The dramatic story of black holes is one filled with mysteries, intrigue and paradoxes. From the obscure birth at the beginning of the twentieth century to their rise in importance at the end of the past century, black holes have constantly made us question the very foundations of what we thought we understood about physics. In the past three years, their properties have come to bamboozle us once more, making us question what we thought we understood.
The essential problem is how to talk about gravity and quantum mechanics consistently when discussing black holes. One might ask why this is important . The foremost and fundamental reason is simple. Quantum mechanics underlies all of reality and we understand the framework that describes quantum origin of all the forces except gravity. Thus if you claim to understand the quantum origin of gravity then the first and important test is to describe black holes.This would then bring us one step closer to finding the one framework that describes all the forces.
This story centers around the work of Stephen Hawking; more specifically his work that applied quantum mechanics to black holes. He made the bold prediction that black holes violated the laws of quantum mechanics. Everyone saw no mistakes in his logic, but no one believed the conclusion. About twenty years later, a solution was proposed to this conundrum but physicists in the last three years have realized that this proposed solution does not make sense.
General Relativity
When Newton proposed his famous law of gravity, one of the profound mysteries was how this force was transmitted. Consider the following: suppose we have two masses separated by one meter. The one feels the gravitational force of the other and this force is calculable. But now assume one of those masses moves to two meters away from the second mass. According to Newton's law, the force should reduce to a quarter of its original value; however, this value disturbingly changes instantaneously, i.e., the other mass immediately feels a quarter of the force.
This might not sound like a big problem and in fact physicists learned not to worry about it, but there is indeed a problem. Suppose these two masses are separated by more than a light year. Newton's theory of gravitation predicts the force will be transmitted instantaneously over a distance of a light year. But this can't be so, because this violates the principle that messages can't be sent faster than the speed of light. This is the problem that Einstein realized and took him 10 years to solve. The solution is contained in his magnum opus, “The General Theory of Relativity†or in physicist speak, “GRâ€.
General Relativity is, to my mind, the most austere and regal of theories in all of physics. As one opens up a textbook on the subject, one can almost see the phrase floating in the air, “All ye without any knowledge of Differential Geometry abandon all hope.†And so with a sunken head one turns around and spends a mandatory month familiarizing oneself with tensor calculus and differential geometry before one is allowed to enter the areopagus.
According to Einstein, space-time(3 dimensions of space and 1 dimension of time) is not a stage on which everything happens, but rather another Shakespearean player, intricately involved in the drama. It bends, stretches, undulates and warps according to the way matter is distributed on it. And as one body of mass feels the curvature of spacetime caused by another body, it tries to find the straightest path possible as it moves.For example this is what the earth is doing as it orbits the sun. The warps and bends caused by matter move at the speed of light replacing Newton's instantaneous transmission.
It is the curvature of spacetime that we feel as gravity. To see how, imagine yourself on a trampoline with a massive bowling ball at the center. If you try to move around you will have to make effort not to walk towards the bowling ball. There will a balancing act on your part. With this analogy in mind, Newton described the feeling of being pulled towards the massive object as the force of gravity, but Einstein understood that the feeling came from the curvature of the trampoline caused by the bowling ball at the center.
It is in the General Theory of Relativity that this tale begins, because a few months after Einstein presented his theory, Karl Schwarzschild found the first solution to the fundamental equation of General Relativity. His solution had a rather strange feature. He noted that if there was a body that was massive and dense enough, there was a distance beyond which one could not escape the gravitational pull of the body and would be inevitably pulled to the center of the body. This was the first discussion of what we now call black holes.
