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.

Particle field in bimetric general relativity

Abstract: The field equations of the bimetric general relativity theory proposed recently by one of the authors (N. Rosen) are put into a static form. The equations are solved near the Schwarzschild sphere, and it is found that the field differs from that of the Einstein general relativity theory: instead of a black hole, one has an impenetrable sphere. For larger distances the field is found to agree with that of ordinary general relativity, so that solar system observations cannot distinguish between the two theories. For very large distances one gets a cosmic contribution to the field which may affect the dynamics of clusters of galaxies.

Einstein energy associated with the Friedmann–Robertson–Walker metric

Abstract: Following Einstein’s definition of Lagrangian density and gravitational field energy density (Einstein in Ann Phys Lpz 49:806, 1916, Einstein in Phys Z 19:115, 1918, Pauli in Theory of Relativity, B.I. Publications, Mumbai, 1963), Tolman derived a general formula for the total matter plus gravitational field energy (P 0) of an arbitrary system (Tolman in Phys Rev 35:875, 1930, Tolman in Relativity, Thermodynamics & Cosmology, Clarendon Press, Oxford, 1962, Xulu in hep-th/0308070, 2003). For a static isolated system, in quasi-Cartesian coordinates, this formula leads to the well known result $${P_0 = \int \sqrt{-g} (T_0^0 - T_1^1 - T_2^2 - T_3^3) d^3 x,}$$ where g is the determinant of the metric tensor and $${T^a_b}$$ is the energy momentum tensor of the matter. Though in the literature, this is known as “Tolman Mass”, it must be realized that this is essentially “Einstein Mass” because the underlying pseudo-tensor here is due to Einstein. In fact, Landau–Lifshitz obtained the same expression for the “inertial mass” of a static isolated system without using any pseudo-tensor at all and which points to physical significance and correctness of Einstein Mass (Landau, Lifshitz in The Classical Theory of Fields, Pergamon Press, Oxford, 1962)! For the first time we apply this general formula to find an expression for P 0 for the Friedmann–Robertson–Walker (FRW) metric by using the same quasi-Cartesian basis. As we analyze this new result, it transpires that, physically, a spatially flat model having no cosmological constant is preferred. Eventually, it is seen that conservation of P 0 is honoured only in the static limit.

Einstein's Hole Argument

Abstract: In general relativity, space and time are inseparable from a gravitational field: no field, no spacetime. This is a lesson of Einstein’s hole argument. We use a simple transformation in a Schwartzschild spacetime to illustrate this. On the basis of the general theory of relativity … space as opposed to “what fills space” … has no separate existence. … There is no such thing as an empty space, i.e., a space without [a gravitational] field. … Spacetime does not claim existence on its own, but only as a structural quality of the field. Albert Einstein, 1952.

New numerical scheme to compute three-dimensional configurations of quasiequilibrium compact stars in general relativity: Application to synchronously rotating binary star systems

Abstract: We develop a new numerical scheme to obtain quasiequilibrium structures of binary neutron star systems and nonaxisymmetric compact stars as well as the space time around those systems in general relativity. Although, strictly speaking, there are no equilibrium states for binary configurations in general relativity, the time scale of changes in orbital motion due to gravitational wave radiation is long compared with the orbital period. Thus, we can assume that binary neutron star systems, and nonaxisymmetric systems in general are in ``quasiequilibrium'' states. Concerning the quasiequilibrium states of binary systems in general relativity, several investigations have been already carried out by assuming conformal flatness of the spatial part of the metric. However, the validity of the conformally flat treatment has not been fully analyzed except for axisymmetric configurations. Therefore, it is desirable to solve for the quasiequilibrium states by developing totally different methods from the conformally flat scheme. In this paper, we present a new numerical scheme to solve the Einstein equations for three-dimensional configurations directly, without assuming conformal flatness, although we use the simplified metric for the space time. This new formulation is an extension of the scheme which has been successfully applied for structures of axisymmetric rotating compact stars in general relativity. It is based on the integral representation of the Einstein equations, and takes into account the boundary conditions at infinity. We have checked our numerical scheme by computing equilibrium sequences of binary polytropic star systems in Newtonian gravity and those of axisymmetric polytropic stars in general relativity. We have applied this numerical code to binary star systems in general relativity and have succeeded in obtaining several equilibrium sequences of synchronously rotating binary polytropes with the polytropic indices $N=0.0,$ $0.5,$ and $1.0.$ It should be noted that our equilibrium sequences are not those of constant baryon mass star models because there is no unique choice of parameters to keep the baryon mass constant for our polytropic relation.