Physics 105 – Spring 2011Astrophysics

SingularitiesJennifer MolnarApril 25, 2011

1 Abstract

It is possible, given a large enough mass within a small enough volume, for an object to collapse under itsown gravitational weight and form a singularity in space-time. Because gravity warps space-time, an effectdescribed by Einstein’s General Theory of Relativity, there is a distance at which even light will be unable tomove quickly enough to escape the singularity; this distance marks the event horizon. It has been proposed(by Penrose, 1969) that it is impossible for any singularities to exist that are not “clothed” by an eventhorizon—in other words, that all singularities must be black holes. The existence of black holes themselveswas only confirmed relatively recently by such discoveries as the galaxy M87 and the stellar companion toCygnus X-1. Given that black holes exist, the possibility of naked singularities is then considered. Severalmathematical models which suggest that naked singularities may be possible, given conditions where theblack hole has sufficient angular momentum, electrical charge, or a small enough mass.

Contents

1 Abstract 1

2 Introduction 2

3 Conclusion 13

A Bibliography 13

List of Figures

1 Space-time Diagram of the Trajectory of Light . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Light Cones Near a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Proportional Relationship of Gravitational Force and Radius . . . . . . . . . . . . . . . . . . 6

4 Model for Increasing the Rotation of a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . 11

1

2 Introduction

Einstein published two papers on his Theories of Relativity: one in 1905 and the other in 1916. The first

was on his Special Theory of Relativity, which described the way space and time necessarily conform to the

requirement that the speed of light be constant in all inertial frames. The second added a correction to the

first, incorporating the effect of gravity on space, time, and light. This was called the theory of General

Relativity. According to Einstein, gravity can be thought of as a description of the topology of space-time

instead of as a force drawing two objects together. Since the consequences of a curved space-time are difficult

to imagine, a classic analogy is used to illustrate: two pilots are flying their airplanes due north. They start

out a fixed distance apart, moving (presumably) parallel to each other. However, they find themselves

getting inexplicably closer and closer together as they continue to fly north. This, clearly, is not the result

of a mysterious force drawing the two sideways, but a natural consequence of the curvature of the earth.

Similarly, gravity can be considered the natural consequence of the topology of space-time, which is affected

by the distribution of mass and energy. This means that light can be affected by gravity, despite being

composed of photons, which are massless particles.

The curvature of space-time can be conveniently described using four-by-four matrices called tensors,

where each row in the tensor maps the dimension variables into another coordinate system. While relativistic

dimensions are generally introduced in Cartesian coordinates (dx, dy, dz, dt), it is often more convenient to

describe the topology of space-time due to gravitational effects using spherical coordinates (dr, dφ, dθ, dt).

This can be done as follows, using the invariant ds2 as an example:

ds2 = c2dt2 − dx2 − dy2 − dz2 (1)

= c2dt2 − dr2 − r2dθ2 − r2sin2θdφ2 (2)

= −gµνdxµdxν (3)

This equation can be rewritten as

dx′0

dx′1

dx′2

dx′3

= −

−1 0 0 0

0 1 0 0

0 0 r2 0

0 0 0 r2sin2θ

dx0

dx1

dx2

dx3

(4)

where dx0 = cdt, dx1 = dx, dx2 = dy, dx3 = dz, and dx′0 = cdt, dx′1 = dr, dx′2 = dθ, and dx′3 = dφ.

2

According to Einstein’s field equation, Gij = −8πGc4 Tij . In other words, the Einstein gravitational tensor

(Gij), which is related to the curvature of space, is proportional to the energy/momentum tensor (also

known as the stress/energy tensor, Tij). Although this looks like a simple equation, the many simultaneous

equations hidden within the tensors make it complicated to solve. The simplest case is the case of a large,

spherically symmetrical object with mass M , first solved by Swartzschild in 1915. Since space-time is no

longer flat in the region of the object, the tensor gµν changes somewhat:

gµν =

−(1− 2GMr2 0 0 0

0 (1− 2GMr2 )−1 0 0

0 0 r2 0

0 0 0 r2sin2θ

(5)

