Robert W. Smith

Honors Physics
Black Holes

Introduction
Most people have heard of these things, or at least read about them. They were used in a variety of movies such as "The Black Hole", and "Super Nova". Many space movies love to add these things to them because of their strange obscure nature. But most people feel that these things do not exist, and never have, and never will. They just think that it was something we put in movies to make the plots more interesting. Well, from what scientist have researched today, there is a great deal of viable evidence that could lead us to believing that black holes are real, and that they are an essential part of the balance of the universe.

Black Hole Theories

Schwartzschild Black Holes

The first thing that we should know about black holes is that there is a great deal of theories backing them up. One of the biggest theories is named the Schwartzschild black holes, also known as the Singularity Theory. Another major theory about black holes is the Kerr black holes.
A black hole can be defined by the Schwartzschild theory as "A mass that is compressed into a sufficiently small radius, where space-time becomes so severely distorted that even light can't escape from its gravity well."1 That is a basic description of the singularity that Schwartzschild had talked about. Also the singularity of the Schwartzschild black hole is a spherical shape, and very attuned to being more like space as opposed to time.

The Schwartzschild Black Hole

Kerr Black Holes

Another theory for black holes is the Kerr theory. There are very few differences in the Kerr black hole and the Schwartzschild black hole. One of the major differences is that with the Kerr black hole, it rotates around an axis point. Once a traveler begins to rotate around the axis point, they have surpassed something called the Stationary limit.
At this point there is no way that one would be able to escape from the black hole unless they were able to travel faster than the speed of light. In the Kerr black hole, there is a space between the event horizon, and the stationary limit called the ergosphere. Theoretically it is possible to escape once in the ergosphere, but yet that remains impossible as of now.
Once the black hole has pulled you through the ergosphere, and past the event horizon, which is the point of no return. Another major difference between the Kerr and Schwartzschild black holes is that the singularity of a Kerr black hole happens to be a ring shape as opposed to a spherical shape. The singularity of the Kerr black hole happens to be more like time than space as the Schwartzschild theory had put it.
Part of the Kerr theory talks about something called a Naked Singularity. Quoted from a website Article that I found online about black holes:
"One of the most unusual characteristics of a Kerr black hole is the possibility that it could evolve into a naked singularity. Due to the law of conservation of angular momentum, a rotating black hole should rotate ever faster as its radius decreased. Once the object's angular momentum increased beyond it's mass, the event horizon of the hole would be moving in excess of the speed of light. At this point, the event horizon would simply vanish from the universe, exposing the singularity. The absence of the event horizon means that we could travel freely into and out of the singularity. While no one has yet to prove that naked singularities cannot exist, most physicists are strongly inclined to believe that such is the case. Safely within the event horizon, a singularity is effectively shut out of the universe. When it is naked, this region of utter disregard for the known laws of nature is free to interact with the rest of the universe. To illustrate just how disruptive such an object might be, the simple act of going into orbit around a naked singularity would enable one to travel to any point in the past."1

The Kerr Black Hole

Other Theories

There is also a great deal of other theories about black holes. Some of these come from Einstein, Stephen Hawking, and Newton. Einstein's theories consist of the Special Theory of Relativity, and the General Theory of Relativity. Stephen Hawking's special theory is the theory of Hawking Radiation, and Newton's theory is the one of the Universal Law of Gravitation.

Einstein's Special Theory of Relativity

I'll start off with Einstein's theories. His Special Theory of Relativity happens to come from two various statements. These happen to be: 1. The speed of light is the same for all observers, no matter what their relative speeds. 2. The laws of physics are the same in any inertial (non- accelerated) frame of reference.2 As this website calls them, both of these statements happen to be postulates. The first of these two postulates states and possibly proves that the Speed of Light is a constant in nature, and will never change, whether some strange force acts on it or not.2 "Einstein developed a theory of motion that could consistently contain both the same speed of light for any observer and the familiar addition of velocities described for the slow moving objects."2 This is the basis of Einstein's Special Theory of Relativity. Also, we should remember that this theory is quite different from the General Theory of Relativity due to the fact that this theory deals with the relativity of motion, while the General Theory of Relativity deals with the relativity of Gravity.
Now, as an explanation to the Special Theory of Relativity, there are some basic definitions that you should know, and understand. The first definition that you should know is Gamma. Gamma is what the measurable effects of relativity are based on.2 According to the website that I have gained this information from, the equation for Gamma is:

