Nonexistence of axially symmetric, stationary solution of Einstein Vacuum Equation with disconnected symmetric event horizon

1991 ◽  
Vol 73 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Yan Yan Li ◽  
Gang Tian
Keyword(s):  
Event Horizon ◽  
Stationary Solution ◽  
Vacuum Equation ◽  
Axially Symmetric ◽  
Einstein Vacuum Equation ◽  
Einstein Vacuum

scholarly journals GRAVITATIONAL FIELD OF A ROTATING GRAVITATIONAL DYON

2002 ◽  
Vol 17 (15n17) ◽  
pp. 1091-1096 ◽  
Author(s):  
N. DADHICH ◽  
Z. YA. TURAKULOV
Keyword(s):  
Gravitational Field ◽  
Gordon Equation ◽  
Equations Of Motion ◽  
Polar Angle ◽  
Vacuum Equation ◽  
Axially Symmetric ◽  
Einstein Vacuum Equation ◽  
Einstein Vacuum ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation

We have obtained the general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein - Gordon equation are separable. It can be interpreted to describe the gravitational field of a rotating dyon, a particle endowed with both gravoelectric (mass) and gravomagnetic (NUT parameter) charges. Further, there also exists a duality relation between the two charges and the radial and the polar angle coordinates which keeps the solution invariant. The solution can however be transformed into the known Kerr - NUT solution indicating its uniqueness under the separability of equations of motion.

INFLATION IN KALUZA-KLEIN COSMOLOGY I: TRANSFORMATION TO FOURTH-ORDER GRAVITY

1990 ◽  
Vol 05 (24) ◽  
pp. 4661-4669 ◽  
Author(s):  
HANS-JÜRGEN SCHMIDT
Keyword(s):  
Fourth Order ◽  
Arbitrary Dimension ◽  
Internal Space ◽  
Vacuum Equation ◽  
Scale Invariant ◽  
Einstein Vacuum Equation ◽  
Higher Dimensional ◽  
Einstein Vacuum ◽  
Klein Model ◽  
Kaluza Klein

The higher-dimensional Einstein vacuum equation with Λ-term is shown to be conformally equivalent to the four-dimensional field equation of scale-invariant fourth-order gravity. This holds for a general warped product between space-time and internal space of arbitrary dimension m which turns out to be an Einstein space. (The limit m → ∞ makes sense!) Thus, the results concerning the attractor property of the power-law inflationary solution derived for fourth-order gravity hold for the Kaluza-Klein model, too.

scholarly journals A Central Mass in a Stationary Vacuum without Spherical or Axial Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
William Davidson
Keyword(s):  
Black Hole ◽  
Event Horizon ◽  
Stationary Solution ◽  
Axial Symmetry ◽  
Axially Symmetric ◽  
Central Mass ◽  
Stationary Vacuum ◽  
Symmetry Properties ◽  
Null Congruence ◽  
Vacuum Spacetime

A vacuum spacetime with a central mass is derived as a stationary solution to Einstein's equations. The vacuum metric has a geodesic, shear-free, expanding, and twisting null congruence k and thus is algebraically special. The properties of the metric are calculated. In particular, it is shown that the spacetime has an event horizon inside which there is a black hole. The metric is neither spherically nor axially symmetric. It is therefore in interesting contrast with the majority of metrics featuring a central mass which have one or more of these symmetry properties. The metric reduces to the Schwarzschild case when a certain parameter is set to zero.

