Thermodynamic of Black Holes & Universe

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; by this I mean that all their neutronness is wiped out. However their total mass-energy remains.

Another way of stating this is that outside of the event horizon all properties of the matter that formed it are gone except for the total mass-energy, rotation, and electric charge: this is sometimes called the Black Hole Has No Hair theorem.

The total mass-energy is manifested as the curvature of spacetime around the singularity.

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. The possible wave states are very similar to the standing waves on a circular drum head that we saw earlier; they aren't exactly the same because the waves exist in three dimensions instead of just the two of the drumhead, but they are very close to the same.

Note that I just said "three dimensions." This is correct; we are using non-relativistic quantum mechanics.

The energy represented by a particular wave state is related to the frequency and amplitude of its oscillation. As we saw for the standing waves on a drumhead, the higher "overtones" have a higher frequency and thus these Quantum Mechanical 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 calculated 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.

But we have seen that the entropy is just a measure of the probability. Thus we can calculate the entropy of a black hole.

We have also seen that the entropy measures the heat divided by the absolute temperature. The "heat" here is just the total mass-energy of the black hole, and if we know that and we know the entropy, we can calculate a temperature for the black hole.

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

In a previous section we saw that 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.

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, strengthens our belief that black holes radiate their energy away and evaporate.

Technical note: if we measure the mass-energy M of a black hole in units where the mass of our Sun is one, then the absolute temperature of the black hole is 6 × 10^-8 / M Kelvin and its lifetime, in seconds, is: 10⁷¹ M³.

Consider the universe. It has a size of about 15 billion light years or so. It also has a total amount of mass-energy. If we represent this mass-energy as quantum mechanical standing waves, just as we did for black holes, we can calculate the total entropy of the universe.

It turns out that the entropy of either a black hole or the universe is proportional to its size squared.

Thus for a given amount of total mass-energy, the larger the object the higher the entropy.

But the universe is expanding, so its size is increasing. Thus the total entropy of the universe is also increasing.

This leads us to the idea that the Second Law of Thermodynamics may be a consequence of the expanding universe. Thus cosmology explains this nineteenth century principle.

Put another way, recall that we have realised that the direction of time, "time's arrow," can come either from the fact that the universe is expanding or from the Second Law of Thermodynamics. We have now found a relationship between these two indicators of the direction of time.

It is amusing to speculate about what will happen to the Second Law of Thermodynamics if the universe is closed, so that at some point the expansion stops and reverses.

Even more wild is the idea that if the expansion of the universe determines the direction of time's arrow, then if the universe starts to contract the direction of time will also reverse.