Black Hole Thermodynamics



Entropy of a black hole

Black holes confront us with a fundamental problem : 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 entropy by falling into a black hole is a violation of the second principle of thermodynamics. This law states that entropy is an always increasing function in a closed system - and the universe is a closed system, as nothing can escape it.

In 1972, Stephen Hawking showed that the area of the horizon of a black hole can not decrease.
(note : what we call here 'the area of the black hole' is in fact, the surface defined by its horizon)

Merging of black holes
In particular, in the case of the merging of two black holes, the area of the final horizon can't be less than the sum of the areas of their initial horizons.

Hence, Jacob Bekenstein identified the area of the black hole, which can never decrease, with entropy : if this area represents a measure of the entropy of the black hole, the second principle is no longer violated.

The entropy of a black hole is 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.

Fine, but a new problem arises : if the black hole has an entropy, it must have a temperature, too.
Every body which has a temperature is able to radiate energy, according to a spectrum which must correspond to its temperature. But, in its classical definition, nothing can escape a black hole.


The extraction process of Penrose

Yet, at the end of the 1960's, physicist Roger Penrose had proposed a way to extract energy from a Kerr black hole.
Let us remember that, once inside the ergosphere, nothing can stay still : it is dragged by the spinning of the black hole. One result of this fact - the calculation is a bit laborious - is the existence of negative energies inside the ergosphere.

If a particle enters the ergosphere and breaks apart into two new particles, one of the two particles is able to fall inside the black hole with a negative energy.

Penrose extraction process
The energy of the outgoing particle increases, as E2+En = E1
(energy conservation).

Of course, such a process is only available on very precise trajectories. From a physical point of view, it's a very unlikely phenomenon.

Taken as a whole, it seems like the energy has increased. But energy is not a free quantity. If the energy of the outgoing particle has increased, it means that the energy of the black hole has been cut down.
There is a limit for the energy that can be extracted from a black hole : when it has no more rotation, the exosphere will disapear, and this process will no longer be useful.
The calculation shows that we can extract a theoretical maximum of 29% of the total energy of a spinning black hole.


Fluctuations of the quantum vacuum

A vacuum is a place which is anything but empty ! In fact, there is a continuous creation - and destruction - of pairs of particles/antiparticles (which are called virtual particles, because they can't be directly measured by particle detectors) during very short times.
This fact is possible with help of Heisenberg's uncertainty principle : the energy of the vacuum, that we suppose equal to zero, can only be defined with an uncertainty ΔE during a time ΔT, with the relationship ΔE*Δ>T > h/4π (h is the Planck's constant).
Pairs of particles/antiparticles, with a ± Δ>E energy, are thus constantly created, for a lifetime of about h/ΔT. Notice that one of the two particles has a positive energy, and the other particle has a negative one, so the total energy remains unchanged.
This phenomenon is called quantum fluctuations of the vacuum .


The Hawking radiation

Let us imagine such a creation of particles in the vicinity of the horizon of a black hole.
If one of these antiparticles falls beyond the horizon, the remaining particle can escape to a large distance from the hole, carrying a positive energy. As it can not annihilate with its antiparticle, it becomes a real particle, and for a distant observer, it will seem to have been emitted by the black hole.
In order to compensate for this energy, carried away by the particle, the black hole has to lose the same amount of energy.

Note : the opposite phenomenon is impossible.
If the particle which falls back into the black hole carries a positive energy, then the other particle will also have to fall, because a particle can not exist in our universe with a negative energy.

Quantum fluctuations
Pairs of virtual particles/antiparticles are continuously created in the vicinity of the horizon of the black hole.
Among these pairs, some of them won't be able to annihilate, because one of the particles has fallen back into the black hole.
The outgoing particle carries energy with it, giving the illusion that the black hole radiates.

So, an evaporating radiation does appear, coming from the black hole. The calculation shows that this radiation exhibits a typical black body spectrum.
The heavier the black hole is, the lower is its temperature. A stellar black hole of 6 solar masses has a temperature of 10-8 K.
Indeed, the smaller the black hole is, the shorter is the distance for the virtual particle to travel before it becomes a real particle. The emission rate and the temperature are hence higher for a small - ie light - black hole.

Since the black hole radiates, it evaporates. Hence its lifetime is finite. For our 6 solar mass black hole, its lifetime is about 2*1068 years.

The temperature of a black hole whose mass is M is given by : -
where - is the reduced Planck constant -

And its lifetime is -x 1066 years

Obviously, with such a weak value, it is completely impossible to try to measure the radiation as it escapes the black hole. We can't have a direct experimental confirmation.

At the end of its life, the mass of the black hole becomes smaller and smaller, and hence its temperature tends towards infinity. The black hole disappears in a fantastic explosion. The current physics is unable to explain the last phases of the evaporation of the black hole.

Non eternal black hole
The Penrose diagram of the black hole is shown on the left : the singularity has a beginning and an end. After its end, the universe again becomes the same as is was before.

There is another explanation for this radiation, more rigorous, and it was found by Hawking himself in 1975 ; it is based on an analogy with the Unruh radiation.
William Unruh showed in 1976 that a uniformly accelerating observer in a vacuum will find himself surrounded by a thermal bath, the "Unruh radiation", whose temperature T is proportional to acceleration γ (this effect is quite weak : T~1 K for γ =1019m/s2).

This effect implies a close relationship between acceleration, gravitation, thermodynamics and quantum mechanics.
We won't go into details of the necessary calculations ; they rest on a semi-classical approach, with a quantization of an existing field.

With the Hawking radiation, there is a decreasing of the area of the black hole, due to the decreasing of its mass with the evaporation. We've seen that this area is comparable to the entropy, but as the loss of entropy of the black hole is exactly compensated with the increase of entropy of the thermal radiation, there is no violation of the second principle of thermodynamics.


References :
An introduction to black hole evaporation (J. Traschen)
Introductory Lectures. on Black Hole Thermodynamics. (T. Jacobson)
The Thermodynamics of Black Holes (R. Wald)
Quantum Vacuum Structure and Cosmology (Rafelski, Labun & Hadad)
The entropy of black holes : a primer (T Damour)