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.

On factorization of Einstein's formalism into a pair of quaternion field equations

Abstract: It is proposed that a full exploitation of the principle of general relativity in the costruction of the metrical field equations implies that the fundamental variables should be quaternion fields rather than the metric tensor field of the conventional formulation. Thus, the tensor property of Einstein's formalism is replaced here by a formalism that transforms as a quaternion—a vector field in co-ordinate space and a second-rank spinor field of the type η ⊗ η* in spinor space. The geometrical field variables of the Riemann space are derived in quaternion form. The principle of least action (with the Palatini technique) is then used to derive a pair of time-reversed quaternion field equations, from the (quaternionic form of ) Einstein's Lagrangian. It is then shown how the conventional tensor form of the Einstein formalism is recovered from a particular combination of the derived time-reversed quaternion equations.

A class of algebraically special perfect fluid space-times

Abstract: Solutions of the Einstein field equations are considered subject to the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the Weyl tensor is algebraically special, (3) the corresponding repeated principal null congruence is geodesic and shearfree. If in addition, the repeated principal null congruence is non-expanding, it follows that the twist of this congruence must be non-zero (for a physically reasonable fluid). The general line element subject to this additional restriction is derived. Furthermore, it is shown that all solutions of the Einstein field equations which satisfy (1) and exhibit local rotational symmetry, necessarily satisfy (2) and (3).

An optimal transport formulation of the Einstein equations of general relativity

Abstract: The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

Henri Poincare and Einstein's Theory of Relativity

Abstract: Contrary to the opinion of some who have written on the history of relativity, the author argues that Henri Poincare did not anticipate Einstein's special theory of relativity; but rather, that Poincare was intent on an entirely different program—the perfection of the Lorentz theory of electrons. It is also suggested that Poincare was not consistent in his use of the so-called “conventional” point of view.

A complex vectorial formalism in general relativity.

Abstract: Complex vectorial formalism in general relativity, discussing Riemann connection related to representation of Lorentz group by three- dimensional linear space