Stationary Axially Symmetric Fields and the Kerr Metric

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.

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Exact Axially Symmetric Solution inf(T)Gravity Theory

Advances in High Energy Physics ◽

10.1155/2014/857936 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 1

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.

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SEPARATING STATIONARY AXIALLY-SYMMETRIC ASYMPTOTICALLY FLAT METRICS

International Journal of Modern Physics A ◽

10.1142/s0217751x89001163 ◽

1989 ◽

Vol 04 (12) ◽

pp. 2953-2958 ◽

Cited By ~ 1

Author(s):

Z. YA. TURAKULOV

Keyword(s):

Gordon Equation ◽

Separation Of Variables ◽

Vacuum Solution ◽

Complete Separation ◽

Kerr Metric ◽

Axially Symmetric ◽

Spherical System ◽

Asymptotically Flat ◽

Klein Gordon ◽

Klein Gordon Equation

Stationary axially-symmetric asymptotically flat metrics allowing the complete separation of variables in the Klein-Gordon equation are considered. It is shown that if such metrics coincide at infinity with the metric of spherical system of coordinates, the variables for them in the Einstein equation are completely separable and the only vacuum solution is the Kerr metric.

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Photon regions and shadows of accelerated black holes

International Journal of Modern Physics D ◽

10.1142/s0218271815420249 ◽

2015 ◽

Vol 24 (09) ◽

pp. 1542024 ◽

Cited By ~ 65

Author(s):

Arne Grenzebach ◽

Volker Perlick ◽

Claus Lämmerzahl

Keyword(s):

Black Holes ◽

Cosmological Constant ◽

Boundary Curve ◽

Kerr Metric ◽

Axially Symmetric ◽

De Sitter ◽

Acceleration Parameter ◽

Type D ◽

Analytical Formulas ◽

Photon Regions

In an earlier paper, we have analytically determined the photon regions and the shadows of black holes of the Plebański class of metrics which are also known as the Kerr–Newman–NUT–(anti-)de Sitter metrics. These metrics are characterized by six parameters: Mass, spin, electric and magnetic charges, gravitomagnetic NUT charge and the cosmological constant. Here, we extend this analysis to the Plebański–Demiański class of metrics which contains, in addition to these six parameters, the so-called acceleration parameter. All these metrics are axially symmetric and stationary type D solutions to the Einstein–Maxwell equations with a cosmological constant. We derive analytical formulas for the photon regions (i.e. for the regions that contain spherical lightlike geodesics) and for the boundary curve of the shadow as it is seen by an observer at Boyer–Lindquist coordinates (rO, ϑO) in the domain of outer communication. Whereas all relevant formulas are derived for the whole Plebański–Demiański class, we concentrate on the accelerated Kerr metric (i.e. only mass, spin and acceleration parameter are different from zero) when discussing the influence of the acceleration parameter on the photon region and on the shadow in terms of pictures. The accelerated Kerr metric is also known as the rotating C-metric. We discuss how our analytical formulas can be used for calculating the horizontal and vertical angular diameters of the shadow and we estimate these values for the black holes at the center of our Galaxy and at the center of M87.

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Distinguishing Brans–Dicke–Kerr type naked singularities and black holes with their thin disk electromagnetic radiation properties

The European Physical Journal C ◽

10.1140/epjc/s10052-020-7736-x ◽

2020 ◽

Vol 80 (2) ◽

Cited By ~ 4

Author(s):

Shahab Shahidi ◽

Tiberiu Harko ◽

Zoltán Kovács

Keyword(s):

Black Holes ◽

Comparative Investigation ◽

Compact Objects ◽

Electromagnetic Properties ◽

Kerr Metric ◽

Massless Scalar ◽

Massless Scalar Field ◽

Axially Symmetric ◽

Field Equations ◽

Naked Singularities

Abstract The possible existence of naked singularities, hypothetical astrophysical objects, characterized by a gravitational singularity without an event horizon is still an open problem in present day astrophysics. From an observational point of view distinguishing between astrophysical black holes and naked singularities also represents a major challenge. One possible way of differentiating naked singularities from black holes is through the comparative study of thin accretion disks properties around these different types of compact objects. In the present paper we continue the comparative investigation of accretion disk properties around axially-symmetric rotating geometries in Brans–Dicke theory in the presence of a massless scalar field. The solution of the field equations contains the Kerr metric as a particular case, and, depending on the numerical values of the model parameter $$\gamma $$γ, has also solutions corresponding to non-trivial black holes and naked singularities, respectively. Due to the differences in the exterior geometries between black holes and Brans–Dicke–Kerr naked singularities, the thermodynamic and electromagnetic properties of the disks (energy flux, temperature distribution and equilibrium radiation spectrum) are different for these two classes of compact objects, consequently giving clear observational signatures that could discriminate between black holes and naked singularities.

