A Brief Introduction

30 August 2021

- Abstract

Black holes are regions in spacetime that have immense gravity, such that light cannot escape. There are two different types of black holes that will be discussed in this paper: Kerr and Schwarzschild black holes. Schwarzschild black holes have zero angular momentum (non-rotating) whereas Kerr black holes have angular momentum. In this paper, the concepts of metric, event horizon and ergosphere radii, spacetime intervals, singularities, and frame-dragging to compare and contrast the two different black holes are explored.

^{[1]}

- Introduction

In 1915, Albert Einstein introduced his theory of general relativity to the scientific world. Einstein’s publication presented a theory of how gravity influences spacetime using a model that combines the three dimensions of space and the fourth dimension of time. Using this idea, scientists and mathematicians have derived multiple equations and found a region of spacetime called a black hole. A black hole has gravity so immense that within a certain distance light cannot escape.

^{[2]}

When considering black holes, spacetime is a major component. It can be expressed mathematically with a metric tensor * _{μν}*:

In the equation, the metric tensor describes how spacetime changes. In addition,

*and*

^{μ}*are spacetime coordinates, and*

^{ν}*and*

^{μ}*are small changes in those coordinates. These variables tell us where the object is located in spacetime. The spacetime interval*

^{ν}*is the distance in spacetime between two events.*

^{2}Note that

*defines the volume of an object and is symmetrical:*

_{μν}When the spacetime interval is written out for a particular set of coordinates, it looks as follows:

The tensor being symmetrical is important. It allows the spacetime equation expressed above to be simplified. Simplify the terms with the same coordinates by adding them as shown below.

^{[3] [4]}

Other important aspects of a black hole are the two boundaries: the event horizon and ergosphere. A black hole can have an outer and inner radius for both boundaries. Inside the outer event horizon’s radius, light cannot escape the black hole. The outer ergosphere is the outer bound of a region that makes it possible to escape the black hole. The inner event horizon and ergosphere are some concepts that would be further explained later in the paper.

^{[5] [6][17]}

The Schwarzschild radius

*is also an important parameter of the black hole. It is the radius of the event horizon for a non-rotating, uncharged black hole. It is defined as , where is the Newtonian constant of gravitation (6.67×10*

_{s}^{-11}m

^{3}/(kg s

^{2})), is the mass of the black hole, and is the speed of light in a vacuum (2.99×10

^{8}m/s).

^{[7] [8]}

- Similarities and Differences Between Black Holes

The two uncharged black holes’ metrics that are solutions to Einstein’s field equations from his theory of general relativity are that of the Schwarzschild and Kerr black holes. In 1915, Karl Schwarzschild discovered a non-rotating, uncharged black hole. It was not until 1963 that Roy Kerr discovered a rotating, uncharged black hole. Each black hole would be named after the discoverer.

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A key difference between the two black holes is that the Schwarzschild black hole has no angular momentum, whereas the Kerr black hole has a nonzero angular momentum. This difference is shown by the spin parameter or angular momentum . There is no angular momentum when or Physicists believe the maximum angular momentum can be observed when . The effects on the black hole when will be discussed later.

^{[9]}

Angular momentum creates an effect on spacetime around the black hole. In other words, when the black hole rotates, it drags spacetime along with it. This phenomenon which occurs outside the outer event horizon is called frame-dragging.

^{[10] [17]}

- Metrics

The Kerr black hole can be summarised in a metric form. Below, the Kerr metric describes spacetime with mass and angular momentum .

Defined variables that are useful to understand the Kerr metric are:

^{[11]}

The coordinates in the equation are time , and the three-dimensional spatial coordinates: * , , .* The zenith angle is the angle measured from the z-axis and the azimuthal angle is the angle from the x-axis in the xy plane.

^{[9]}

Converting from these three-dimensional coordinates to Cartesian coordinates: , , is done using,

^{[9]}

The Kerr metric can be simplified to one that fits a Schwarzschild black hole — the Schwarzschild metric. When looking at the metric for the non-rotating black hole, certain parts of the Kerr metric vanish or change value such as the term and due to .

The Schwarzschild metric describes spacetime with mass . The metric can be written as

^{[12]}

- Event Horizons

The event horizon can be found when looking at the Kerr metric. When the radial component of the metric * ^{2}* term is undefined,

or ,

Where ,

Solving for yields the event horizon radius. It is the following:

Notice that due to the plus-minus sign, there can be up to two event horizons.