In this equation, the V= the speed of the object in question, and C= the speed of light. According to this, when the object's speed is much less than the speed of light, the gamma number is closer and closer to one. According to this website, this could be called and considered a Non- Relativistic Situation (Newtonian).2
Next, we will cover the term Momentum. The standard equation for this happens to be P=mv. According to this website, we will be able to modify this equation, and make it so that p=gamma(mv). 2 What this equation tells us is that when the speed increases, and nears the speed of light, the mass increases, as well as the momentum. When it reaches the speed of light, there must be an infinite amount of momentum to keep it moving, and since the object needs an infinite amount of force to keep it moving, we have to end it saying that massive particles, can't move anywhere near the speed of light.2 Basically what it boils down to is that we cannot move at the speed of light due to the amount force needed to keep it moving at an infinite momentum.
The final term that you must know for Einstein's Special Theory of Relativity is Energy. The equation for this happens to be Einstein's most famous equation. E=mc2. Energy equals mass times the speed of light squared. According to the website where this information came from, the energy is equal to the "mass at rest"2 times the speed of light squared. "Einstein also showed that the correct relativistic expression for the energy of a particle of mass m with momentum p is E2=m2c4+p2c2. This is a key equation for any real particle, giving the relationship between its (E), momentum (p), and its rest mass (m)."2
Now to substitute P into the equation for E, and due to some algebra which was not said, we come to the conclusion that E=gamma mc2. In essence this says that Energy is equal to gamma times rest energy. Now according to the site where this information is coming from, they take the equation for Kinetic Energy (KE), and place it into the equation for resting energy. It comes out looking like E=KE+mc2. But when they make the assumption that KE= gamma (1/2) mv2, it turns out that the gamma statement will only work with the Total Energy, and not just a single part of energy.2
"Another interesting fact about the expression that relates E and p above (E=m2c4+m2c2), is that it is also true for the case where a particle has no mass (m=0). In this case, the particle always travels at a speed c, the speed of light. You can regard this equation as a definition of momentum for such a mass-less particle. Photons have kinetic energy and momentum, but no mass!"2 Thus, we can stated that a Photon is able to travel at the speed of light due to the fact that there is no mass, and this is why we are able to take photons, and understand the reasons that they are light particles. This is because they are defined as "mass-less"2.
According to SLAC's website, which is where the information used in this tutorial came from, they say that mass and kinetic energy happen to be interchangeable. Two of the situations they give are: 1. In case of an atomic explosion, mass energy is released as kinetic energy of the resulting material, which has slightly less mass than the original material. 2. In any particle decay process, some of the initial mass energy becomes kinetic energy of the products.2
Now, to explain the units used when doing these calculations, we should start off with the units for mass. Many people will think, that we would use kilograms to measure the mass, but that is a false statement. For measuring mass, we will use a unit of energy.2 This unit of energy is the electron volt, or the energy gained by one electron when it moves through a potential difference of one volt. "By definition, one electron volt (eV), is equivalent to 1.6x10-19 joules."2
The next part of Einstein's Special Theory of Relativity that we will visit is Length Contraction and Time Dilation. In order to define these, we must use and example of two spaceships, with two astronauts, both holding a meter stick in their hands, and having a clock next to them. Remember that both spaceships are moving at the same speed, and are parallel to each other. Their speed is close to the speed of light. Length Contraction states that each observer will see the meter stick of the other as shorter then their own. Time Dilation states that each observer will see the clock of the other ticking slower than his or her own. Both of these statements are by the same factor of gamma.
For Time Dilation, we should look at a particle created at SLAC, called Tau. A Tau's lifetime is approximately 3.05x10-13 s. The website as SLAC calculates the distance traveled by Tau, by using the distance equals speed times time traveled. This allows us to find the distance by taking 3.05x10-13 and multiplying it by c, which is the speed of light, which is 3.00x108. The distance traveled ends up to be 9.15x10-5 meters. But for whatever reason, when SLAC measured the distance that the Tau particle traveled in the accelerator, it turns out that it traveled farther than what the calculations said it should have. In order to make the Time Dilation statement work correctly, we would times the distance traveled by the standard gamma value. SLAC uses 20 as that value, and when we multiply 9.15x10-5 with that, we come to 1.8x10-3, which is approximately 1.8 mm.2 "Observations particles with a variety of velocities have shown that time dilation is a real effect. In fact, the only reason cosmic ray muons ever reach the surface of the earth before decaying is the time dilation effect."2
We could call Length Contraction, an opposition to Time Dilation. With Length contraction, instead of us looking at the Tau particle, we should be the Tau particle looking at us. This will put us in the Tau particle's frame of reference. Normally, we would feel that the Tau particle moved 1.8 mm, but in all actuality, the view that the Tau particle would see would be 1.8mm/gamma, which is approximately equal to 0.09 mm. That is how Length Contraction works.