This is called the Swartzschild metric, and is actually a very useful case to have a solution for since

celestial bodies with masses significant enough to distort space-time tend to pull themselves into spheres by

their own gravity. (The limits of this formula lie more substantially in the fact that it assumes that the

spherical body is non-rotating—an unrealistic assumption for actual systems. However, it still acts as a

good approximation for slowly rotating bodies, such as the Earth and Sun.) This formula enables not only

spatial distortion, but time distortion due to gravity to be quantified, much like the Lorentz transformation

equations enable the quantification of space-time distortion due to motion.

While light always moves at a constant speed according to Einstein’s Theory of Special Relativity, his

General Theory of Relativity states that the time distortion due to gravity may still be significant enough

to prevent light from escaping from a gravitational field, due to frequency distortion. The mathematical

argument proceeds thusly: If a photon is emitted from our object of mass M , it will have a locally observed

wavelength of λ0 = cdτ , where dτ is the proper time (the time as measured by an observer in the inertial

frame of the object). An observer at a good distance away will find that the space-time curvature is essentially

unaffected by the mass of M and can be treated as flat. His measurement of the time between wavelengths

(dt) will be equal to the proper time multiplied by the first term of the Schwartzschild metric. In other

words,

cdt =

(1− 2GM

c2rs

)−1/2cdτ (6)

λobs =

(1− 2GM

c2rs

)−1/2λ0 (7)

where rs is equal to the radius at the surface of mass M . This shift in wavelength is usually very small, but in

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extreme cases (where M is very large and rs is very small), the wavelength can be substantially lengthened.

In particular, as rs approaches 2GMc2 , the observed wavelength approaches infinity and the associated energy

(E = hν = hc/λ) drops to zero. In other words, no light from this radius will be visible to an observer,

and any photons emitted from within this radius will not be able to pass this radius. A little more math

shows that the relativistic escape velocity, found by equating the kinetic and potential energy of a particle,

Thus, the Schwarzschild radius (rs = 2GMc2 ) marks the boundary called the “event horizon” of the spherical

object—which, since it cannot emit any light, is known as a black hole.

This effect can be visualized using space-time diagrams. Imagine space as a flat sheet in the x-y plane,

with progression in time marked along the z-axis. (For ease of visualization, only two dimensions of space are

considered.) When a brief flash of light is emitted from a point source, it extends outwards from the point

equally in all directions, at the speed of c = 3.0× 108 m/s. This trajectory forms a cone on our space-time

diagram, as shown in Figure 11.

Figure 1: Space-time Diagram of the Trajectory of Light

In the presence of a strong gravitational field such as a neutron star, the light cone extending from the

point source starts to tilt toward the center of the star—closer proximities yield greater tilts. In an object as

massive as a black hole, there comes a point where the light cone tilts so much that one side of it is entirely

parallel to the world-line of the point mass (see Figure 2, where the thick line represents the world-line of

1taken from Wikimedia Commons, http://en.wikipedia.org/wiki/File:World line.svg on April 16, 2011

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the black hole and the thin dotted line represents the event horizon, past which no light can escape):

Figure 2: Light Cones Near a Black Hole

Since nothing can travel faster than light, all events in the future of an object or particle must correspond

to a world line that falls entirely within a light cone centered at that object. It is then possible to see, from

Figure 2, that any particle within the event horizon of a black hole must move further toward the center of

mass of the black hole. Even light must converge to this point; there is no other possibility. This rule applies

even to the matter that makes up the black hole itself: it must compress and continue compressing, until

it reaches a point of infinite density and infinitesimal size. There is no coordinate system possible in which

this phenomenon does not occur; the point at which this system reaches a point of infinite density is known

as a “singularity.”

