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# The Kerr Metric of Spinning Black Holes

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This is a short article providing an overview of the Kerr Metric of spinning black holes which I did in the third year of my course in the University of Nottingham.
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The Kerr Metric of Spinning Black Holes
by Ivo Plamenov Petkov
The physics of black holes has been a compelling field of scientific study in the
twentieth century beginning with Einstein's formulation of the theory of general relativity
which predicted that a sufficiently large star will deform spacetime after its collapse to form
a black hole. This article provides a brief overview of spinning black holes and the metric
which is used to describe their properties which is attributed to Roy Kerr, a New Zealand
mathematician. Kerr described the geometry of spacetime around an uncharged black
hole with a spherical event horizon by solving Einstein's field equations of general
relativity, thus providing a generalisation of the Schwarzshild metric, which describes the
gravitational field outside a spherical mass [3,6].
The present article does not aim to explain the Kerr metric and the mathematics
which is used in detail but to introduce four new features which do not appear in the
abovementioned Schwarzshild metric. A spinning black hole is solely described by only
two quantities, its mass M and the respective angular momentum J, according to the no-
hair theorem [2]. A simple case of study presents spacetime and particle motion in the
equatorial plane of a black hole, i.e., the plane through the centre of the black hole which
is perpendicular to the spin axis. Equation {1} gives the Kerr metric expressed in Boyer-
Lindquist coordinates [7], where a = JM-1 is the so-called Kerr parameter, which appears so
often that it has been given its own name [4].
The Kerr metric introduces a new r-value for the horizon, which depends on the
value of the Kerr parameter. In comparison, dr2 in {1} increases without a limit for the
Schwarzshild metric. The presence of the product dtdφ of two spacetime coordinates,
called the cross product, is another new feature introduced by Kerr. A direct implication
which can be made from this is that these coordinates are closely related. This creates a
frame dragging [5] effect which is nothing else but spacetime around a spinning black hole
being on the move itself!
The third new feature of the Kerr metric is the static limit [4,5]. A spinning black hole is
characterised by several structural elements, as seen in Figure 1. While particles in the
spherical event horizon obey the same rules as within any black hole, i.e., even light
cannot escape after crossing that boundary, and the ergosphere is a region where space
rotates so fast that it is impossible for a body to hover in such a way to appear stationary
to a distant observer, the static limit is the limit where a particle moving against the flow of
space, defined by the spin axis, would appear static to a distant observer. The horizon of a
spinning black hole lies at an r-value which is less than 2M, or in other words, where the
metric coefficient of dr2 blows up. Comparing this to the equatorial plane, the coefficient of
dt2 goes to zero at r = 2M, just as it behaves in the Schwarzschild metric for a static black
hole. The r-value in the equatorial plane at which the coefficient of the dt2 term goes to
zero represents the static limit. Examining equation {1} verifies that the expression for the
static limit in the equatorial plane is the same irregardless of the value of the Kerr
parameter a.
Figure 1. The structure of a spininng black hole, with its event horizon, beyond which time slows
down and not even light can escape the black hole, the ergosphere ("ergon", from Greek, which means
work) where no object will appear stationary to an observer due to the spin of the black hole, and the static
limit, which is the third new feature in the Kerr metric.
available energy, which is implied if angular momentum exists. While no net energy can be
extracted from a static black hole in general, an energy of rotation is present in the case of
a spinning one.
The spinning black hole is a remarkable stellar object which exhibits multiple
properties which vary from the ones of a static black hole. The Kerr metric introduces new
features that describe a black hole's properties and distinguishes them from the
Schwarzshild sloutions. While the new metric is only applicable when a neutrally charged,
spinning black hole is under investigation, it provides a deeper understanding of a group of
celestial objects which have been of particular interest to the astrophysicists for more than
a century. The presence of angular momentum has many implications, one of which being
the modification of the spacetime fabric in such a way that even it is no longer stationary,
thus allowing matter to orbit this class of black holes at a closer distance. This is also
verified by the intimate relationship of the spacetime coordinates and their cross product,
which justifies the frame dragging effect. The existence of the static limit shows that the
metric behaves similarly to the Schwarzshild metric and set a relationship in the Kerr's
solution which is independent of the Kerr parameter. Finally, the presence of rotaional
energy is a feature completely foreign to a static black hole, which poses a multitude of
questions and paves the way of further investigation of the Universe and one of its most
mysterious objects.
References:
1. Chambers, C.M. , Hiscock, W.A. , Taylor, B.E., "The "Ups" and "Downs" of a Spinning Black Hole",
Department of Physics, Montana State University, Bozeman, MT 59717-3840, USA.
2. "Chapter 3, The Kerr Solution", http://www.roma1.infn.it/teongrav/leonardo/bh/bhcap3.pdf
3. Kerr, R. P., “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,”
Physical Review Letters, Volume 11, pages 237–238 (1963).
4. Perepelitsa, D. V., "Spinning Black Hole Energetics", MIT Department of Physics (2007)
5. Taylor, E. F., "Project F The Spinning Black Hole", http://www.eftaylor.com/pub/SpinNEW.pdf
6. van der Wijk, P. C., "The Kerr-Metric: describing Rotating Black Holes and Geodesics", 1469037,
Rijksuniversiteit Groningen (2007).
7. Visser, M., "The Kerr spacetime: A brief introduction", School of Mathematics, Statistics, and
Computer Science Victoria University of Wellington (2008)

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Algebraically special solutions of Einstein's empty-space field ; equations that are characterized by the existence of a geodesic and shear-free ; ray congruence are considered. A class of solutions is presented for which the ; congruence is diverging and is not necessarily hypersurface orthogonal. (C.E.S.);
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Comment: This is a draft of an introductory chapter on the Kerr spacetime that is intended for use in the book “The Kerr spacetime”, currently being edited by Susan Scott, Matt Visser, and David Wiltshire. This chapter is intended as a teaser and brief introduction to the mathematics and physics of the Kerr spacetime — it is not, nor is it intended to be, a complete and exhaustive survey of everything in the field. Comments and community feedback, especially regarding clarity and pedagogy, are welcome.
Spinning Black Hole Energetics
• D V Perepelitsa
Perepelitsa, D. V., "Spinning Black Hole Energetics", MIT Department of Physics (2007)
Project F The Spinning Black HoleThe Kerr-Metric: describing Rotating Black Holes and Geodesics
• E F Taylor
• P C Wijk
Taylor, E. F., "Project F The Spinning Black Hole", http://www.eftaylor.com/pub/SpinNEW.pdf 6. van der Wijk, P. C., "The Kerr-Metric: describing Rotating Black Holes and Geodesics", 1469037, Rijksuniversiteit Groningen (2007).
Project F The Spinning Black Hole
• E F Taylor
• P C Van Der Wijk
Taylor, E. F., "Project F The Spinning Black Hole", http://www.eftaylor.com/pub/SpinNEW.pdf 6. van der Wijk, P. C., "The Kerr-Metric: describing Rotating Black Holes and Geodesics", 1469037, Rijksuniversiteit Groningen (2007).