The Kerr solution

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Kerr Black Hole ◽  
Geodesic Equation ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Axially Symmetric ◽  
Observable Universe ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
The Many

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

scholarly journals Axially Symmetric and Stationary Solution of New General Relativity

1981 ◽  
Vol 66 (4) ◽  
pp. 1500-1503 ◽  
Author(s):  
M. Fukui ◽  
K. Hayashi
Keyword(s):  
General Relativity ◽  
Stationary Solution ◽  
Axially Symmetric

scholarly journals Integrability and vesture for harmonic maps into symmetric spaces

2016 ◽  
Vol 28 (03) ◽  
pp. 1650006 ◽  
Author(s):  
Shabnam Beheshti ◽  
Shadi Tahvildar-Zadeh
Keyword(s):  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Dressing Method ◽  
Special Cases ◽  
Einstein Vacuum ◽  
Completely Reducible

After formulating the notion of integrability for axially symmetric harmonic maps from [Formula: see text] into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding [Formula: see text]-solitonic harmonic maps into a non-compact Grassmann manifold [Formula: see text] is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases [Formula: see text] and [Formula: see text] and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

Introduction

2020 ◽  
pp. 1-23
Author(s):  
Sergiu Klainerman ◽  
Jérémie Szeftel
Keyword(s):  
General Relativity ◽  
Initial Data ◽  
Nonlinear Stability ◽  
Kerr Solution ◽  
Data Sets ◽  
Axially Symmetric ◽  
Kerr Spacetime ◽  
Closed Orbits ◽  
Mathematical Question ◽  
Einstein Vacuum

This introductory chapter provides a quick review of the basic concepts of general relativity relevant to this work. The main object of Albert Einstein's general relativity is the spacetime. The nonlinear stability of the Kerr family is one of the most pressing issues in mathematical general relativity today. Roughly, the problem is to show that all spacetime developments of initial data sets, sufficiently close to the initial data set of a Kerr spacetime, behave in the large like a (typically another) Kerr solution. This is not only a deep mathematical question but one with serious astrophysical implications. Indeed, if the Kerr family would be unstable under perturbations, black holes would be nothing more than mathematical artifacts. The goal of this book is to prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, namely, solutions of the Einstein vacuum equations for asymptotically flat 1 + 3 dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield Z with closed orbits.

scholarly journals Pure Gauss–Bonnet NUT black hole with and without non-central singularity

2021 ◽  
Vol 81 (5) ◽  
Author(s):  
Sajal Mukherjee ◽  
Naresh Dadhich
Keyword(s):  
Black Hole ◽  
Event Horizon ◽  
Higher Dimensions ◽  
Radial Symmetry ◽  
Cosmological Horizon ◽  
Vacuum Equation ◽  
Closed Timelike Curves ◽  
Central Singularity ◽  
Horizon Topology ◽  
Nut Solution

AbstractIt is known that NUT solution has many interesting features and pathologies like being non-singular and having closed timelike curves. It turns out that in higher dimensions horizon topology cannot be spherical but it has instead to be product of 2-spheres so as to retain radial symmetry of spacetime. In this letter we wish to present a new solution of pure Gauss–Bonnet $$\Lambda $$ Λ -vacuum equation describing a black hole with NUT charge. It has three interesting cases: (a) black hole with both event and cosmological horizons with singularity being hidden behind the former, (b) a regular spacetime free of both horizon and singularity, and (c) black hole with event horizon without singularity and cosmological horizon. Singularity here is always non-centric at $$r \ne 0$$ r ≠ 0 .

Type-D solutions describing the collision of plane-fronted gravitational waves

1988 ◽  
Vol 417 (1853) ◽  
pp. 417-431 ◽  
Keyword(s):  
Shock Waves ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Event Horizon ◽  
Surface Acting ◽  
Null Geodesics ◽  
Type D ◽  
Null Surface ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum

Some exact solutions of the Einstein vacuum equations describing the collision of plane-fronted gravitational shock waves accompanied by impulsive waves which produces a type-D geometry in the region of interaction are presented. The collision results in the development of a null surface acting like an event horizon, and the metric has been analytically extended beyond it by using Kruskal coordinates properly adapted to the problem. The extension shows that all null rays emerging from the interaction region escape to infinity: no focusing is present on the horizon. The connection between focusing and creation of singularities has also been investigated by analysing the behaviour of a particular congruence of null geodesics.

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