A calculation of infinity in a physics formula is usually an indication that a formula has reached a limit

at which it ceases to apply to the real world. The infinite amount of energy that was calculated to exist

in a black body box was an indication that some information was missing from the classical formulas—

information that was eventually supplied by quantum mechanics. In this case, the “infinity” comes from the

gravitational formula,

F =GMm

r2(8)

or in other words, F ∝ 1r2 , where r is the distance from little mass m to the center of gravity. As the distance

from the center of gravity decreases, the force increases to infinity. But this formula no longer applies once

the distance is smaller than the radius of the object itself. Once the small mass gets close enough to the

center of mass of the large object to be below its surface, a different formula holds (F ∝ r), because the net

gravitational pull on the object is only due to the mass beneath it. (Mass above it will pull on it equally in

all directions, creating a net zero gravitational force.) The general relationship between gravitational force

and radius is illustrated in Figure 3.

Clearly, the infinity shown in the graph only can occur in real life if the radius of the large mass, M , is

equal to zero. At first glance this appears to be an unrealistic scenario, but observe: by keeping the mass

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Figure 3: Proportional Relationship of Gravitational Force and Radius

constant and increasing the density (thereby decreasing the radius), the point at which the transition is made

between the F ∝ 1/r2 formula and the F ∝ r formula moves farther to the left, rising higher and higher

along the dotted line. The star must output a tremendous amount of thermal energy to maintain a high

enough pressure to prevent collapse. As a star uses up fuel and creates less thermal energy, it is conceivable

for the star’s pressure to decrease, the radius to shrink, and the gravitational force to reach a point (the

Schwartzschild radius) where particles on the surface of the star are being pulled inward too strongly for

thermal energy to resist them. This shrinks the star even further, creating a cycle that will inevitably lead

to a black hole.

Any mass can become a singularity if compressed tightly enough. The question then is, is there a feasible

process by which matter can be compressed to this point? Above a certain threshold mass, this level of

compression is practically inevitable. It begins with a main-sequence star, composed mostly of hydrogen

(the most abundant element in our universe), with a mass at least 25 times that of our sun. The star

experiences two opposing forces, which keep it generally in stasis—the force of its own intense gravity,

trying to pull the molecules inward, and the pressure from the thermal energy in its interior, pushing the

particles outwards. If at any time one of these forces becomes more powerful than the other, the star’s size

shifts—either by compressing or expanding, if the shift is gradual, or by explosion/implosion if the change is

rapid (potentially expelling outer layers of gas, further shifting the star’s size), until the star again reaches

equilibrium between the two forces.

At early stages of the star’s life, the thermal energy that prevents the star from collapsing is supplied by

the nuclear fusion of hydrogen atoms. There are several possible reactions that can and do take place, but

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by far the most common reaction is the PPI chain, which proceeds according the following sequence:

11H +1

1 H → 21H + e+ + νe (9)

21H +1

1 H → 32He + γ (10)

32He +3

2 He → 42He + 211H (11)

The net reaction is the synthesis of four hydrogen atoms into one helium atom and two electrons, two

neutrinos, and two photons. Energy is produced by this reaction and carried away by the neutrinos and

photons.

As more of the hydrogen gets converted to helium, the average particle mass increases and so does the

density. Along with the density, the pressure and temperature also rise, until the star is able to produce

the activation energy necessary to overcome the Coulomb barrier between two helium atoms and fuse them

to produce carbon. This is called the triple alpha reaction, and can only occur at temperatures that are 64

times the temperature required to burn hydrogen. This process again produces energy that is carried away

by photons, and causes the average particle mass (and thus star density and temperature) to continue to

increase.

Not all stars have enough mass to reach the stages at which helium can be fused; fewer still are able to

proceed to the next step to burn carbon and oxygen. At every stage, the amount of energy that needs to be

present for the reaction to proceed increases, and the energy produced diminishes. At a certain size, further

fusion reactions actually absorb energy instead of producing it. The final product of nuclear fusion is 56Fe.

At the point when 56Fe is being generated, the star is running out of fuel and does not have very long to

live.

