Energy Extraction of Kerr Black Holes

T. Zheng

Department of Physics,

University of British Columbia

(Dated: December 5, 2016)

## **ABSTRACT**

This paper covers two theoretical mechanisms that seeks to explain the possibility of extracting energy directly from Kerr black holes. This paper begins with a self-contained introduction to black holes, followed by a discussion on the space-time features of Kerr black holes. The unique space-time features of Kerr black holes will then be used to explain the two mechanisms for energy extraction: the Penrose process, and the Blandford-Znajek process.

## **I. INTRODUCTION**

The study of rotating astrophysical objects is an active area of research in general relativity. Rotating black holes (also known as Kerr black holes) provide much insight into the space-time physics of strong, rotating gravitational fields. The space-time geometry and gravitational effects lead to two theoretical explanations for the energy extraction of Kerr black holes: the Penrose process, and the Blandford-Znajek process. The Penrose process is a mechanism that is capable of extracting energy from the rotational energy of a Kerr black hole via the exchange of energy between a Kerr black hole and in-falling matter. The Blandford-Znajek process relies on the existence of surrounding bulk matter to accumulate charge via the rotation of the Kerr black hole. In recent literature, the theoretical results of the Blandford-Znajek process has demonstrated great potential for explaining the generation of relativistic jets in the observable universe.

## **II. BLACK HOLES: THE BASICS**

A black hole is a region of space-time formed by a collapsed mass, such that the escape velocity of the collapsed mass exceeds the speed of light. This is only possible if the radius of the black hole is less than its Schwarzschild radius (Kutner, 2003)

R
s

{\displaystyle R\_s}

:

R
S

=

2
G
M

c
2

{\displaystyle R\_S = \frac{2GM}{c^2}}

Since the speed of light is the cosmic speed limit of causality, there is a region of the black hole where events do not affect outside observers. This particular region of spacetime is called the event horizon.

The composition of black holes is uncertain, and it should remain so since it is impossible for observers to retrieve information from the event horizon of a black hole. To an outside observer, a hypothetical probe captured by the gravitational pull of a black hole would fall towards the black hole with slowing speed. This is caused the by the ever-growing protraction of time under extreme gravity, or gravitational time-dilation. When the hypothetical probe crosses the event horizon, it would appear to be hovering in stasis at the boundary of the event horizon. To the outside observer, the probe will remain to be seen stationary forever. This can be explained by the fact that all events within the event horizon are causally disconnected with events outside the event horizon. The hypothetical probe has indeed moved past the event horizon; however, all subsequent events that the probe experiences beyond the event horizon cannot be observed by any outside observer. All information beyond the event horizon is therefore privately safeguarded by the black hole.

Nevertheless, black holes are not entirely opaque. It is known that black holes can be characterized by three observables: mass, charge, and angular momentum (Bekenstein, 2004). Mass is the most intrinsic physical observable of black holes. All black holes possess the mass of the original collapsed object. However, a black hole would not possess charge or angular momentum unless the original object (such as a collapsing star) contained excess charge, or is rotating. Black holes are rather lacklustre in constitution, and this is even given a funny name: the no-hair principle. However, due to the extreme distortion of space-time, the behaviour of black holes is considerably exotic compared to other massive astrophysical objects.

Table I exhibits the four types of black holes with their proper names. This paper will focus entirely on Kerr black holes (rotating black holes with not charge).

## **III. SPACE-TIME FEATURES OF KERR BLACK HOLES**

In 1963, Roy Kerr solved the Einstein equations for a black hole with the presence of angular momentum. Kerr’s solution for a rotating black hole is called the Kerr metric (Visser, 2008):

where

# a
J

/

m

{\displaystyle a = J/m }

 (the angular momentum to mass ratio),

ρ
2

=

r
2

a
2

cos
2

⁡

(
θ
)

{\displaystyle \rho^2 = r^2 + a^2\cos^2{(\theta)} }

, and

# Δ

r
2

−
2
M
r

a
2

{\displaystyle \Delta = r^2 -2Mr + a^2 }

.

Since general relativity describes gravity using non-Euclidean geometry, it comes to no surprise that the unique gravitational effects of Kerr black holes arise from severe space-time curvature. Under Newtonian gravity, one would have thought only mass affects the gravitational field. This is only partially true in the framework of general relativity. One consequence of general relativity states that mass and energy can distort spacetime. Consequently, the rotation of a black hole affects its space-time as well.

Unique to massive rotating objects (including rotating black holes) is a phenomenon called frame dragging. Frame dragging is an interesting behaviour of rotating objects with strong gravitational fields. For a rotating black hole, the effects are so striking that it drags objects around a region called the ergosphere. This action is different from the usual trajectory of objects falling into a static black hole. The ergosphere is a region of space-time surrounding a Kerr black hole that possesses negative energy, and it is situated outside the event horizon. Since the ergosphere is outside the event horizon, it is possible to counter the frame dragging trajectory and escape the black hole. The rotation in which formed the ergosphere essentially stalls an entering object from its impending demise.

