scholarly journals Soliton-Like Spherical Symmetric Solutions to the Electromagnetic and Scalar Nonlinear Induction Field Equations in the General Relativity Theory

2022 ◽  
Vol 08 (01) ◽  
pp. 147-163
Author(s):  
Arnaud E. Yamadjako ◽  
Alain Adomou ◽  
Yélomè J. F. Kpomahou ◽  
Jonas Edou ◽  
Siaka Massou
Keyword(s):  
General Relativity ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Induction Field

Exact solutions of the field equations: twist-free pure radiation fields

1962 ◽  
Vol 270 (1342) ◽  
pp. 328-334 ◽  
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Radiation Field ◽  
Relativity Theory ◽  
Vacuum Field ◽  
Field Equations ◽  
General Relativity Theory ◽  
Pure Radiation ◽  
Radiation Fields ◽  
Null Congruence

This note is intended to give a rough survey of the results obtained in the study of twist-free pure radiation fields in general relativity theory. Here we are using the following Definition. A space-time ( V 4 of signature +2) is called a pure radiation field if it contains a distortion-free geodetic null congruence (a so-called ray congruence ), and if it satisfies certain field equations which we will specify below (e.g. Einstein’s vacuum-field equations). A (null) congruence is called twist-free if it is hypersurface-orthogonal (or ‘normal’). The results listed below were obtained by introducing special (‘canonical’) co-ordinates adapted to the ray congruence. Detailed proofs were given by Robinson & Trautman (1962) and by Jordan, Kundt & Ehlers (1961) (see also Kundt 1961). For the sake of completeness we include in our survey the subclass of expanding fields, and make use of some formulae first obtained by Robinson & Trautman.

scholarly journals On Hilbert's world-function

1927 ◽  
Vol 113 (765) ◽  
pp. 496-511 ◽  
Keyword(s):  
Dynamical System ◽  
General Relativity ◽  
Minimum Principle ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Least Action ◽  
Principle Of Least Action ◽  
The Universe ◽  
World Function

The well-known theorem that the motion of any conservative dynamical system can be determined from the “Principle of Least Action” or “Hamilton’s Principle” was carried over into General Relativity-Theory in 1915 by Hilbert, who showed that the field-equations of gravitation can be deduced very simply from a minimum-principle. Hilbert generalised his ideas into the assertion that all physical happenings (gravitational electrical, etc.) in the universe are determined by a scalar “world-function” H, being, in fact, such as to annul the variation of the integral ∫∫∫∫H√(−g)dx 0 dx 1 dx 2 dx 3 where ( x 0 , x 1 , x 2 , x 3 ) are the generalised co-ordinates which specify place and time, and g is (in the usual notation of the relativity-theory) the determinant of the gravitational potentials g v q , which specify the metric by means of the equation dx 2 = ∑ p, q g vq dx v dx q . In Hilbert’s work, the variation of the above integral was supposed to be due to small changes in the g vq 's and in the electromagnetic potentials, regarded as functions of x 0 , x 1 , x 2 , x 3 .

Non-linear Lagrangians and Palatine's device

1960 ◽  
Vol 56 (4) ◽  
pp. 396-400 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
General Relativity ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Invariant Action ◽  
Action Integral ◽  
Independent Variation ◽  
Non Linear

ABSTRACTField equations in general relativity theory have sometimes been generated by subjecting, in an invariant action integral, the components of linear connexion and the components of a covariant tensor of valence 2 to independent variation. The conceptual objections to this process, and some of the manifold formal difficulties inherent in it, are discussed in some detail. At the same time certain results obtained elsewhere are strengthened and in part corrected.

scholarly journals Reciprocal Static Solutions of the Equations of the Gravitational Field

1956 ◽  
Vol 9 (1) ◽  
pp. 13 ◽  
Author(s):  
HA Buchdahl
Keyword(s):  
General Relativity ◽  
Equation Of State ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Remarkable Property ◽  
Reciprocal Transformation ◽  
Total Energy Density ◽  
Static Solutions ◽  
Line Elements