^{[4]}

In the graph below, the event horizon radius

*r*as a function of spin parameter is shown. Say that mass is 10 solar masses (1 solar mass is approximately 2 x 10

^{30}kg). This means that

*is 8.928 x 10*

_{s}^{12}meters.

^{[13]}

**Graph 1: The outer and inner event horizon as a function of the spin parameter. When the spin parameter is zero, the outer event horizon is equal to the Schwarzschild radius while the inner event horizon is zero (this is a point singularity). As the spin parameter goes from zero to half of the Schwarzschild radius, the outer event horizon and inner event horizon touch or are at the same point in spacetime. Technically there is only one event horizon at this special spin parameter.**

- Ergospheres

The ergosphere is defined by an outer and inner ergosphere. In some black holes, there is a region bounded by the outer ergosphere (outer bound) and the outer event horizon (inner bound) in which it is possible to escape with more energy and mass due to frame dragging. In addition, the inner ergosphere is supposedly unstable and typically has a radius less than the inner event horizon.

^{[5][17]}

When looking at the Kerr metric, the ergosphere radii are found when the * ^{2}* term is zero,

or .

Using these equations, solve for the radius of the ergosphere .

Like the event horizon, there can be two ergospheres. The outer ergosphere is a boundary outside the outer event horizon in which an object can steal energy from the black hole and escape. The inner ergosphere is at a radius less than or equal to the inner event horizon depending on the zenith angle .

^{[4]}

In the graph below, the ergosphere radius as a function of spin parameter is shown. As with the event horizon graph, the values are calculated given that is 10 solar masses and zenith angle . The Schwarzschild radius is 8.928 x 10

^{12}meters.

^{[13]}

**Graph 2: Outer and inner ergosphere as a function of spin parameter at . When the spin parameter is zero, the outer ergosphere is equal to the Schwarzschild radius while the inner ergosphere is zero. As the spin parameter goes from zero to half of the Schwarzschild radius, the outer and inner ergospheres touch.**

Note that when , the radius of the ergosphere is the radius of the event horizon because . This means that at the poles of a rotating black hole (along the z-axis), the ergosphere and event horizon touch.

In the graph below, the ergosphere radius as a function of the zenith angle is shown. It is useful to note that and the mass is 10 solar masses.

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**Graph 3: The outer and inner ergosphere as a function of the zenith angle. As the zenith angle goes from 0 (on the z-axis) to 90 degrees (on the xy plane), the outer event horizon and inner ergosphere go from the same radius to the Schwarzschild radius and a radius of zero, respectively.**

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- Spacetime Intervals

With the Kerr and Schwarzschild metrics, the sign of * ^{2}* is important. When

*is positive ( ), it behaves as a time-like interval, which means an object must travel faster than the speed of light to get from one event to the other. When it is negative ( ), it behaves as a space-like interval, and an object can travel slower than the speed of light to get from one event to the other. When it is zero ( ), it behaves as a light-like interval because something traveling at the speed of light will reach the other event.*

^{2}^{[9]}

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In the graphs below, the sign of the areas around the event horizon radius are shown. In figure 1, the spin parameter is zero which means that it is a Schwarzschild black hole.

**Figure 1: The regions of spacetime for a Schwarzschild black hole are graphed. The area between the inner and outer event horizon yields a .**

*Whereas, the area outside the outer event horizon has a .*In figure 2, the spin parameter is .

**Figure 2: The regions of spacetime for a Kerr black hole are graphed.**

*The area inside the inner event horizon has a .*The area between the inner and outer event horizon yields a .*Whereas, the area outside the outer event horizon has a*And in figure 3, the spin parameter is .

**Figure 3: The regions of spacetime for a Kerr black hole are graphed.**

*T*he inner and outer event horizon radii are equal. The*area inside and outside the event horizons have a .*^{[9]}

When , there is a naked singularity which means that the black hole does not have an event horizon. This is due to an imaginary number being in the square root of the event horizon equation.

Note that when looking at Figures 1-3, the sign of

*, more specifically determines the behavior of the radial coordinate . This is due to:*

_{rr}The defined variables used are:

and

Since only can be negative, use

When the coefficient of the coordinate differential squared is negative, the coordinate behaves like time so it can only move in one direction which is inwards due to the object having to travel faster than light to escape. If it is positive, it behaves like a spatial coordinate so it can move in whatever direction due to it being able to travel slower than light and escape.