Einstein's General Theory of Relativity

Now, I am going to explain in depth, Einstein's general theory of relativity. "Here is Einstein's brilliant Idea: Identify freely falling bodies in a gravitational field with the inertial observers of special relativity."3 According to the author of the book that this information is being gained, Spacetime is curved. According to Einstein's theory, the presence of a gravitational field corresponds to the curvature of the spacetime geometry.3
Somehow this had to be explained through the equations that Einstein wrote up and postulated. He needed to somehow relate the spacetime gravitational field to the equation of special relativity. This he did by stating, "curvature of spacetime" = "energy density of matter"3 Now to explain that statement, I will use an excerpt from Space, Time, and Gravity, by Robert M. Wald.
"The description of Einstein's equation is somewhat of an oversimplification, and some words of caution should be given. First, the left side of the equation is not the entire curvature of spacetime but only a part of it. Thus, outside the matter distribution (where the right side of Einstein's equation will be zero) spacetime will in general still be curved; that is, a gravitational field will be present. Furthermore, gravitational radiation-ripples in the curvature which propagate through spacetime-can exist. Second, the right side contains contributions from other properties of matter besides energy density. In particular, pressures and stresses contribute to the curvature of spacetime in general relativity."3
To turn the theory of general relativity into the theory of special relativity, all we have to do is put a zero in place of the right side of Einstein's statement. That will make it so that there is no mass, in which then we can state that it is able to travel at the speed of light, in a straight line, on a flat plane. That is how we are able to relate Einstein's two theories, because one cannot coexist without the other. Our author of the book that this information is from, kindly states that "The 'geodesic hypothesis' actually follows as a consequence of Einstein's equation and does not have to be postulated separately" where the "geodesic hypothesis" is the equation that proves spacetime to be curved, and relates it to Einstein's energy equation.3
According to what the author of this book is describing, Einstein's Theory, like all theories, has four key predictions or flaws, that nature does not respond well to. These are The Motion of the Planets, Light "Bending", The Gravitational Redshift, and Time Delay. Due to the fact that this theory has not been tested with objects that hold a massively strong gravitational field, such as black holes, we cannot state that his theory is without flaw.3
First, the motions of the planets are described to having related Newton's law of universal gravitation, to Einstein's theory of general relativity. But in closer examination, we have found out that when following Kepler's law which states that the planets move in an elliptical manner, we find that Einstein's law drifts away from Newton's law. The author of this book gives an example, which is as follows:
"In particular, the elliptical orbits of planets don't quite close but precess instead. This precession is much too small to observe for all planets except mercury, where general relativity predicts an orbital precession of 43 seconds of arc per century. (A second of arc is 1/3,600 of a degree.) Indeed, it had been observed long before the development of general relativity and had been an unexplained mystery until then."3
Second is Light "Bending". General relativity states that the route of a photon, or light ray, in spacetime is a null (lightlike) geodesic. "Because spacetime is not flat in the vicinity of the sun, a light ray passing near the sun will appear to a distant observer to be deflected."3 Because of the gravitational pull from the sun, it seems to bend the light. This in fact is true, because the light will travel in the straightest line possible, and according to Einstein's theory, the geodesic curvature of space can be described as the straightest line possible.
The third fallacy, or prediction that attempts to prove Einstein's theory incorrect, is The Gravitational Redshift. In order to explain this point, I am going to take another excerpt from Robert M. Wald's book Space, Time, and Gravity.