While the internal structure of the star can be fairly complex at this point, with layers of iron, silicon,

carbon and oxygen near its core, all the way up to helium and hydrogen in its outer layers—each potentially

fusing into heavier particles where densities remain high enough to allow it, near the boundaries of each

level—the important part of this discussion is that nuclear burning cannot go on interminably. At a certain

point, each layer no longer has sufficient temperature and pressure for any more fusion to occur, and the

star runs out of thermal energy.

Without the fuel to sustain it, gravity rapidly overcomes the opposing thermal forces keeping it at bay.

The star collapses quickly and dramatically to a hot and tightly-compressed ball of heavier atoms, rotating

extremely quickly because of the decreased rotational inertia. Depending on the original mass, a star may

compress to a white dwarf or a neutron star, supported against the force of gravity by electron or neutron

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degeneracy (respectively). However, at masses exceeding 25 solar masses, the star will almost certainly be

too massive for even neutron degeneracy to sustain it. The star shrinks below the Schwartzschild radius

into a black hole, and from then on it is past the point of no return. All light cones point inward, towards

its center, and the matter must remain within the boundaries that those light cones set until the matter

compresses entirely into a point of infinite density and infinitesimal space: the singularity. In this case, the

singularity is known as a space-like singularity.

Roger Penrose proposed another type of singularity, the time-like singularity, in January of 1965. If a

group of celestial bodies is brought close together, the mutual gravity between them all will attract them

towards their center of mass. With a little translational velocity, the bodies will be mostly prevented from

colliding and will instead be on orbital trajectories around this center, subject to deflection by nearby objects.

Occasionally, these deflections impart enough energy to one of the objects to allow it to actually leave the

cluster; conservation of energy states that the remaining matter will then have less energy, move slower, and

be closer together. If the initial mass is high enough and the “evaporation” of matter cooling enough, it

does not matter if the leftover masses do not eventually compress into a single point: the only requirement

for a black hole is that there exist a radius inside of which light cannot escape, and this conglomeration of

stars may provide that.

Despite having plausible mechanisms for how a black hole could be created, scientists have been cautious

to accept their existence. A singularity is a point at which the universe apparently ceases to exist. There

are no known physical laws that describe such points, and no way of testing any hypotheses that allows for

the experimental results to return to the rest of humanity. However, significant evidence has been found

for the existence of black holes. M87, a galaxy in the Virgo constellation, shows an extremely dense cluster

of stars in its center—evidence of a stronger attraction than could be accounted for by the stars’ collective

mass. The existence of a black hole at the center of the galaxy became a well-founded explanation when the

Hubble telescope provided images that showed spiral-shaped gas rotating rapidly around a central object.

The angular speed of the gas at its outer edges was 550,000 m/s, suggesting that the mass of the central

object was over three billion times the mass of our Sun and yet occupied a space no larger than the Milky

Way. This was the first definitive discovery of a black hole.

Other potential black holes have been discovered in binary star systems. Cygnus X-1 is a star in the

Cygnus constellation, which orbits its companion every five-and-a-half days, as is determined by analyzing

the periodicity of its red- and blue-shifts. The companion star is dark, but a heavy emitter of x-rays; this and

the low periodicity of Cygnus X-1, which is a blue supergiant of high mass, suggest that the companion star

is small but massive: either a neutron star or a black hole. Cygnus X-1’s luminosity and the hue distortion

8

due to light absorption by interstellar dust reveal its radius and distance from Earth, which can be used to

estimate the radius of its companion star. The estimation places this mass at somewhere between 8 and 18

solar masses, which is much more massive than a neutron star: Cygnus X-1’s stellar companion is almost

certainly a black hole.

Singularities, then, do exist. The implications of this statement are unknown. At a point where space-time

no longer exists, what can occur? Speculation ranges from worm-holes to entrances to alternate universes,

but regardless, there is no way for us to find out. Whatever strange things go on near a singularity do not

affect the rest of the universe, because the singularity is “clothed” by its event horizon. Roger Penrose, in his

paper on gravitational collapse in 1969, made the famous speculation that singularities may all be clothed—

that there may be some sort of “cosmic censor” that prevents them from ever being visible (”naked”) to a

far-away observer. However, he stated that mathematically, solutions can be found that allow light to escape

from a singular point in space-time—there is a possibility that naked singularities exist. However, it is yet

to be determined whether or not these solutions have any physical basis.

