Black Holes

Saturday, December 08, 2018

Black Holes - String Theory, Hawking Radiation And Determinism

Black Holes have 3 properties. They have mass, force charges and a rate of spin.
These properties are weirdly similar to those of the elementary particles and so we have to ask the question; what would a black hole with the mass of an elementary particle look like?
General relativity allows black holes of any mass and so this seems like a sensible question. At these distances however, quantum mechanics is required and so a unified theory is needed to answer this kind of question.

Calabi-Yau Space Transitions

When looking at mirror symmetry, we looked at spheres with 2D surfaces that tore (flop-transitions). There are also spheres with 3D surfaces which we cannot visualize, but mathematics allows us to work in higher dimensions. If 3D spheres in Calabi-Yau spaces did collapse, we would actually see the catastrophe which we had avoided previously because 1D strings cannot protect us from the tear. Luckily, we have M-Theory now and this changes things. We have found that the universe may actually be made up of 3-branes and these could wrap around the tear and protect us. The 3D version of the flop-transition, which is called a conifold transition, results in no catastrophe despite the massive change in topology.

The interesting thing about conifold transitions is that when the 3 brane warps around the sphere, it sets up a gravitational field that looks like that of a black hole.

The the mass of a 3-brane or black hole is proportional to the 3D sphere it wraps and so when the 3D sphere is a point, the black hole is massless.

Another interesting point relates to the number of holes in the Calabi-Yau space after a conifold transition. The number of holes dictates the number of possible low energy vibrations which directly influence the elementary particles and force carriers that we see. Flop-transitions do not affect the number of holes, but conifold transitions actually change the number of holes and so changes the particles in a Calabi-Yau space.
When the 2D sphere replaces the 3D sphere at the point of tear, the change means the number of possible vibrations corresponding to massless particles increases by one.

The new pattern of vibration from a conifold transition is the microscopic description of a massless particle into which the black hole becomes. As a Calabi-Yau shape undergoes a transition, a big black hole becomes smaller and smaller until it becomes as massless particle, e.g, a photon. This offers a connection between black holes and the elementary particles.

It shows that we can continuously change Calabi-Yau shapes to change the elementary particles. By changing the SCC and Calabi-Yau shapes, the 5 string theories are further connected by performing conifold transitions.

Black Hole Entropy

Entropy is a measure of disorder. The 2nd law of thermodynamics states that overall entropy must always increase over time in the universe.
In 1970, Jacob Bekenstein suggested that black holes have entropy.

Ask yourself the question, what happens if you clean you desk near a black hole? Usually entropy increases because some energy used to clean your desk is transferred to air particles, which become more disordered. Near a black hole, you could suck these disordered particles into the black hole and then you have reduced entropy because your desk is cleaner? You could actually just suck the hole room into the black hole and you have also reduced entropy this way.

The only way to satisfy the 2nd law of thermodynamics would be for the black hole to have entropy. This entropy would have to increase while matter was sucked into the black hole to offset the decrease in entropy that we see.

Steven Hawking showed that the area of a black hole (the event horizon) increased in any physical interaction and so Bekenstein said that the event horizon provides a measure of entropy.

There are some big problems with this however.

1. Black holes seem like the most orderly object in the universe. How can something with only spin, mass and charge have entropy. This is like trying to muddle up 3 letters in a word, there are simply not enough ways to rearrange the letters to make the word seem disordered.

2. Entropy is a quantum mechanical process while black holes are in the general relativity section of physics.

Hawking had also thought and decided that for a black hole to have entropy, then a black hole must have a non zero temperature. This means it must emit radiation. He found that by adding in some quantum mechanics to black hole physics, it allowed radiation to be emitted. This is what is known as Hawking Radiation.

Quantum field theories let virtual particles pop into and out of existence if we look for short periods of time. If 2 virtual photons are near the event horizon, gravity can suck one photon in and not the other. To someone watching, it looks like you have a black hole emitting photons.

The temperature of the black hole is therefore given by the strength of it's gravity. A stronger field means it is less likely to suck 1 virtual photon in and not the other and so the black hole is cooler.

The problem is normal black holes have temperatures of a billionth of a degree above absolute zero and so the radiation is impossible to detect. As the mass of a black hole decreases however, the temperature increases and a black hole the size of an asteroid would emit as much radiation as hydrogen bombs (gamma rays). We are yet to see this Hawking Radiation however.

String Theory And Hawking Radiation

String theory agrees with Hawking's prediction as well. Theorist found that for extremal black holes (ones with force charges), they could build them up using BPS branes. They therefore could count the number of ways they could leave the overall charge, mass and rate of spin the same while changing the order of the components. This number was compared with the event horizon (the entropy predicted by Hawking and Bekenstein) and this agreed perfectly.

Determinism

Classical determinism states that if we knew everything about every particle in the universe right now, we could perfectly predict the future (including human behavior because this ultimately comes down to the interaction of atoms governed by the laws of physics).
Heisenberg's uncertainty principle undercuts classical determinism because you can't know the precise properties of every particle in the universe. Instead these particles are replaced by wave functions which evolve precisely overtime using equations like the Dirac and Schrodinger equations. Using this, we can predict how the wave function of particles evolve over time and from this know the probability that an event will occur. This is what is known as quantum determinism because wave functions evolve in a deterministic manner.

Quantum determinism may be violated by black holes however. When something falls in a black hole, the wave function may be lost.
This means information in the universe can be lost and determinism will not hold. The uncertainty principle implies that each region of space is full of tiny black holes. If this is the case and information is lost forever, then information may be lost in our universe (very slowly) through these tiny black holes and determinism goes out the window. It is likely that this loss of information through the uncertainty principle is very rare now, but at the start of the universe may have been significant, and so lead to unpredictable early evolution.

Black holes probably radiate however and so determinism may still be able to hold. This is because as they radiate, their masses slowly decrease over time and the event horizon shrinks. This means regions that previously were inaccessible reenter the cosmic playing field.

If wave functions do reemerge as the event horizon shrinks, then information will always be conserved in the universe.

The second black hole mystery is what happens to space-time at the center of a black hole? All mass is drawn towards the center and the huge mass and energy there results in space-time being infinitely curved. No one really has a good guess of what happens here. Some physicists say that matter has no future there and so time comes to an end at the center of a black hole. Some say that it may be a gateway to another universe. Some might even go as far as saying it is a question we should not bother asking because it is pointless. What is certain is that at these short distances and high masses, a unified theory of quantum gravity will be key to have any chance of answering these questions.

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