"Suppose and observer O1 sends out two signals with a time interval ?t1 between them. Let observer O2 receive these signals. In curved spacetime-or even in flat spacetime if there is relative motion between O1 and O2-there is no reason why the time interval ?t2 between O2's reception of the signals must equal ?t1. Thus, in particular, if O1 emits light at frequency v1, O2 will, in general, observe it to have frequency v2?v1. If O1 and O2 are "at rest" and v2?v1, this effect is known as the Gravitational Redshift. One can show that light emitted in a region of strong gravitational attraction will be seen by a distant observer to have a lower frequency."3
According to Robert M. Wald, the Gravitational Redshift will occur when energy is conserved. He uses the equation for the energy of a photon to describe this. E=hv where h is Planck's constant.3 "If the frequency of light remained unchanged as it left a region of strong gravitational attraction, its energy would not be decreased. One could then convert the energy of the photons in the light beam to rest mass energy and lower the mass back into the strong-field region using the gravitational attraction to do work. If one then converts the rest mass energy back into photons, one will return to the same configuration as one began with, but energy will have been gained in the process of lowering the mass. Thus, the quantum formula E=hv, together with conservation of energy, requires the existence of the Gravitational Redshift phenomenon predicted by general relativity."3
Finally, the fourth prediction is Time Delay. When looking directly at a passing by laser, or light, you would see that the speed of the light is the speed of light. But, when the light ray passes through a region of high spacetime curvature, it can affect the total time taken to travel the route.3 "General relativity predicts that light rays emitted from a planet or spacecraft as it passes behind the sun will suffer a small, additional "gravitational time delay" as compared with what would be predicted using Newtonian Theory, due to the spacetime curvature near the sun."3
Hawking Radiation
Hawking Radiation defies all beliefs about black holes. The basis of the word black hole which describes a super dense singularity that has such a strong gravitational force, that not even light can penetrate its event horizon. A black hole is black, isn't it? So much for that belief, because according to Stephen Hawking, black holes actually emit a glow, that many have come to call, Hawking Radiation.
According to Hawking, black holes emit a small radiation aura formed from photons, neutrinos, and other massive particles of radiation. According to the essay which this information is being extracted from, although never observed due to the fact that the star's radiation that is being sucked into the black hole blocks it out, there is a certain amount of heat radiating outwards. "If a mass of a black hole is M solar masses, Hawking predicted it should glow like a blackbody of temperature: (6x10- 8/M) Kelvin, so only for very small black holes would this radiation be significant."4
According to this essay, "The most drastic consequence is that a black hole, left alone and unfed, should radiate away its mass, slowly at first but then faster and faster as it shrinks, finally dying in a blaze of glory like a hydrogen bomb. However, the total lifetime of a black hole of M solar masses works out to be 1071M3 seconds, so don't wait around for a big one to give up that ghost."4
The essay explains Hawking Radiation like this: "Virtual particle pairs are constantly being created near the horizon of the black hole, as they are everywhere. Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other. But near the horizon of a black hole, it's possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking Radiation."4
Many people find this statement hard to prove, because it would naturally be thought that both particles could not escape the gravitational force of the black hole. But think about it this way, if one particle is an antiparticle, and the other is a normal particle, wouldn't the one that is opposite of the one being pulled into the black hole be able to deflect the black hole's gravitational field, because it is of the opposite polarity as the one that was sucked in? One would think that could be a way of proving Hawking correct with that matter.
And other sources might take the phenomenon known as Dark Matter into hand of this. If Dark Matter were able to slow down a satellite that should be speeding up, wouldn't it be able to speed up something that is supposed to be slowing down? These are the unanswered questions that people will most likely be directing towards the ways to prove Hawking Radiation as law of Black Holes.