The Murky Realm of Quantum Mechanics
As strange as General Relativity may seem, it is nothing compared to the conceptual shift needed to understand quantum mechanics. To get an inkling of what I mean by this, consider the fact that one genius worked for about a decade to come up with GR, while quantum mechanics required 10 geniuses over a decade and a half to grasp . Let us then begin with the fundamental equation, Schrödinger’s equation, drum rolls please….here it is:
You are not expected to understand the symbols – all that I ask is you take note of the symbol . This is what physicists call the wave function. The equation above describes how it changes in space and time. What exactly is the wave function? Almost a century after it was first proposed no one really knows if it is just a mathematical tool or if it represents something real out there in the world (I will be damned if I try to take a stab at that question). This brings us to what I will call the first axiom of quantum mechanics namely
“The wave function contains all the information about the system you are allowed to know.â€
But what do I mean by “information� Roughly speaking, its square , 
, gives you the probability distribution of seeing a particle at any point in space at a certain time. In other words, quantum mechanics does not tell you that a particle is at some position at a certain time, rather it tells you the probability of seeing it at that position at a certain time. If that does not sound obvious, welcome to the club, I will be your guide.
The right answer to the interpretation of the wave function won the physicist, Max Born, a Nobel prize in 1954. Even Schrödinger did not know how to interpret his own equation! I can almost you hear you protest, “But I want to know where the particle is.†Ah! Remember my first axiom! The wave function contains all the information you are allowed to know,and since all it gives you are probabilities, those probabilities are all you are allowed to know. End of story.
The wave function allows for systems to be in extremely peculiar states. Imagine a coin; it has a “heads†side and a “tails†side. It makes no sense to say a coin shows both heads and tails. But it does in quantum mechanics. This strange state of being both heads and tails is called a superposition state.
The superposition state is what makes quantum mechanics unintuitive because it has no classical analogue. So asking completely common sense questions, like “what side of the coin was right side up before I looked ?†are not appropriate.
These superposition states have one more puzzling behavior that is relevant to our story and that is quantum entanglement. This behavior was first predicted to happen by Albert Einstein in his attempt to show that quantum mechanics could not be a complete theory of the atomic world.
Suppose I have two “quantum pens†that are both in the superposition state. For this thought experiment, superposition will mean the pen can be in a quantum state of being black and red at the same time (the sum of being black and red where black and red are both solutions to Schrödinger’s equation). Next, I make the pens interact(entangle). This interaction could be as simple as the pens touching each other. I keep one pen and give the other to my friend John. John then goes to Alpha Centauri, our nearest star about 4 light years from earth. On earth, I now observe my quantum pen and find it to be a black pen. Quantum mechanics tells me John is guaranteed to find his pen to be red. Then suppose he comes back and we put the pens in superposition states. They interact and again he goes to Alpha Centauri. If I observe the pen being red, quantum mechanics guarantees John will be observe his pen to be black. Whatever color I observe he is guaranteed to see the other.
Quantum Mechanics and General Relativity: Profiles of an uneasy friendship
The information paradox: Quantum Mechanics spoils the party
Between 1915 and 1958, mathematicians and physicists realized that at some distance around extremely dense objects,there is a surface called the event horizon. Beyond this distance, one's fate is sealed and nothing, not even light, can escape. Stephen Hawking and Roger Penrose did beautiful work showing that these regions of space-time, now called black holes, were an inevitable feature of General Relativity. These results are called the singularity theorems. During 1974-75 Stephen Hawking went one step further. He did what no one had dreamed of doing. He applied quantum mechanics near the event horizon of the black hole.
What did he find? He found that, because of the intense gravitational fields, a pair of matter and antimatter particles can be created from the vacuum (empty space around black hole). This is because gravitational fields have energy in them and they excite particles out of the vacuum.
These particles are in an quantum mechanical entangled state, just like the pens we previously discussed. One of the two gets sucked into the black hole while the other escapes and is observed by some observer outside. As a consequence black holes are not actually completely black. What the outside observer sees is called Hawking radiation. The outside observer sees hawking radiation to be classical radiation (think of light from a bulb).
This phenomena has the added effect that the black hole actually shrinks in size. This is because the gravitational energy of the black hole is expended to take the antiparticle and particle pair out of the vacuum and by E=mc2 this reduces the mass. Eventually no mass will be left.
