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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.

The final feature of the Kerr metric which is worth mentioning in this article is the

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)