Black hole thermodynamics

In physics a special field has been created entirely centered around the thermodynamics of black holes. Within black holes many of the physical laws which we accept should break down, including those of thermodynamics.

Problem one

Classical thermodynamics states as its second principle that entropy is an always increasing function in a closed system - and the universe is a closed system, as nothing can escape it. So we ask; what happens to the information when a particle falls inside a black hole? Remember that only three parameters are required to fully describe a black hole: its mass, its electrical charge, and its angular momentum. But, in order to describe a physical system, we need other information, especially entropy, which is a measurement of its disorder, losing this information would be a violation of thermodynamics second principle.

We imagine a black hole as the singularity in the center surrounded by a spherical event horizon.We know that when a black hole is created by a collapsing neutron star that the neutrons are crushed out of existence; They cease to be neutrons. We have seen that all matter has a wave aspect, and Quantum Mechanics describes the behavior of these waves. So, we shall think about representing the mass-energy inside the event horizon as waves. Now, what kind of waves are possible inside the black hole? The answer is standing waves, waves that "fit" inside the black hole with a node at the event horizon. We know that the energy represented by a particular wave state is related to the frequency and amplitude of its oscillation, higher frequency waves contain more energy.

Assume that the total mass-energy inside the event horizon is fixed. So, we have various standing waves, each with a certain amount of energy, and the sum of the energy of all these waves equals the total mass-energy of the black hole. There are a large number of ways that the total mass-energy can distribute itself among the standing waves. We could have it in only a few high energy waves or a larger number of low energy waves. It turns out that all the possible standing wave states are equally probable. Thus, we can calculate the probability of a particular combination of waves containing the total mass-energy of the black hole the same way we calculate the probability of getting various combinations for dice. Just as for the dice, the state with the most total combinations will be the most probable state.

Entropy is just a measure of the probability. and can be expressed as:

Where A is the area of the black hole, k the Boltzman's constant, h the Planck's constant, c the speed of light and G the gravitational constant.

Thus we can calculate the entropy of a black hole which solves our first problem. However entropy measures the heat divided by the absolute temperature.(In this context "heat" is just the total mass-energy of the black hole.) If we know that and we know the entropy, we can calculate a temperature for the black hole. But, thermodynamics states that all bodies with temperature above absolute-zero radiate heat and we know that nothing escapes the event horizon.

Problem two

Any body with a temperature above absolute zero will radiate energy. And we have just seen that a black hole has a non-zero temperature. Thus thermodynamics says it will radiate energy and evaporate. We can calculate the rate of radiation for a given temperature from classical thermodynamics. We can calculate the rate of radiation for a given temperature from classical thermodynamics. We can also use the following formula to calculate the black hole's temperature:

So, how is this possible? Nothing can get across the event horizon, so how can the black hole radiate? The answer is via virtual pair production.

Consider a virtual electron-positron pair produced just outside the event horizon. Once the pair is created, the intense curvature of spacetime of the black hole can put energy into the pair. Thus the pair can become non-virtual; the electron does not fall back into the hole. There are many possible fates for the pair. Consider one of them: the positron falls into the black hole and the electron escapes. According to Feynman's view we can describe this as follows:

The electron crosses the event horizon travelling backwards in time, scatters, and then radiates away from the black hole travelling forwards in time.

Using the field of physics that calculates virtual pair production etc., called Quantum Electrodynamics, we can calculate the rate at which these electrons etc. will be radiating away from the black hole. The result is the same as the rate of radiation that we calculate using classical thermodynamics. The fact that we can get the radiation rate in two independent ways, from classical Thermodynamics or from Quantum Electrodynamics, is another indicator of the strngth attributed to this theory; that black holes radiate their energy away and evaporate.

So, as Hawking realised, "we can apply all of Thermodynamics to a black hole."