The Schwartzschild metric that considers the case of an uncharged, non-rotating black hole has already

been discussed. Realistically, however, neither of these conditions are likely to be met. Any system is likely

to have some initial rotational velocity, and as this matter gets compressed, the decrease in rotational inertia

causes the angular velocity to speed up. The effect is exactly comparable to an ice skater pulling her hands

in to her chest during a spin. The presence of charge also adds some conditions to the metric, since charge

presumably must still be conserved, even in the vicinity of black holes. The more complicated equations

that apply to this type of singularity are known as the Kerr-Newman solutions, where Kerr’s contribution

was to consider rotation and Newman’s was to take charge into account.

For reference, the equation describing the Schwartzschild metric is shown below:

ds2 = −[1− 2m

r

]dt2 +

dr2

1− 2m/r+ r2dθ2 + r2sin2θdφ2 (12)

where m is the mass of the central object and ds2 refers to the proper distance between two points. For

reference, compare this to Equation 2, the formula describing ds2 for flat space. When m = 0, the formula

reduces to the flat space formula.

The original form for Kerr’s equations was written

ds2 =

[1

2mr

r2 + a2cos2θ

](du+a sin2θdφ)2+2(du+a sin2θdφ)(dr+a sin2θdφ)+(r2+a2cos2θ)(dθ2+sin2θdφ2)

(13)

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Again, by plugging in m = 0, one again derives the formula as it corresponds to flat space-time. Plugging

in a = 0, where a is the parameter describing the rotation of the main mass body (am represents the

magnitude of the angular momentum), yields the original Schwartzschild metric. Arriving at this solution

was no mean feat: the Kerr solution was published by Roy Kerr in 1963, 48 years after Einstein published his

general theory of relativity. Karl Schwartzschild’s solution was first published within two months. Such an

eminent person as Chandrasekhar extolled Kerr’s solution as “splendorous, joyful, and immensely ornate,”

recognizing its importance in astrophysics.

The Newman solution handles the case of a charged mass. In Newman’s paper, published with five

contemporaries in 1964, the Newman metric was given in matrix form:

gµν =

x(−a2sin2θ x(r2 + a2) 0 −xa

. x[2mr − (r2 + a2)− e2] 0 xa

. . −x 0

. . . x(−sin−2θ)

(14)

gµν =

1 + x(e2 − 2mr) 1 0 x(asin2θ)(2mr − e2)

. 0 0 −asin2θ

. . −x−1 0

. . . −sin2θ(r2 + a2 + ag03)

(15)

where x = (r2 + a2cos2θ)−1. Again observe the connection with the previously shown formulas.

The Kerr-Newman metric predicts an event horizon for all cases in which m ≥ a2 + e2. In cases where

m < a2 + e2, there could conceivably be a singularity which allowed some light to escape and reach outside

observers. Because of the powerful gravity nearby the singularity, the light would be distorted and convo-

luted, so an observer’s ability to interpret the outgoing information might be in question. Regardless of its

intelligibility, however, a naked singularity would allow whatever strangeness can occur there the capability

of extending causal influence on outside parts of the universe. Again, scientists are skeptical that such a

system actually exists; no evidence for one has yet been discovered. On the other hand, without knowing

what goes on in the vicinity of a singularity, it is hard to know what to look for. The question that scientists

ask instead is one of feasibility: what would the structure of the singularity have to be in order for it to be

naked, and is there a mechanism by which that structure can form?