But trouble is on the way. Imagine some particles escape while others get trapped into the interior past the event horizon where they are forever lost to the outside world and all the while the black hole is getting smaller. At some point, the black hole completely disappears out of existence, taking with it all the other particles that went in it. Now comes the question, can we reconstruct the whole process just by analyzing the particles that escaped?
In other words, is there a way of figuring out what the initial state of the black hole was from the observed hawking particles on the outside? Hawking's calculation showed the answer to be no, because the outside observer just sees particles that have no quantum mechanical properties, although the initial state of the particles was quantum mechanical. This means we have lost information, or more precisely, quantum information. Now you may or may not find this troublesome but what has just happened is that we applied quantum mechanics and got a result forbidden by quantum mechanics.
Holographic Principle and Black Hole Complementarity to the Rescue
This paradox bothered physicists like Gerard't Hooft and Leonard Susskind, who were inspired to propose a radical solution. The information needed to specify the state of hawking quanta, like the amount of entropy of the hawking particles created, is in some sense both inside (beyond the event horizon) the black hole and outside.
Roughly speaking, all the information needed to specify everything happening inside the black hole can be entirely encoded on a surface near the event horizon and the outside observer sees the information accumulate on this surface. This outside observer is able to account for all the information after the last hawking radiation is emitted just before the black hole disappears, while any infalling observer sees all the information located on the inside. The inside observer (one past the event horizon) can't send messages out thus any message sent in from the outside is destroyed at the singularity with the unfortunate soul. So the two observers will never find out that they do not agree on where the information is.
The idea that the information is both inside and outside is called black hole complementarity. The information is encoded in two different ways. The claim that three dimensional information (what is inside the black hole) can be entirely encoded on a two dimensional surface (surface on the event horizon) is called the holographic principle.
Firewalls
What was not done explicitly was to show how to recover the information from the point of view of the outside observer. With the firm belief that information was preserved, i.e, not lost when the black hole disappeared, a group of researchers, Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully set about trying to do this. They ran into snag that never went away. The snag is now called the AMPS paradox (name comes from the first letters of the authors' last names). The source of the snag is again quantum mechanics.
Remember that when the pair of matter and antimatter particles are created, they are in an entangled state. Now imagine waiting until the black hole has lost half its mass and emits a hawking particle (one of the pair of particles). Not only is it fully entangled with its other half, but it turns out that it is fully entangled with other hawking particles that were produced in the early history of the black hole, as authors of the, AMPS paradox showed. This turns out to be forbidden by quantum mechanics. A particle can not be fully entangled with more than one particle; this is called the monogamy of entanglement.
A way out of this is to suggest that once the pair of antimatter and matter particles are created there is some process that destroys the entanglement. This would release a high amount of energy creating a huge wall of high energy particles from the vacuum(around the event horizon) called a firewall.
One might then be tempted to throw one's hand in the air in resignation and accept the existence of firewalls – except that this firewall would represent a violation of Einstein's equivalence principle in General Relativity. The equivalence principle says that free falling in a gravitational field is equivalent to floating in empty space which would clearly not be the case near the event horizon.
Why? One of the particles created at the event horizon free falls into the black hole and so by the equivalence principle they other should float past a smooth and empty spacetime away from the black hole but instead a wall of high energy particles is created from the vacuum around the event horizon.
Where are we left?
Thus we have two immensely unsavory choices: I) violate Quantum Mechanics or II) violate General Relativity. Quantum mechanics is used to understand a vast array of our world from cellphones to laptops while General Relativity is the framework we use to understand Cosmology and Astrophysics. We can't give up either.
We must understand how these two theories work together ,and anyone that figures this out has potentially found the much sought after theory of quantum gravity. Thereby bringing us that much closer to unifying gravity with all the other forces.
About the Author:
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Amara Katabarwa is a PhD candidate in the Physics and Astronomy Department at the University of Georgia studying His research focus is understanding Decoherence in Quantum Circuits and near term application of first generation Quantum Computers. In his free time he likes to read or guiltlessly laze about. He can be contacted at akataba@uga.edu. |
