Exact solutions of the Bach field equations of general relativity

1980 ◽  
Vol 17 (1) ◽  
pp. 15-36 ◽  
Author(s):  
B. Fiedler ◽  
R. Schimming
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Field Equations

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 How Extra Symmetries Affect Solutions in General Relativity

Universe ◽  
2020 ◽  
Vol 6 (10) ◽  
pp. 170
Author(s):  
Aroonkumar Beesham ◽  
Fisokuhle Makhanya
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Einstein's Field Equations

To get exact solutions to Einstein’s field equations in general relativity, one has to impose some symmetry requirements. Otherwise, the equations are too difficult to solve. However, sometimes, the imposition of too much extra symmetry can cause the problem to become somewhat trivial. As a typical example to illustrate this, the effects of conharmonic flatness are studied and applied to Friedmann–Lemaitre–Robertson–Walker spacetime. Hence, we need to impose some symmetry to make the problem tractable, but not too much so as to make it too simple.

scholarly journals ENERGY–MOMENTUM DISTRIBUTION: SOME EXAMPLES

2007 ◽  
Vol 22 (10) ◽  
pp. 1935-1951 ◽  
Author(s):  
M. SHARIF ◽  
M. AZAM
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Momentum Distribution ◽  
Special Class ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Degenerate Solution ◽  
Dust Solution

In this paper, we elaborate the problem of energy–momentum in General Relativity with the help of some well-known solutions. In this connection, we use the prescriptions of Einstein, Landau–Lifshitz, Papapetrou and Möller to compute the energy–momentum densities for four exact solutions of the Einstein field equations. We take the gravitational waves, special class of Ferrari–Ibanez degenerate solution, Senovilla–Vera dust solution and Wainwright–Marshman solution. It turns out that these prescriptions do provide consistent results for special class of Ferrari–Ibanez degenerate solution and Wainwright–Marshman solution but inconsistent results for gravitational waves and Senovilla–Vera dust solution.

scholarly journals Nonstatic Spherically Symmetric Solutions for a Perfect Fluid in General Relativity

1976 ◽  
Vol 29 (2) ◽  
pp. 113 ◽  
Author(s):  
N Chakravarty ◽  
SB Dutta Choudhury ◽  
A Banerjee
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Perfect Fluid ◽  
Dynamical Behaviour ◽  
Symmetric Distribution ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Spherically Symmetric Solutions ◽  
Spherically Symmetric Distribution ◽  
General Method

A general method is described by which exact solutions of Einstein's field equations are obtained for a nonstatic spherically symmetric distribution of a perfect fluid. In addition to the previously known solutions which are systematically derived, a new set of exact solutions is found, and the dynamical behaviour of the corresponding models is briefly discussed.

scholarly journals Extension Rules of Newman–Janis Algorithm for Rotation Metrics in General Relativity

2020 ◽  
pp. 1-14
Author(s):  
Yu-Ching, Chou
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Kerr Metric ◽  
Tensor Structure ◽  
De Sitter ◽  
Field Equations ◽  
Null Tetrad ◽  
Axisymmetric Solutions ◽  
Metric Function ◽  
De Sitter Metric

Aims: The aim of this study is to extend the formula of Newman–Janis algorithm (NJA) and introduce the rules of the complexifying seed metric. The extension of NJA can help determine more generalized axisymmetric solutions in general relativity.Methodology: We perform the extended NJA in two parts: the tensor structure and the seed metric function. Regarding the tensor structure, there are two prescriptions, the Newman–Penrose null tetrad and the Giampieri prescription. Both are mathematically equivalent; however, the latter is more concise. Regarding the seed metric function, we propose the extended rules of a complex transformation by r2/Σ and combine the mass, charge, and cosmologic constant into a polynomial function of r. Results: We obtain a family of axisymmetric exact solutions to Einstein’s field equations, including the Kerr metric, Kerr–Newman metric, rotating–de Sitter, rotating Hayward metric, Kerr–de Sitter metric and Kerr–Newman–de Sitter metric. All the above solutions are embedded in ellipsoid- symmetric spacetime, and the energy-momentum tensors of all the above metrics satisfy the energy conservation equations. Conclusion: The extension rules of the NJA in this research avoid ambiguity during complexifying the transformation and successfully generate a family of axisymmetric exact solutions to Einsteins field equations in general relativity, which deserves further study.

scholarly journals Revisiting cosmologies in teleparallelism

2021 ◽  
Author(s):  
Fabio D'Ambrosio ◽  
Lavinia Heisenberg ◽  
Simon Kuhn
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Symmetry Reduction ◽  
Affine Geometry ◽  
Linear Extensions ◽  
Field Equations ◽  
Torsion Scalar ◽  
Friedmann Equations ◽  
General Field ◽  
Theories Of Gravity

Abstract We discuss the most general field equations for cosmological spacetimes for theories of gravity based on non-linear extensions of the non-metricity scalar and the torsion scalar. Our approach is based on a systematic symmetry-reduction of the metric-affine geometry which underlies these theories. While for the simplest conceivable case the connection disappears from the field equations and one obtains the Friedmann equations of General Relativity, we show that in $f(\mathbb{Q})$ cosmology the connection generically modifies the metric field equations and that some of the connection components become dynamical. We show that $f(\mathbb{Q})$ cosmology contains the exact General Relativity solutions and also exact solutions which go beyond. In $f(\mathbb{T})$~cosmology, however, the connection is completely fixed and not dynamical.

scholarly journals Some exact solutions of certain field equations in general relativity

1976 ◽  
Vol 22 (2) ◽  
pp. 167-172
Author(s):  
L. K. Patel
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Einstein Space ◽  
Field Equations

SummaryA general metric is considered. Some solutions correspoding to the field equations Rik λgik = −8πEik are obtained as particular cases. Details of these solutions are also discussed. An Einstein space corresponding to the field equations Rik = λ gik is also constructed as a particular case of the general metric.

scholarly journals Generic off-diagonal solutions and solitonic hierarchies in (modified) gravity

2015 ◽  
Vol 30 (19) ◽  
pp. 1550090 ◽  
Author(s):  
Sergiu I. Vacaru
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Modified Gravity ◽  
Field Equations ◽  
Physical Data ◽  
Modified Gravity Theories ◽  
De Vries ◽  
Einstein Spaces ◽  
Curve Flows ◽  
Gravity Theories

We have summarized our recent results on encoding exact solutions of field equations in Einstein and modified gravity theories into solitonic hierarchies derived for nonholonomic curve flows with associated bi-Hamilton structure. We argue that there is a canonical distinguished connection for which the fundamental geometric/physical equations decouple in general form. This allows us to construct very general classes of generic off-diagonal solutions determined by corresponding types of generating and integration functions depending on all (spacetime) coordinates. If the integral varieties are constrained to zero torsion configurations, we can extract solutions for the general relativity (GR) theory. We conclude that the geometric and physical data for various classes of effective/modified Einstein spaces can be encoded into multi-component versions of the sine-Gordon, or modified Korteweg–de Vries equations.

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