scholarly journals The Axially Symmetric Stationary Vacuum Field Equations in Einstein's Theory of General Relativity

1978 ◽  
Vol 31 (5) ◽  
pp. 427 ◽  
Author(s):  
A Sloane
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Vacuum Field ◽  
Stationary Case ◽  
Axially Symmetric ◽  
Field Equations ◽  
Asymptotically Flat ◽  
Stationary Vacuum ◽  
Flat Metric

All formulations of Einstein's vacuum field equations for which exact solutions have been found in the axially symmetric stationary case are compared, and the inherent restrictions of each are displayed. A measure of their usefulness as theoretical tools is gained by the ease with which they admit Kerr's (1963) solution (it being the simplest asymptotically flat metric of this kind). The solutions found using each formulation are listed and, where possible, suitably classified.

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 A class of axially symmetric stationary exact solutions of Einstein's vacuum field equations

1978 ◽  
Vol 20 (3) ◽  
pp. 344-351
Author(s):  
M. D. Patel
Keyword(s):  
Gravitational Field ◽  
Exact Solutions ◽  
Coordinate System ◽  
Stationary Distribution ◽  
Vacuum Field ◽  
Axially Symmetric ◽  
Field Equations ◽  
System A ◽  
Oblate Spheroidal

AbstractEinstein's vacuum field equations of an axially symmetric stationary rotating source are studied. Using the oblate spheroidal coordinate system, a class of asymptotically fiat solutions representing the exterior gravitational field of a stationary rotating oblate spheroidal source is obtained. Also it is proved that an analytic axisymmetric and stationary distribution of dust cannot be the source for the gravitational field described by the axisymmetric stationary metric.

scholarly journals New Solutions of Einstein Equations Representing Spinning Masses

1974 ◽  
Vol 64 ◽  
pp. 191-191
Author(s):  
Humitaka Sato ◽  
Akira Tomimatsu
Keyword(s):  
Exact Solutions ◽  
Event Horizon ◽  
Equatorial Plane ◽  
Space Time ◽  
Einstein Equations ◽  
Vacuum Field ◽  
Kerr Metric ◽  
Field Equations ◽  
Asymptotically Flat

We found new, stationary axisymmetric, asymptotically flat exact solutions to Einstein's vacuum field equations, which are classified by an integer δ and Kerr metric is the solution of δ = 1. The number of ring singularity on the equatorial plane is δ. The odd δ metrices contain the surfaces of event horizon but the even δ metrices do not. Except the Kerr metric, however, the space-time becomes singular at the poles on these surfaces.

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 Axially Symmetric, Asymptotically Flat Vacuum Metric with a Naked Singularity and Closed Timelike Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Debojit Sarma ◽  
Faizuddin Ahmed ◽  
Mahadev Patgiri
Keyword(s):  
Naked Singularity ◽  
Empty Space ◽  
Space Time ◽  
Axially Symmetric ◽  
Field Equations ◽  
Closed Timelike Curves ◽  
Asymptotically Flat ◽  
Type D ◽  
Initial Hypersurface ◽  
Petrov Classification

We present an axially symmetric, asymptotically flat empty space solution of the Einstein field equations containing a naked singularity. The space-time is regular everywhere except on the symmetry axis where it possesses a true curvature singularity. The space-time is of type D in the Petrov classification scheme and is locally isometric to the metrics of case IV in the Kinnersley classification of type D vacuum metrics. Additionally, the space-time also shows the evolution of closed timelike curves (CTCs) from an initial hypersurface free from CTCs.

The general axially symmetric static solution of Einstein’s vacuum field equations

1982 ◽  
Vol 382 (1783) ◽  
pp. 467-470 ◽  
Keyword(s):  
General Solution ◽  
Potential Function ◽  
Line Element ◽  
Static Solution ◽  
Vacuum Field ◽  
Axially Symmetric ◽  
Field Equations ◽  
Axis Of Symmetry ◽  
Axisymmetric Solutions ◽  
Associated Function

The general solution in closed form, including all the static axisymmetric solutions of Weyl, is presented in the canonical coordinates ρ and z of his line element. This general solution is constructed from an arbitrary function f ( z ), which coincides with his potential function along the axis of symmetry. To illustrate how the solution may be used, a particular function f , one resulting from a Newtonian solution, is used to find both the potential function and its associated function in the line element.

Numerical relativity. II. Numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space‒times with two Killing vectors

1983 ◽  
Vol 386 (1791) ◽  
pp. 373-391 ◽  
Keyword(s):  
General Relativity ◽  
Numerical Methods ◽  
Plane Wave ◽  
Numerical Solution ◽  
Initial Value Problem ◽  
Vacuum Field ◽  
Field Equations ◽  
Initial Value ◽  
Wave Space ◽  
Characteristic Initial Value Problem

This is the second of a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity. Although the equations to be integrated have regular coefficients, the nonlinearity leads to the occurrence of singularities after a finite evolution time. In this paper we first discuss some novel techniques for integrating the equations right up to the singularities. The second half of the paper presents as examples the numerical evolution of the Schwarzschild and certain colliding plane wave space‒times.

Rotating fluid masses in general relativity. II

1966 ◽  
Vol 62 (3) ◽  
pp. 495-501 ◽  
Author(s):  
R. H. Boyer
Keyword(s):  
General Relativity ◽  
Variational Principle ◽  
First Integrals ◽  
Constant Angular Velocity ◽  
Hydrodynamic Equations ◽  
Rotating Fluid ◽  
Axially Symmetric ◽  
Field Equations ◽  
Full Field ◽  
Specific Entropy

AbstractWe describe some properties of a stationary, isolated, axially symmetric, rotating body of perfect fluid, according to general relativity. We first specialize to the case of constant specific entropy and constant angular velocity. The latter condition is equivalent to rigidity in the Born sense; both conditions are consequences of a simple variational principle. The hydrodynamic equations can then be integrated completely. Analogous first integrals are given also for the case of differential rotation. No use is made of the full field equations.

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.

Sign in / Sign up

Export Citation Format

Share Document