According to the Kerr-Newman equations, all that really has to occur for a naked singularity to form is

for a black hole to gain enough rotation or charge to exceed its mass parameter. However, creating such a

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black hole or manipulating a pre-existing one in such a way that it is able to meet these conditions seems

to be impossible. In principle, it seems that one could increase the rotation of a black hole by firing a

stream of particles toward its event horizon tangentially, or by dropping a rapidly-rotating mass into the

black hole from the top. However, the rotational inertia acquired by the black hole due to the increase

actually overcompensates for the gain in angular momentum. An attempt to fire massless particles (such

as photons) into the black hole to speed it up will not contribute mass, but will contribute energy, which

still adds to the mass of the hole. However, photons also have spin. An alternative experiment could be

tried wherein photons are not shot tangentially into the black hole but dropped down into it with the spin

in the appropriate direction—much like the mass in Figure 4. By increasing the wavelength, the energy

imparted is decreased, so that the gain in spin is more significant than the gain in energy. Unfortunately,

the wavelengths necessary for this are so large that the photons are more likely to scatter off of the black

hole than to be absorbed by it.

Likewise, it seems possible to increase the charge of a black hole by firing charged particles into it until

a naked singularity forms. However, as the charge inside the black hole builds up, it takes more and more

energy to fire additional charged particles into it, until the energy input into the black hole again outweighs

the increase in charge.

Figure 4: Model for Increasing the Rotation of a Black Hole

Let us abandon for the moment the practical difficulties in trying to defeat the “cosmic censor.” One

purely mathematical example of a naked singularity is E.P.T. Liang’s infinite cylinder. Liang showed that

collapsing matter into a line of infinite density instead of a single point would create a string singularity

without an event horizon. It is difficult to determine whether or not this possibility has any realistic analogs:

the model needs to be validated for cases where the cylinder is finite or has small deviations from being

perfectly symmetrical. Because of the complexity of Einstein’s field equations, this has not yet been done.

Another mathematical model for a naked singularity was generated by Dr H. Muller zum Hagen. His

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theoretical naked singularity occurs if the outside of a collapsing star moves inward faster than the inside

does. If this is the case, the outer layers may be able to meet the inner layers at a certain radius; then

an infinitely dense shell exists that allows a spherical naked singularity to develop. This model also makes

several idealizations which may make it inapplicable to the real universe, including the assumption that the

matter is able to fuse without producing any pressure to prevent all the layers from combining together at

the critical radius.

One last example of a hypothetical system producing a naked singularity is to remove energy or mass

from a black hole to decrease its Schwartzschild radius. If the mass or energy is reduced to zero, then the

Schwartzschild radius is also zero, and whatever singularity is left will be visible to a far-away observer. This

last model has some promise to it, as black holes do lose mass over time. This mass does not come from

matter that has fallen into the hole; that matter is unable to escape the event horizon. Instead, Stephan

Hawking in 1974 discovered that a moderately-sized black hole can be approximately the size of an atomic

nucleus. At distances so small, quantum mechanics needs to be taken into account, and quantum mechanics

allows a non-zero probability for subatomic particles to pop in and out of existence. These particles include

small units of negative mass-energy. If two particles are created near the event horizon—one with positive

mass-energy and one with negative mass-energy—and the negative particle is absorbed by the black hole

while the positive one escapes, then the mass of the black hole has effectively been reduced slightly. Hawking

further discovered that the radiation of particles out of a black hole increases as the black hole’s size decreases.

This predicts that as the black hole begins to lose mass, the process of its evaporation speeds up more and

more in a positive feedback loop. Thermodynamically speaking, the black hole emits blackbody radiation

with a temperature that increases exponentially as the hole evaporates. This self-propagating cycle implies

that the black hole will eventually evaporate away. When it does, with a final explosion as its temperature

sky-rockets, a naked singularity may be left.

There are theoretical difficulties with this solution. John Wheeler’s no-hair theorem suggests that black

holes are defined by three and only three characteristics from the outside: their mass, their angular momen-

tum, and their charge. This makes black hole radiation independent of whatever specific objects originally

fell into the black hole—the only thing that determines any aspect of the radiated particles is the black

hole’s mass. If the black hole is able to shrink to nothing via radiation, then this violates the principle

of Conservation of Information, which assumes that if the complete state of a system is known, then it is

theoretically possible to reconstruct all its past and future states. The loss of information that occurs when

masses fall into a black hole is called the “information paradox,” and has not yet been resolved.