This paper deals with reciprocal static line-elements, previously defined by the author, the condition that their Ricci tensors vanish being no longer imposed. If the indices i, k run from 1 to n with the exception of the fixed index a (the line-elements being static with respect to xa) a certain quantity appears with the remarkable property that in a reciprocal transformation is invariant, whilst merely changes sign. is closely related to the Hamiltonian derivative of the Gaussian curvature, so that the general results obtained may be applied to the field equations of General Relativity Theory, with n=a=4. is then the total energy density; and formally every static distribution of matter has a " reciprocal distribution" associated with it. In particular, the equation of state of a distribution of fluid reciprocal to a distribution of fluid possessing a given equation of state may be obtained directly from the latter, i.e. without the solution of the field equations being known.

scholarly journals Charged and Non-Charged Black Hole Solutions in Mimetic Gravitational Theory

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 559 ◽  
Author(s):  
Gamal Nashed
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Black Hole ◽  
Gravitational Theory ◽  
Relativity Theory ◽  
Charged Black Hole ◽  
Field Equations ◽  
General Relativity Theory ◽  
Mimetic Theory ◽  
Black Hole Solutions

In this study, we derive, in the framework of mimetic theory, charged and non-charged black hole solutions for spherically symmetric as well as flat horizon spacetimes. The asymptotic behavior of those black holes behave as flat or (A)dS spacetimes and coincide with the solutions derived before in general relativity theory. Using the field equations of non-linear electrodynamics mimetic theory we derive new black hole solutions with monopole and quadrupole terms. The quadruple term of those black holes is related by a constant so that its vanishing makes the solutions coincide with the linear Maxwell black holes. We study the singularities of those solutions and show that they possess stronger singularity than the ones known in general relativity. Among many things, we study the horizons as well as the heat capacity to see if the black holes derived in this study have thermodynamical stability or not.

On The Motion of Particles in General Relativity Theory

1949 ◽  
Vol 1 (3) ◽  
pp. 209-241 ◽  
Author(s):  
A. Einstein ◽  
L. Infeld
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Equations Of Motion ◽  
Relativity Theory ◽  
Fundamental Importance ◽  
Approximation Procedure ◽  
High Approximation ◽  
Field Equations ◽  
General Relativity Theory ◽  
First Time

The gravitational field manifests itself in the motion of bodies. Therefore the problem of determining the motion of such bodies from the field equations alone is of fundamental importance. This problem was solved for the first time some ten years ago and the equations of motion for two particles were then deduced [1]. A more general and simplified version of this problem was given shortly thereafter [2].Mr. Lewison pointed out to us, that from our approximation procedure, it does not follow that the field equations can be solved up to an arbitrarily high approximation. This is indeed true.

The Coordinate Conditions and the Equations of Motion

1953 ◽  
Vol 5 ◽  
pp. 17-25 ◽  
Author(s):  
L. Infeld
Keyword(s):  
General Relativity ◽  
Coordinate System ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Coordinate Conditions

The problem of the field equations and the equations of motion in general relativity theory is now sufficiently clarified. The equations of motion can be deduced from pure field equations by treating matter as singularities, [2; 3], or from field equations with the energy momentum tensor [4]. Recently two papers appeared in which the problem of the coordinate system was considered [5; 8]. The two papers are in general agreement as far as the role of the coordinate system is concerned. Yet there are some differences which require clarification.

Radiation and Gravitational Equations of Motion

1951 ◽  
Vol 3 ◽  
pp. 195-207 ◽  
Author(s):  
L. Infeld ◽  
A. E. Scheidegger
Keyword(s):  
General Relativity ◽  
Equations Of Motion ◽  
Classical Field ◽  
Relativity Theory ◽  
Field Theories ◽  
Field Equations ◽  
General Relativity Theory ◽  
Principle Of Superposition ◽  
Classical Field Theories ◽  
Gravitational Equations

Among the classical field theories, general relativity theory occupies a somewhat peculiar place. Unlike those of most other field theories, the field equations in relativity theory are non-linear. This implies that many facts, well known in linear theories, have no analogues in general relativity theory, and conversely. The equations of motion of the sources of the gravitational field are contained in the field equations, a fact which does not apply for the motion of an electron in the electromagnetic field. Conversely, it is difficult to define the notion of a wave (familiar in electrodynamics) in relativity theory; for, the linear principle of superposition is crucial for the existence of waves (at least in the sense that the notion of a wave is normally used).

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