## **IV. ENERGY EXTRACTION OF KERR BLACK HOLES**

Black holes can act as a catalyst for extracting the rest mass energy of a particle; however, this energy does not come from the black hole itself. If a black hole has angular momentum (i.e. is rotating), there are classical processes in which energy can be extracted from the black hole itself. In this paper, we will focus on the energy extraction mechanisms of Kerr black holes. The first mechanism is called the Penrose process. The second mechanism is called the Blandford-Znajek process.

## **V. PENROSE PROCESS**

In 1969, Roger Penrose proposed a method in which energy is extracted from the Kerr black hole from its rotational energy. This method is called the Penrose process. The process begins with matter being gravitated towards a Kerr black hole. The matter is then dragged into the rotating ergosphere (the aforementioned region of negative energy). Once the matter enters the ergosphere, it will separate into two particles. One particle will escape to infinity, while the other particle will eventually fall past the event horizon. The matter that is captured by the black hole will possess negative energy. An outside observer would notice ejected particles carrying positive energy that is greater than the energy of the matter that initially entered the ergosphere. Consequently, the captured particle will decrease the angular momentum of the Kerr black hole.

The inward falling particle possesses negative energy in the ergosphere. One consequence of the Penrose process suggests that the Kerr black hole will experience a reduction of angular momentum; hence, a reduction of its rotation. This reduction in the rotation is converted into the extracted energy that is carried by the ejected matter. As the Penrose process continues, the gradual reduction of angular momentum will eventually convert a rotating black hole into a static black hole (Schwarzschild black hole). In 1983, Subrahmanyan Chandrasekhar calculated that the maximum efficiency of the Penrose process is roughly 20.7 percent (Perepelitsa, 2007). This efficiency is rather disappointing to astronomers, as it is insufficient for powering the plasma emissions emanating out of Kerr black holes.

## **VI. BLANDFORD-ZNAJEK PROCESS**

Although the Penrose process is capable of extracting energy from a Kerr black hole, it is incapable of explaining certain astronomical observations surrounding Kerr black holes. The most notable of astronomical observations surrounding Kerr black holes are relativistic jets. Relativistic jets are bursts of plasma propelled at speeds near the speed of light. These jets are a stunning feature of quasars, stellar black holes, and pulsars. The Penrose process is too inefficient to power such powerful emissions of plasma. It is more realistic to find matter accumulating around a Kerr black hole (Komissarov, 2014). This bulk of matter around a Kerr black hole is called the accretion disk (see Fig. 4). The appearance of accretion disks are commonly found around large rotating masses. A good example would include super-massive black holes that are typically seated at the heart of galaxies.

The density of matter in the accretion disk rises sharply closer to the black hole. Kerr black holes are, by definition, electrically neutral; however, the surrounding matter of the accretion disk becomes magnetized over time. This magnetization generates strong magnetic fields. The strong magnetic fields are then dragged into the ergosphere. Eventually, the black hole will swallow the accretion disk, threading the magnetic fields carried by the bulk matter through the black hole. The rotating magnetic fields then induce strong electromagnetic forces that accelerate charged plasma at relativistic speeds along the axis of rotation. If Blandford and Znajek are correct, then Kerr black holes may explain how many relativistic jets are formed and powered. Physicists and astronomers now believe that the Blandford-Znajek process is responsible for not only the relativistic jets observed from rotating black holes.

## **VII. CONCLUSION**

The rotation of Kerr black holes has provided many avenues of studying astrophysics. From a theoretical standpoint, Kerr black holes present a wealth of interesting physical phenomenon. In section III, we introduced the space-time features of Kerr black holes (most notably the ergosphere and frame dragging). From the ergosphere and the frame dragging dynamics, two theoretical mechanisms of energy extraction have been discussed since the 1960s. The Penrose process is capable of extracting energy directly from the rotation of the Kerr black hole, despite its lower efficiency. The Blandford-Znajek process is much more efficient, and is a good theoretical candidate for explaining the production of relativistic jets.

## **VIII. ACKNOWLEDGEMENTS**

I would like to express special thanks to Dr. Janis McKenna for teaching PHYS 348. I learned a lot about the importance of communication in science, in particular physics, and the importance of giving presentations with confidence and clarity. I would like to thank my peers for their efforts disseminating a wide array of challenging topics in physics and astronomy. I learned a lot about physics as a discipline, and its potential for spurring innovation.

## **IX. REFERENCES**

References and Further Reading

Perepelitsa, D., 2007, Spinning Black Hole Energetics. MIT Department of Physics.

Komissarov, Sergei., 2014, Blandford-Znajek Mechanism: Event Horizon or Ergoregion?. Guest Lecture. Perimeter Institute for Theoretical Physics.

Bekenstein, J. D., 2004, Black holes and information theory. CONTEMPORARY PHYSICS. Volume: 45, Issue: 1, Pages: 31-43.

Jacobson, T., 1996, Introductory Lectures on Black Hole Thermodynamics. Institute of Theoretical Physics University of Utrecht.

Visser, M., 2008, The Kerr spacetime: A brief introduction. Victoria University of Wellington.

Kutner, M., 2003, Astronomy: A Physical Perspective. Second Ed. Cambridge University Press