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Interaction of Scalar, Spinorial, Vectorial, Spin-Vectorial and Tensorial Particles with the Axially-Symmetric Gravitational Field Described by the Kerr Metric

Free and Interacting Quantum Fields ◽

10.1142/9789813145474_0008 ◽

2017 ◽

pp. 481-553

Keyword(s):

Gravitational Field ◽

Kerr Metric ◽

Axially Symmetric ◽

Symmetric Gravitational Field

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Kerr metric Killing bundles

The European Physical Journal C ◽

10.1140/epjc/s10052-021-08986-0 ◽

2021 ◽

Vol 81 (3) ◽

Author(s):

D. Pugliese ◽

H. Quevedo

Keyword(s):

Black Holes ◽

Rotation Axis ◽

Outer Region ◽

Kerr Metric ◽

Axially Symmetric ◽

Kerr Geometry ◽

Inner Horizon ◽

Killing Horizon ◽

Extended Plane ◽

Orbital Frequency

AbstractWe provide a complete characterization of the metric Killing bundles (or metric bundles) of the Kerr geometry. Metric bundles can be generally defined for axially symmetric spacetimes with Killing horizons and, for the case of Kerr geometries, are sets of black holes (BHs) or black holes and naked singularities (NSs) geometries. Each metric of a bundle has an equal limiting photon (orbital) frequency, which defines the bundle and coincides with the frequency of a Killing horizon in the extended plane. In this plane each bundle is represented as a curve tangent to the curve that represents the horizons, which thus emerge as the envelope surfaces of the metric bundles. We show that the horizons frequency can be used to establish a connection between BHs and NSs, providing an alternative representation of such spacetimes in the extended plane and an alternative definition of the BH horizons. We introduce the concept of inner horizon confinement and horizons replicas and study the possibility of detecting their frequencies. We study the bundle characteristic frequencies constraining the inner horizon confinement in the outer region of the plane i.e. the possibility of detect frequency related to the inner horizon, and the horizons replicas, structures which may be detectable for example from the emission spectra of BHs spacetimes. With the replicas we prove the existence of photon orbits with equal orbital frequency of the horizons. It is shown that such observations can be performed close to the rotation axis of the Kerr geometry, depending on the BH spin. We argue that these results could be used to further investigate black holes and their thermodynamic properties.

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Solar Internal Rotation and Dynamo Waves: A Two-Dimensional Asymptotic Solution in the Convection Zone

International Astronomical Union Colloquium ◽

10.1017/s0252921100064861 ◽

2000 ◽

Vol 179 ◽

pp. 379-380

Author(s):

Gaetano Belvedere ◽

Kirill Kuzanyan ◽

Dmitry Sokoloff

Keyword(s):

Magnetic Field ◽

Asymptotic Solution ◽

Internal Rotation ◽

Convection Zone ◽

General Idea ◽

Dynamo Model ◽

Axially Symmetric ◽

Dynamo Action ◽

Regeneration Rate ◽

Dynamo Waves

Extended abstractHere we outline how asymptotic models may contribute to the investigation of mean field dynamos applied to the solar convective zone. We calculate here a spatial 2-D structure of the mean magnetic field, adopting real profiles of the solar internal rotation (the Ω-effect) and an extended prescription of the turbulent α-effect. In our model assumptions we do not prescribe any meridional flow that might seriously affect the resulting generated magnetic fields. We do not assume apriori any region or layer as a preferred site for the dynamo action (such as the overshoot zone), but the location of the α- and Ω-effects results in the propagation of dynamo waves deep in the convection zone. We consider an axially symmetric magnetic field dynamo model in a differentially rotating spherical shell. The main assumption, when using asymptotic WKB methods, is that the absolute value of the dynamo number (regeneration rate) |D| is large, i.e., the spatial scale of the solution is small. Following the general idea of an asymptotic solution for dynamo waves (e.g., Kuzanyan & Sokoloff 1995), we search for a solution in the form of a power series with respect to the small parameter |D|–1/3(short wavelength scale). This solution is of the order of magnitude of exp(i|D|1/3S), where S is a scalar function of position.

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