Therefore, when the region of spacetime has a , it means that something moving at or around light speed would not be able to escape the black hole and would go inwards. When the region of spacetime has a , it means it would be able to escape the black hole.

^{[9]}

The outer and inner event horizon are both light-like intervals due to . But crossing over the outer event horizon normally means that an object cannot escape due to the area inside the outer event horizon having (the exception is when ). For the inner horizon, it is not fully known what this will be like to cross, but it has been predicted that the inner horizon is inherently unstable.

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- Singularities

For Kerr black holes, the singularity is a ring as opposed to a point with the Schwarzschild black hole. It can be calculated when .

Note:

The radius of the ring singularity is equal to .

With the Schwarzschild black hole, its singularity can be found at

^{[9] [4]}

- Rotation and Frame Dragging

The rotational or angular speed of a Kerr black hole is given by

Where _{tφ }and _{φφ} are defined as

and

This equation becomes

With the Schwarzschild black hole, when angular momentum , the term in Kerr metric equals zero ( ). Therefore, there is no frame-dragging in this type of black hole.

The rotating mass of the black hole causes frame-dragging.

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- Conclusion

In this review paper, the two uncharged black holes were compared and contrasted. It can be argued that the major difference between the Kerr and Schwarzschild black holes is in how they are defined — a Kerr black hole rotates with an angular momentum which leads to them having different metrics, event horizon and ergosphere radii, and singularities. However the Kerr black hole is a more generalized version of Schwarzschild black hole and due to , the simplification of the event horizon, ergosphere, singularity, and the Kerr to the Schwarzschild metric is not merely random. The Kerr and Schwarzschild black holes are not so different.

Although the review paper is on the Kerr and Schwarzschild black holes, physicists and mathematicians have discovered four in total. A direction for further exploration may entail comparing the Reissner-Nordström and Kerr-Newman black holes with the Schwarzschild and Kerr black holes. Due to the nonzero charge of the Reissner-Nordström and Kerr-Newman black holes, it would be interesting to analyse the event horizons, ergospheres, and metrics as the spin parameter and other factors change when a charge is added.

# References

[1] Visser, Matt. “The Kerr Spacetime: A Brief Introduction.” *Cornell University* General Relativity and Quantum Cosmology (2007).

[2] “History Topics.” Mathematical Physics Index. School of Mathematics and Statistics University of St Andrews, Scotland, June 2002. https://web.archive.org/web/20150204231934/http://www-history.mcs.st-and.ac.uk/Indexes/Math_Physics.html.

[3] Brown, Robert G. The Metric Tensor, December 28, 2007. https://webhome.phy.duke.edu/~rgb/Class/phy319/phy319/node131.html.

[4] Herman, Dr. Russell L. “Notes on the Kerr Metric.” *PHY 490 The Physics of Black Holes*. Lecture presented at the PHY 490 The Physics of Black Holes, April 6, 2021.

[5] Griest, Kim. “Physics 161: Black Holes: Lecture 22: 26 February 2010.” Lecture, February 26, 2010.

[6] Curiel, Erik. “The Many Definitions of a Black Hole.” *Nature Astronomy* 3, no. 1 (2019): 27–34. https://doi.org/10.1038/s41550-018-0602-1.

[7] Pacucci, Fabio. “Black Hole Calculator (Formulas), Fabio Pacucci – Harvard University & Sao.” Black Hole Calculator: Formulas, July 20, 2019. https://www.fabiopacucci.com/resources/black-hole-calculator/formulas-black-hole-calculator/.

[8] “The NIST Reference on Constants, Units, and Uncertainty.” Fundamental physical constants from NIST, October 1994. https://physics.nist.gov/cuu/Constants/.

[9] Flournoy, Alex. *General Relativity Topic 24: Rotating Black Holes*. *YouTube*, 2019. https://www.youtube.com/watch?v=IWLtlQru3Mo.

[10] Iorio, Lorenzo, and Alberto Morea. “The Impact of the New Earth Gravity Models on the Measurement of the Lense-Thirring Effect.” *SpringerLink*, 2004, 1321–33. https://doi.org/https://doi.org/10.1023/B:GERG.0000022390.05674.99.

[11] Weber, Kyhl. “Kerr Geometry and Rotating Black Holes.” *Physics391*. Lecture, December 13, 2018.

[12] Siopsis, George. “Lecture 2: Schwarzschild Black Hole.” Lecture, 2010.

[13] Woo, Marcus. “What Is Solar Mass?” Space.com. Space, December 6, 2018. https://www.space.com/42649-solar-mass.html.