The other difficulty with this hypothesis is that no evidence has been found of such black hole explosions;

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this does not rule out the possibility, though, as they may be extremely uncommon. A lower limit to

their prevalence can even be established: if there were more than a few nucleus-sized black holes (about a

billion tons in mass) per 1027 km3, then their mass would exceed the mass of the galaxies of the universe,

producing very obvious effects. Discovery of an exploding black hole is also limited by the sensitivity of our

equipment: even if the explosion is comparable to the detonation of a million megaton bomb, its effects will

be indiscernible unless it is quite close to our solar system. Martin Rees of Cambridge University points

out that it may still be possible to detect an exploding black hole by detecting a byproduct of the explosion

itself: the disturbance in the local electromagnetic field will reach the surface of Earth in the form of a

radio wave. Radio astronomers have found nothing yet, which suggests that if black hole explosions exist,

they occur at a rate of less than one per million cubic light years per year. Any definitive conclusions await

further research.

3 Conclusion

Einstein’s General Theory of Relativity predicted that masses are able to affect the very structure of space-

time, allowing large masses to influence even massless particles such as light. When masses are compressed

to very high densities, the inverse-square law predicts that they may be able to curve space to such an

extent that all matter and energy within an “event horizon” must converge to an infinitely-dense point,

known as a singularity. Though the existence of singularities was debated for some time, current evidence

shows strong evidence for the existence of black holes—singularities that do exhibit this event horizon.

Speculation on naked singularities (singularities without this event horizon) has led to some interesting

models and predictions, but research has not yet been able to provide any evidence of any violations of

Penrose’s “cosmic censorship” hypothesis: that at locations as strange and unpredictable as singularities, it

is impossible for any causal connection to exist between the singularity and the outside universe.

A Bibliography

• Class notes (Physics 105, Duke University, Professor Mark Kruse, Spring 2011)

• Davies, P. C. W. The Edge of Infinity: Where the Universe Came from and How It Will End. New

York: Simon and Schuster, 1981. Print.

• Penrose, Roger. ”Gravitational collapse: the role of general relativity” Rivista del Nuovo Cimento, 1

(1969) pp. 252276

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• Shapiro, Stuart L., and Saul A. Teukolsky.Black Holes, White Dwarfs, and Neutron Stars: the Physics

of Compact Objects. Weinheim: Wiley, 2004. Print.

• ”Kerr Metric.” Wikipedia, the Free Encyclopedia. Web. 21 Apr. 2011. <http://en.wikipedia.org/wiki/

Kerr metric>.

• Visser, Matt. ”The Kerr Spacetime: A Brief Introduction.” Apr. 2011. <http://arxiv.org/abs/0706.0622>.

• Shapiro and Teulosky. Formation of Naked Singularities: A Violation of Cosmic Censorship. (1991)

• ”Theory of Relativity.” Wikipedia, the Free Encyclopedia. Web. 21 Apr. 2011. <http://en.wikipedia.org/

wiki/Theory of relativity>.

• Griffiths, David Jeffrey. Introduction to Electrodynamics. Upper Saddle River, NJ [u.a.: Prentice Hall,

1999. Pages 500-504.

• ”HyperPhysics.” Web. Apr. 2011. <http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html>.

• ”HubbleSite - NewsCenter - Hubble Confirms Existence of Massive Black Hole at Heart of Active

Galaxy (05/25/1994) - Release Text.” HubbleSite - Out of the Ordinary...out of This World. Web.

Apr. 2011. <http://hubblesite.org/newscenter/archive/releases/1994/23/text/>.

• E. T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash and R. Torrence: Journ. Math.

Phys., 6, 918 (1965).

• “Eric Weisstein’s World of Physics.” Web. Apr. 2011. <http://scienceworld.wolfram.com/physics/

BlackHoleNoHairTheorem.html>.

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