scholarly journals Disforming the Kerr metric

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Timothy Anson ◽  
Eugeny Babichev ◽  
Christos Charmousis ◽  
Mokhtar Hassaine
Keyword(s):  
Event Horizon ◽  
Compact Object ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Candidate Event ◽  
Kerr Spacetime ◽  
Constant Degree ◽  
Tensor Theory ◽  
Timelike Hypersurface ◽  
Null Congruence

Abstract Starting from a recently constructed stealth Kerr solution of higher order scalar tensor theory involving scalar hair, we analytically construct disformal versions of the Kerr spacetime with a constant degree of disformality and a regular scalar field. While the disformed metric has only a ring singularity and asymptotically is quite similar to Kerr, it is found to be neither Ricci flat nor circular. Non-circularity has far reaching consequences on the structure of the solution. As we approach the rotating compact object from asymptotic infinity we find a static limit ergosurface similar to the Kerr spacetime with an enclosed ergoregion. However, the stationary limit of infalling observers is found to be a timelike hypersurface. A candidate event horizon is found in the interior of this stationary limit surface. It is a null hypersurface generated by a null congruence of light rays which are no longer Killing vectors. Under a mild regularity assumption, we find that the candidate surface is indeed an event horizon and the disformed Kerr metric is therefore a black hole quite distinct from the Kerr solution.

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.

On embeddings of the Kerr geometry

1981 ◽  
Vol 59 (5) ◽  
pp. 688-692 ◽  
Author(s):  
Nigel A. Sharp
Keyword(s):  
Black Hole ◽  
Event Horizon ◽  
Equatorial Plane ◽  
Kerr Metric ◽  
Intrinsic Structure ◽  
Kerr Geometry ◽  
Rotating Black Hole ◽  
Isometric Embeddings

The use of isometric embeddings of curved geometries reveals their intrinsic structure in a way that is readily appreciated. This is done for 3 two-surfaces sliced from the Kerr metric which describes a rotating black hole: the equatorial plane, the event horizon, and the ergosurface.

scholarly journals Exact Axially Symmetric Solution inf(T)Gravity Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Gamal G. L. Nashed
Keyword(s):  
Vacuum Solution ◽  
Gravity Theory ◽  
Kerr Metric ◽  
Radial Coordinate ◽  
Lorentz Transformations ◽  
Axially Symmetric ◽  
Field Equations ◽  
Tetrad Field ◽  
Kerr Spacetime ◽  
Euler’S Angles

A general tetrad field with sixteen unknown functions is applied to the field equations off(T)gravity theory. An analytic vacuum solution is derived with two constants of integration and an angleΦthat depends on the angle coordinateϕand radial coordinater. The tetrad field of this solution is axially symmetric and the scalar torsion vanishes. We calculate the associated metric of the derived solution and show that it represents Kerr spacetime. Finally, we show that the derived solution can be described by two local Lorentz transformations in addition to a tetrad field that is the square root of the Kerr metric. One of these local Lorentz transformations is a special case of Euler’s angles and the other represents a boost when the rotation parameter vanishes.

scholarly journals ORBITS IN A NON-KERR DYNAMICAL SYSTEM

2011 ◽  
Vol 21 (08) ◽  
pp. 2261-2277 ◽  
Author(s):  
G. CONTOPOULOS ◽  
G. LUKES-GERAKOPOULOS ◽  
T. A. APOSTOLATOS
Keyword(s):  
Angular Momentum ◽  
Outer Region ◽  
Kerr Black Hole ◽  
Compact Object ◽  
Kerr Metric ◽  
Integrable Case ◽  
Closed Curves ◽  
Surface Of Section ◽  
Fundamental Frequencies ◽  
Chaotic Orbits

We study the orbits in a Manko–Novikov type metric (MN) which is a perturbed Kerr metric. There are periodic, quasi-periodic, and chaotic orbits, which are found in configuration space and on a surface of section for various values of the energy E and the z-component of the angular momentum Lz. For relatively large Lz there are two permissible regions of nonplunging motion bounded by two closed curves of zero velocity (CZV), while in the Kerr metric there is only one closed CZV of nonplunging motion. The inner permissible region of the MN metric contains mainly chaotic orbits, but it contains also a large island of stability. When Lz decreases, the two permissible regions join and chaos increases. Below a certain value of Lz, most orbits escape inwards and plunge through the horizon. On the other hand, as the energy E decreases (for fixed Lz) the outer permissible region shrinks and disappears. In the inner permissible region, chaos increases and for sufficiently small E most orbits are plunging. We find the positions of the main periodic orbits as functions of Lz and E, and their bifurcations. Around the main periodic orbit of the outer region, there are islands of stability that do not appear in the Kerr metric (integrable case). In a realistic binary system, because of the gravitational radiation, the energy E and the angular momentum Lz of an inspiraling compact object decrease and therefore the orbit of the object is nongeodesic. In fact, in an extreme mass ratio inspiraling (EMRI) system the energy E and the angular momentum Lz decrease adiabatically and therefore the motion of the inspiraling object is characterized by the fundamental frequencies which are drifting slowly in time. In the Kerr metric, the ratio of the fundamental frequencies changes strictly monotonically in time. However, in the MN metric when an orbit is trapped inside an island the ratio of the fundamental frequencies remains constant for some time. Hence, if such a phenomenon is observed this will indicate that the system is nonintegrable and therefore the central object is not a Kerr black hole.