[14] Weisstein, Eric W. “Spherical Coordinates.” MathWorld-A Wolfram Web Resource. Accessed September 9, 2021. https://mathworld.wolfram.com/SphericalCoordinates.html.

[15] Hershey, Joshua. “Spacelike, Lightlike, and Timelike Intervals.” Relativity, March 10, 2021. http://www.faithfulscience.com/relativity/spacetime-intervals.html.

[16] Dafermos, Mihalis. “Stability and Instability of the CAUCHY Horizon for The Spherically Symmetric Einstein–Maxwell-Scalar Field Equations.” *Annals of Mathematics* 158, no. 3 (2003): 875–928. https://doi.org/10.4007/annals.2003.158.875.

[17] Hirata, Christopher. “Lecture XIX: Angular Momentum and Rotating Black Holes.” *Physics 6820*. Lecture, 2019.

[18] Brady, Patrick R., Ian G. Moss, and Robert C. Myers. “Cosmic Censorship: As Strong as Ever.” *Physical Review Letters* 80, no. 16 (1998): 3432–35. https://doi.org/10.1103/physrevlett.80.3432.

- Appendix: Detailed Calculations

## Maximum Angular Momentum

To find the greatest possible angular momentum, use the equation for the event horizon radius.

Inside the square root, there is . When this is equal to zero, .

As discussed earlier, when , there is a nonrotating Schwarzschild black hole. The possible Kerr black holes exist when . When , something called a naked singularity happens. This means that the event horizon is no longer there. The Cosmic Censorship Hypothesis, the idea that the black hole’s singularity is unobservable by an outside observer, disproved the idea of a naked singularity.

^{[9] [18]}

## Simplifying the Kerr Metric to the Schwarzschild Metric

## The Kerr metric simplifies to the Schwarzschild metric in the case where the spin parameter is zero (). These are the steps to derive the Schwarzschild metric from the Kerr metric:

First, write out the original form of the Kerr metric,

Next, set equal to zero. The term cancels due to the in the numerator. Plug in * ^{2}* and which simplify to

*and*

^{2}_{s}) respectively. Then, simplify the

*term by canceling a from the numerator and denominator.*

^{2}Next, simplify the

*by canceling a from the numerator and denominator.*

^{2 }Finally, rewrite the

*term as an inverse.*

^{ 2 }^{[11] [12]}

## Calculating the Event Horizon Radius

From the equations and , the radius of the inner and outer event horizons are found by solving for radius .

Set equal to zero.

Next, subtract the * ^{2}* from both sides of the equation.

After, complete the square by adding to both sides,

Then take the square root of both sides

Now add to both sides to isolate

Next, combine the terms under the square root to have the same denominator, and simplify to a factor of two outside the square root

^{[4]}

## Calculating the Ergosphere Radius

From the equation , the radius of the inner and outer ergosphere are found by solving for radius .

First, set * _{tt }*equal to zero. Note:

Then cancel out the negative sign and the

*. Also plug in*

^{2}*. Note: .*

^{2}Then subtract from both sides of the equation.

Multiply to both sides of the equation.

Next, subtract from both sides of the equation.

After, complete the square by adding to both sides,

Next take the square root of both sides

Now add to both sides to isolate . Also make the terms in the square root have a common denominator.

Finally, simplify the terms which results in a denominator of two.

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## Calculating the Ring and Point Singularities

The ring singularity for the Kerr black hole can be calculated using oblate spheroidal coordinates.

^{[4]}

Solving for yields

By plugging in into the oblate spheroidal coordinates, find that

To find the radius of the singularity , use the equation

Then plug in the values for x and y.

Next, distribute the square and factor,

The is equal to 1 and,

At , the term reduces to

For any azimuthal angle ,

or .

This means the singularity is a circle in the xy plane with the radius

The equation from the Kerr black hole ( ) can be used for the Schwarzschild black hole. As the Schwarzschild black hole has zero angular momentum ( ), the radius of the singularity is also zero ( ). Therefore, the singularity for the Schwarzschild black hole is a point.

^{[9]} ^{[14]}

## Calculating Angular Momentum

Due to angular momentum being proportional to , start with this equation.

Then plug in _{tφ }and _{φφ }from the metric,

Next, cancel out the negative signs, and make both the numerator and denominator have common denominators * ^{2}*.

Then take cancel out the

*and*

^{2}*. It becomes*

^{2}θ^{[17]}