Nöther Conserved Quantities and Entropy in General Relativity

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

It is proved that the combined neutrino gravitational field equations in general relativity admit no nontrivial solutions in a static space-time provided that the energy flow vector of the neutrino field is timelike or null for all observers.

General Relativity: An Introduction for Physicists provides a clear mathematical introduction to Einstein's theory of general relativity. It presents a wide range of applications of the theory, concentrating on its physical consequences. After reviewing the basic concepts, the authors present a clear and intuitive discussion of the mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are then used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is then introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle. Written for advanced undergraduate and graduate students, this approachable textbook contains over 300 exercises to illuminate and extend the discussion in the text.

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed by H. Friedrich (H. Friedrich, “Cauchy problem for the conformal vacuum field equations in general relativity”, Commun. Math. Phys. 91 (1983), 445-472.), (H. Friedrich, “On the existence of n-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure”, Commun. Math. Phys. 107 (1986), 587-609.), (-,“On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations”, Jour. Diff. Geom. 34 (1991), 275-345) using conformal completions techniques. In this conformal approach one has to 1) construct initial data, which satisfy the general relativistic constraint equations, with appropriate behavior near the conformal boundary, and 2) show a local (and perhaps also a global) existence theorem for the associated evolution problem. In this context solutions of the constraint equations can be found by solving a nonlinear elliptic system of equations, one of which resembles the Yamabe equation (and coincides with this equation in some cases), with the system degenerating near the conformal boundary. In the first part of the talk I (PTC) will describe the existence and boundary regularity results about this system obtained some years ago in collaboration with Helmut Friedrich and Lars Andersson. Some new applications of those techniques are also presented. In the second part of the talk I will describe some new results, obtained in collaboration with Olivier Lengard, concerning the evolution problem.

The core of general relativity are the Einstein field equations, which relate curvature to energy and momentum. This chapter introduces curvature and the energy-momentum tensor to derive tensor equations between them that fulfill the correct Newtonian limit—the Einstein field equation. Finally, the small curvature limit in which the Einstein field equations are linear is worked out. This establishes the contact with the other classical field theory, Maxwell electrodynamics.

The metric tensor of a Riemannian space-time is composed quadratically of timelike and spacelike tetrad fields. A system of first-order field equations for the tetrad fields is postulated, which determines the geometric structure of space-time as well as the second-order dynamics of the fields. The energy-momentum content of the fields automatically corresponds to the geometric structure of the space-time according to the Einstein field equations. The spin contribution of the fields is mutually compensated and therefore spin does not influence the space-time geometry. As a consequence, it is not necessary to include torsion into the general theory of relativity and nevertheless the equivalence principle can be used to consistently transfer energy-momentum tensors from flat to curved space.

This book deals with Colombeau solutions to Einstein field equations in general relativity: Gravitational singularities, distributional SAdS BH spacetime-induced vacuum dominance. This book covers key areas of Colombeau nonlinear generalized functions, distributional Riemannian, geometry, distributional schwarzschild geometry, Schwarzschild singularity, Schwarzschild horizon, smooth regularization, nonsmooth regularization, quantum fields, curved spacetime, vacuum fluctuations, vacuum dominance etc. This book contains various materials suitable for students, researchers and academicians of this area.

The aim of this chapter is a detailed study of applicability and applications of distributional geometry as developed in the previous chapters—with its special focus, of course, on the generalized pseudo-Riemannian geometry of Section 3.2.5—to the theory of general relativity. We shall begin with a general introduction into our theme in Section 5.1, where we also introduce our notational conventions concerning general relativity. Section 5.2 is divided into two parts. The first one is concerned with a brief overview of applications of linear distributional geometry to relativity with a strong focus on a “no-go”-theorem by Geroch and Traschen (cf. [Ger87]). In the second part of Section 5.2 we use the nonlinear distributional geometry of Chapter 3 to define the generalized curvature tensor as well its contractions appearing in the field equations of general relativity. Moreover, we present a guideline for applying this setting to the study of singular spacetime metrics and discuss consistency with respect to the classical theory. Finally in Section 5.3 we give a complete distributional description of impulsive gravitational pp-waves based upon a variety of concepts introduced so far.

The covariant wedge products of tetrads in General Relativity are introduced and related field equations are derived. The electromagnetic dual field is treated in detail and it is shown that this field is closely related with one component field, which is eigenfunction of d'Alembertian operator. The formalism is developed to incorporate internal symmetry.

The thin-sandwich conjecture for Einstein's field equations of general relativity is discussed. Three of the four initial-value equations of general relativity are considered. This system of three equations is shown to have an integrability condition when the spatial metric, its time rate of change, and the lapse function N are given. The nature of this integrability condition is discussed in detail. It is also noted that this integrability condition can be expected to play a significant role in any proof of existence for a thin-sandwich problem in which the spatial metric and its time derivative are chosen arbitrarily. A local proof of existence is given for the first three thin-sandwich equations. The proof allows the spatial metric components, their time rate of change, and the lapse function N to be chosen arbitrarily. No coordinate conditions of any kind are used in the proof.

When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. Because metric lengths are the physically measured lengths, their use makes it possible to understand general relativity directly in terms of physical quantities such as energy and momentum within a curved space–time. The metric form of the second law contains gravitational effects in exactly the same manner as occurs in relativity. Its mathematical derivation uses vectors rather than tensors, and nongravitational forces can occur in this modified second law without a tensor form. Because quantum mechanics is based on Newtonian concepts of energy and momentum, it is shown that when metric lengths replace coordinate lengths in Dirac’s wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation. Metric transformations are also used to describe the Dirac equation for the gravitational central field in a Schwarzschild metric.

Inthe present work we show that the existence of non-vanishing torsion field may solve, at least, one of the problems FRW-cosmology, the particle horizons problem. The field equations of general relativity (GR) are written in a space having non-vanishing torsion, the absolute parallelism (AP) space. An AP-Structure, satisfying the cosmological principle, is used to construct a world model. Energy density and pressure, purely induced by torsion, are defined from the building blocks of the AP-geometry using GR. When these quantities are used in the FRW-dynamical equations, we get a world model free from particle horizons.

Rotating cosmological cylindrical wormholes in GR and TEGR sourced by anisotropic fluids

In a preceding paper, (see abstr. A2555 of 1974), the authors obtained a solution of the field equations of general relativity for an elastic sphere of constant density. The present investigation obtains a solution of the charged elastic fluid distribution in general relativity.

An exact solution of the field equations of general relativity is obtained for a static, spherically symmetric distribution of charge and mass which can be matched with the Reissner–Nordström metric at the boundary. The self-energy contributions to the total gravitational mass are computed retaining only the first order terms in the gravitational constant.