scholarly journals Unit-Lapse Forms of Various Spacetimes

2021 ◽  
Author(s):  
Joshua Baines
Keyword(s):  
Initial Velocity ◽  
Mathematical Object ◽  
Kerr Metric ◽  
Coordinate Systems ◽  
Relativity Principle ◽  
Spatial Infinity ◽  
Kerr Spacetime ◽  
Spacetime Curvature ◽  
Stationary Spacetimes ◽  
Better Than

<p><b>Every spacetime is defined by its metric, the mathematical object which further defines the spacetime curvature. From the relativity principle, we have the freedom to choose which coordinate system to write our metric in. Some coordinate systems, however, are better than others. In this text, we begin with a brief introduction into general relativity, Einstein's masterpiece theory of gravity. We then discuss some physically interesting spacetimes and the coordinate systems that the metrics of these spacetimes can be expressed in. More specifically, we discuss the existence of the rather useful unit-lapse forms of these spacetimes. Using the metric written in this form then allows us to conduct further analysis of these spacetimes, which we discuss. </b></p><p>Overall, the work given in this text has many interesting mathematical and physical applications. Firstly, unit-lapse spacetimes are quite common and occur rather naturally for many specific analogue spacetimes. In an astrophysical context, unit-lapse forms of stationary spacetimes are rather useful since they allow for very simple and immediate calculation of a large class of timelike geodesics, the rain geodesics. Physically these geodesics represent zero angular momentum observers (ZAMOs), with zero initial velocity that are dropped from spatial infinity and are a rather tractable probe of the physics occurring in the spacetime. Mathematically, improved coordinate systems of the Kerr spacetime are rather important since they give a better understanding of the rather complicated and challenging Kerr spacetime. These improved coordinate systems, for example, can be applied to the attempts at finding a "Gordon form" of the Kerr spacetime and can also be applied to attempts at upgrading the "Newman-Janis trick" from an ansatz to a full algorithm. Also, these new forms of the Kerr metric allows for a greater observational ability to differentiate exact Kerr black holes from "black hole mimickers".</p>

scholarly journals On the Trapped Surface Characterization of the Black Hole Region in Kerr Spacetime

2018 ◽  
Vol 10 (4) ◽  
pp. 24
Author(s):  
Mohammed Kumah ◽  
Francis Oduro
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Event Horizon ◽  
Surface Characterization ◽  
Local Approach ◽  
Kerr Spacetime ◽  
Trapped Surfaces ◽  
Trapped Surface ◽  
Future Direction

Black holes are classically characterized by event horizon which is the boundary of the region from which particles or photons can escape to infinity in the future direction. Unfortunately this characterization is a global concept as the knowledge of the whole spacetime is needed in order to locate a black hole region and the event horizon. It is therefore important to recognize black holes locally; this has motivated the need to use local approach to characterize black holes. Specifically, we apply covariant divergence and Gauss’s divergence theorems to compute the divergences and the fluxes of appropriate null vectors in the Kerr spacetime to actually determine the existence of trapped and marginally trapped surfaces in its black hole region.

Extremal Kerr Black Holes, Naked Singularity & Wormholes

2020 ◽  
Author(s):  
Deep Bhattacharjee
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Black Hole ◽  
Angular Momentum ◽  
Event Horizon ◽  
Kerr Black Hole ◽  
Naked Singularity ◽  
Kerr Metric ◽  
Spin Angular Momentum ◽  
The Way