Newton's Universal Law of Gravity
Finally, the other theory that ties into black holes happens to be Newton's Universal Law of Gravity. Most people would feel that this should have been explained before the others, as a basis of them, but the reason that it is not being explained before the rest of them, is because it has been disproven, over and over again by the other ones. To some extent, it is true, and that is why it is considered the Universal Law of Gravity, but there are many fallacies that are being questioned in the other theories, that it should come after them, because they will shape and define it.
Now, most people know Newton's Universal Law of Gravity to be:

That is Force is equal to the Gravity times mass one times mass two all divided by the distance squared. G is the universal gravity constant. According to the website where this information was obtained, the units for G happen to be meters squared over kilograms squared, so that it cancels out when Newtons is added. The example that they show is:

"The value of G is so small that explains the reason why forces of a gravitational nature are almost imperceptible in everyday objects and even massive structures and buildings. They are only evidenced in a planetary scale, where masses are in the order of 1030 kg."5

The Mystery of the Black Hole
In this section of the report, we are going to delve into the mystery of the black hole, and the reason that it is eating the star. Or at least the reason our perception is leading us to believe that it is eating the star.
We will start off with the way a black hole operates. As discussed in the previous sections, the Schwartzschild Black Hole will be the one that we look at. We will start off with the start beginning to get sucked in by the Black Hole. This is because of its immense gravitational field. As we all know, Black Holes have the strongest gravitational pull in the universe. So the star will indefinitely be sucked in. As it is sucked in, it will start spiraling downwards due to a concept we discussed in Einstein's General Theory of Relativity. The Curvature of spacetime being the thing that lets us move in a straight line in space. It is because of this that the star looks like it is being drained down a sink, and becoming a giant vortex into the depths of the black hole. That is its quickest route. As the start is being "pulled" into the black hole, it is starting to space out, as discussed in Einstein's Special Theory of Relativity. Because its speed is approximately the speed of light, and it is trying to escape, the processes of Length Contraction, and Time Dilation will begin. As it vortices in past the event horizon, there is no escape from the black hole. The star's gases shall continue to be sucked in until the star has been unwrapped, and completely annihilated. Once it has passed the Event Horizon, then it will be pushed through the obesity, and finally it will hit the singularity. Now, once it is at the singularity, the atoms and all of the gases and materials that have been sucked through will immediately be crushed to the size of a carbon atom. Here is the question though; will the Black Hole ever get full? That is the mystery of the black hole. When will it get full, and that is what leads us onto another topic. As a quick diversion from black holes, I shall quickly talk about White Holes, and Worm Holes.
A White hole is the exact opposite of a black hole. Where a black hole pulls stuff in rapidly, a white hold expels it rapidly. Of course, this is only in theory. We will never know unless we can prove the existence of a particle fountain. Which would also be called a white hole. Once we have proven their existence, then we will be able to prove the theory that what the black hole pulls in, travels through a wormhole, and then is expelled from a white hole. But until then, the Star will just be consumed, and shall sit there for all eternity inside the vastly dense, yet pitifully micro singularity.

Works Cited
, Robert W. Beyond the Event Horizon. Dec. 01, 1993. May 15, 2004.
. Theory: Special Relativity. May 5, 2003. May 15, 2004.

, Robert M. Space, Time, and Gravity. Chicago, Illinois: University of Chicago Press, 1992.
, John. Hawking Radiation. 1994, revised 1997. May 15, 2004.

Academy Site. Newton's Law of Universal Gravitation.