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 the ''derivation'' of Einstein's field equations

Abstract: It is shown that the conventional correspondence procedure used in the transition from Newtonian to the Einsteinian theory of gravitation is not unique. A new procedure of correspondence whereby the theory is required to reduce to a special relativistic limit before it reduces to the Newtonian limit is described. It is shown that the special relativistic correspondence leads to first and second order requirements, whereas the Newtonian correspondence leads only to first order requirements. The resulting theory is logically plausible and experimentally viable.

Conharmonic curvature tensor and the Spacetime of General Relativity

Abstract: The signiflcance of conharmonic curvature tensor is very well known in the difierential geometry of certain F-structures (e.g., complex, almost complex, Kahler, almost Kahler, Hermitian, almost Hermitian structures, etc.). In this paper, a study of conharmonic curvature ten- sor has been made on the four dimensional spacetime of general relativity. The spacetime satisfying Einstein fleld equations and having vanishing conharmonic tensor is considered and the existence of Killing and confor- mal Killing vectors on such spacetime have been established. Perfect ∞uid cosmological models have also been studied.

Connections, loops and quantum general relativity

Abstract: The current status of a programme for nonperturbative canonical quantisation of general relativity is briefly summarized.

Singularities in general relativity.

Abstract: The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.