore divided into three parts corresponding to the different formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms appearing in the " Riemann tensor " according to the " Vessiot structure equations " must not be identified with the quadratic terms appearing in the well known " Cartan structure equations " for Lie groups and a similar comment can be done for the " Weyl tensor ". In particular, " curvature+torsion" (Cartan) must not be considered as a generalization of "curvature alone" (Vessiot). Roughly, Cartan and followers have not been able to " quotient down to the base manifold ", a result only obtained by Spencer in 1970 through the "nonlinear Spencer sequence" but in a way quite different from the one followed by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS SPENCER: The " Ricci tensor " only depends on the nonlinear transformations (called " elations " by Cartan in 1922) that describe the "difference " existing between the Weyl group (10 parameters of the Poincare subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined by a canonical splitting, that is to say without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly, the Spencer sequence for the conformal Killing system and its formal adjoint fully describe the Cosserat/Maxwell/Weyl theory but General Relativity is not coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be " parametrized ", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework, though striking it may look like for certain apparently well established theories such as electromagnetism and general relativity. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.