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.

Spherically Symmetrical Models in General Relativity

Abstract: The field equations of general relativity areapplied to pressure-free spherically symmetrical systemsof particles The equations of motion are determinedwithout the use of approximations and are compared with the Newtonian equations The total energyis found to be an important parameter, determining thegeometry of 3-space and the ratio of effectivegravitating to invariant mass The Doppler shift isdiscussed and is found to contain both the velocity shiftand the Einstein shift combined in a rather complexexpression

Cylindrically symmetric charged dust distributions in rigid rotation in general relativity

Abstract: The paper presents a family of stationary cylindrically symmetric solutions of the Einstein-Maxwell equations corresponding to a charged dust distribution in rigid rotation. The interesting feature of the solution is that the Lorentz force vanishes everywhere and the ratio of the charge density and mass density may assume arbitrary value. The solutions do not seem to have any classical analogue.

Palatini approach to Born-Infeld-Einstein theory and a geometric description of electrodynamics

Abstract: The field equations associated with the Born-Infeld-Einstein action are derived using the Palatini variational technique. In this approach the metric and connection are varied independently and the Ricci tensor is generally not symmetric. For sufficiently small curvatures the resulting field equations can be divided into two sets. One set, involving the antisymmetric part of the Ricci tensor ${R}_{{}_{\ensuremath{\vee}}^{\ensuremath{\mu}\ensuremath{ u}}},$ consists of the field equation for a massive vector field. The other set consists of the Einstein field equations with an energy momentum tensor for the vector field plus additional corrections. In a vacuum with ${R}_{{}_{\ensuremath{\vee}}^{\ensuremath{\mu}\ensuremath{ u}}}=0$ the field equations are shown to be the usual Einstein vacuum equations. This extends the universality of the vacuum Einstein equations, discussed by Ferraris et al., to the Born-Infeld-Einstein action. In the simplest version of the theory there is a single coupling constant and by requiring that the Einstein field equations hold to a good approximation in neutron stars it is shown that mass of the vector field exceeds the lower bound on the mass of the photon. Thus, in this case the vector field cannot represent the electromagnetic field and would describe a new geometrical field. In a more general version in which the symmetric and antisymmetric parts of the Ricci tensor have different coupling constants it is possible to satisfy all of the observational constraints if the antisymmetric coupling is much larger than the symmetric coupling. In this case the antisymmetric part of the Ricci tensor can describe the electromagnetic field.

Three-dimensional classical spacetimes

Abstract: We extend what is known about the structure of (2+ 1)-dimensional gravitational field theories. The non-existence of any Newtonian limit to these theories is investigated in the presence of Brans-Dicke scalar fields and non-linear curvature terms in the gravita- tional action. A number of new exact static and non-static solutions of (2+1) general relativity with scalar field, perfect fluid and magnetic field sources are presented and studied in detail. Some of these possess a correspondence with (3 + 1) solutions of general relativity through a Kaluza-Klein type reduction and exhibit the 'wedge' structure of (3 + 1)- dimensional solutions describing line sources like vacuum strings. An algebraic classification of (2+ 1) gravitational fields is derived using the Bach-Weyl tensor. The description of the general cosmological solution is given in terms of arbitrary spatial functions independently specified on a spacelike surface of constant time together with a series approximation to spacetime in the vicinity of a general cosmological singularity. Various results and conjectures regarding spacetime singularities are given. Two exact cosmological solutions containing self-interacting scalar fields that produce inflationary behaviour are also found.