This paper is totally based on the mathematical physics of the Black holes. In Einstein’s theory of “General Relativity”, Schwarzschild solution is the vacuum solutions of the Einstein Field Equations that describes the gravity potential from outside the body of a spherically symmetric object having zero charge, zero mass and zero cosmological constant[1]. It was discovered by Karl Schwarzschild in 1916, a little more than a month after the publication of the famous GR and the singularity is a point singularity which can be best described as a coordinate singularity rather than a real singularity, however, the drawback of this theory is that it fails to take into account the real life scenario of black holes with charge and spin angular momentum. The black hole is based on event horizon and Schwarzschild radius. However, Physicists were trying to develop a metric for the real life scenario of a black hole with a spin angular momen-tum and ultimately the exact solution of a charged rotating black hole had been discovered by Roy Kerr in 1965 as the Kerr-Newman metric[2][3]. The Kerr metric is one of the toughest metric in physics and is the extensional generalization to a rotating body of the Schwarzschild metric. The metric describes the vacuum geometry of space-time around a rotating axially-symmetric black hole with a quasipotential event horizon. In Kerr metric there are two event hori-zons (inner and outer), two ergospheres and an ergosurface. The most important effect of the Kerr metric is the frame dragging (also known as Lense-Thirring Precession) is a distinctive prediction of General relativity. The first direct observation of the collision of two Kerr Black Holes has been discovered by LIGO in 2016 hence setting up a milestone of General Relativity in the history of Physics. Here, the Kerr metric has been introduced in the Boyer-Lindquist forms and it is derived from the Schwarzschild metric using the Spin-Coefficient formalism. According to the “Cosmic Censorship Hypothesis”, a naked singularity cannot exist in nature as nature always hides the singularity via an event horizon. However, in this paper I will prove the existence of the “Naked Singularity" taking the advantage of the Ring Singularity of the Kerr Black Hole and thereby making the way to manipulate the mathematics by taking the larger root of Δ as zero and thereby vanishing the ergosphere and event horizon making the way for the naked ring singularity which can be easily connected via a cylindrical wormhole and as ‘a wormhole is a black hole without an event horizon’ therefore, this cylindrical connection paved the way for the Einstein-Rosen Bridge allowing particles or null rays to travel from one universe to another ending up in a future directed Cauchy horizon while changing constantly from spatial to temporal and again spatial paving the entrance to another Kerr Black hole (which would act as a white hole) in the other universes. I will not go in detail about the contradiction of ‘Chronology Protection Conjecture” [4]whether the Stress-Energy-Momentum Tensor can violate the ANEC (Average Null Energy Conditions) or not with the values of less than zero or greater than, equal to zero, instead I will focus definitely on the creation of the mathematical formulation of a wormhole from a Naked Ring Kerr Singularity of a Kerr Black Hole without any event horizon or ergosphere. Another important thing to mention in this paper is that I have taken the time to be imaginary[5] as because, a singularity being an eternal point of time can only be smoothen out if the time is imaginary rather than real which will allow the particle or null rays inside a wormhole to cross the singularity and making entrance to the other universe. The final conclusion would be to determine the mass-energy equivalence principle as spin angular momentum increases with a decrease in BH mass due to the vanishing event horizon and ergosphere thereby maintaining the equivalence via apparent and absolute masses in relation to spin J along the orthogonal Z axis. A ‘NAKED SINGULARITY’ alters every parameters of a BH and to include this parameters along with affine spin coefficient, it has been proved that without any spin angular momentum the generation of wormhole and vanishing of event horizon and singularity is not possible.

scholarly journals Superradiance initiated inside the ergoregion

2016 ◽  
Vol 28 (10) ◽  
pp. 1650025 ◽  
Author(s):  
Gregory Eskin
Keyword(s):  
Event Horizon ◽  
Finite Time ◽  
Naked Singularity ◽  
Negative Energy ◽  
Initial Energy ◽  
Kerr Metric ◽  
Large Parameter ◽  
The Cauchy Problem ◽  
Highly Oscillatory ◽  
Energy Formula

We consider the stationary metrics that have both the black hole and the ergoregion. The class of such metric contains, in particular, the Kerr metric. We study the Cauchy problem with highly oscillatory initial data supported in a neighborhood inside the ergoregion with some initial energy [Formula: see text]. We prove that when the time variable [Formula: see text] increases this solution splits into two parts: one with the negative energy [Formula: see text] ending at the event horizon in a finite time, and the second part, with the energy [Formula: see text], escaping, under some conditions, to the infinity when [Formula: see text]. Thus we get the superradiance phenomenon. In the case of the Kerr metric the superradiance phenomenon is “short-lived”, since both the solutions with positive and negative energies cross the outer event horizon in a finite time (modulo [Formula: see text]) where [Formula: see text] is a large parameter. We show that these solutions end on the singularity ring in a finite time. We study also the case of naked singularity.

An interpretation of the Kerr metric in general relativity

1965 ◽  
Vol 61 (2) ◽  
pp. 531-534 ◽  
Author(s):  
R. H. Boyer ◽  
T. G. Price
Keyword(s):  
Vacuum Solution ◽  
Preceding Paper ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Rotating Fluid ◽  
Exterior Field ◽  
Rotating Sphere ◽  
Rotating Body ◽  
Planetary Orbits ◽  
Kerr Field

The preceding paper ((1)) dealt with some general properties of the gravitational field of a rotating fluid mass. An interesting example of a vacuum solution that might be the exterior field of some rotating body was recently found by Kerr ((4)). It was natural to apply the preceding theory to the Kerr solution. This paper deals with other aspects of that solution, particularly the behaviour of its bounded geodesics (planetary orbits). It would seem desirable to know what sort of rotating body could be a source of the Kerr field. It will appear that one of the parameters in Kerr's solution can plausibly be related to the angular momentum per unit mass of a uniformly rotating sphere, the other parameter being a measure of the mass of the sphere.

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.

Sign in / Sign up

Export Citation Format

Share Document