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.

Rotating fluids in General Relativity

Abstract: We define the notion of a surface of revolution in the curved spaces of General Relativity and we present the corresponding Einstein's equations in the case of an anisotropic fluid with bulk and shear viscosity and heat conduction. We indicate a method of integration and some particular solutions.

Fractality field in the theory of scale relativity

Abstract: In the theory of scale relativity, space-time is considered to be a continuum that is not only curved, but also non-differentiable, and, as a consequence, fractal. The equation of geodesics in such a space-time can be integrated in terms of quantum mechanical equations. We show in this paper that the quantum potential is a manifestation of such a fractality of space-time (in analogy with Newton’s potential being a manifestation of curvature in the framework of general relativity).

Finding and using exact solutions of the Einstein equations

Abstract: The evolution of the methods used to find solutions of Einstein’s field equations during the last 100 years is described. Early papers used assumptions on the coordinate forms of the metrics. Since the 1950s more invariant methods have been deployed in most new papers. The uses to which the solutions found have been put are discussed, and it is shown that they have played an important role in the development of many aspects, both mathematical and physical, of general relativity.

Linearized modified gravity theories with a cosmological term: advance of perihelion and deflection of light

Abstract: Two different ways of generalizing Einstein's general theory of relativity with a cosmological constant to Brans-Dicke type scalar-tensor theories are investigated in the linearized field approximation. In the first case a cosmological constant term is coupled to a scalar field linearly whereas in the second case an arbitrary potential plays the role of a variable cosmological term. We see that the former configuration leads to a massless scalar field whereas the latter leads to a massive scalar field. General solutions of these linearized field equations for both cases are obtained corresponding to a static point mass. Geodesics of these solutions are also presented and solar system effects such as the advance of the perihelion, deflection of light rays and gravitational redshift were discussed. In general relativity cosmological constant has no role on these phenomena. We see that for the Brans-Dicke theory the cosmological constant has also no effect on these phenomena. This is because solar system observations require very large values of the Brans-Dicke parameter and the correction terms to these phenomena becomes identical to GR for these large values of this parameter. This result is also observed for the theory with arbitrary potential if the mass of the scalar field is very light. For a very heavy scalar field, however, there is no such limit on the value of this parameter and there are ranges of this parameter where these contributions may become relevant in these scales. Galactic and intergalactic dynamics is also discussed for these theories at the latter part of the paper with similar conclusions.