The black hole’s gravitational field is described by general relativity, while the surrounding vacuum space time is described by quantum field theory. The quantum evaporation process is similar to pair production in a strong magnetic field due to vacuum polarization. In the Fermi sea populated by virtual pairs of particles-antiparticles which create and annihilate themselves, there are four possible processes. Let us have a look at each of them one by one.

Process I – In this process some virtual pairs of particle-antiparticle emerging from the quantum vacuum just annihilate outside the horizon.

Process II – In process II, some pairs of particle-antiparticle are broken, and then the particle escapes the black hole, while the antiparticle is captured by the black hole.

Process III – In process III, it is same as the process II in which some pairs of particle-antiparticle are broken, and then antiparticle escapes the black hole, while the particle is captured by the black hole.

Process IV – In the process IV, some pairs of particle-antiparticle produced in the vicinity of the black hole disappear completely in the event horizon.

According to the calculations, process II is dominant among all the processes, due to the gravitational potential which polarizes the quantum vacuum. As a result, a black hole radiates particles with a thermal spectrum characterized by a blackbody temperature given by the formula :

here h is Planck’s constant. From the above formula we can notice one thing that the temperature is negligible for any astrophysical black hole with mass comparable or greater than the mass of Sun. But for mini-black holes with mass about 10^15 g, the Hawking temperature is 10^12 K.

Since the black hole radiates away energy, it will loose energy and evaporates on a timescale approximately given by the equation :-

Thus, for mini-black holes whose mass is about 10^15 g, will evaporate in about 10^10 years which is on a timescale shorter than the age of the universe. It is also possible that some of them evaporate now and give rise to a huge burst of high energy radiation. But nothing similar has ever been observed till now. Therefore, such an observational constraint limits the density of mini-black holes to be less than about 100/(lightyear)^3.

Now talking about the entropy of black holes, the black hole entropy is given by

where kB is Boltzmann’s constant. On simplifying the above formula we will get S = 10^77 kB (M/Mo )² for a Schwarzschild black hole. Whereas the entropy of a non-collapsed star like the Sun is approximately 1058 kB, which means the black holes are huge entropy reservoirs.

So by Hawking radiation, the event horizon area of a black hole decreases, which is in violation of the Second Law of black hole thermodynamics. Therefore, black hole’s second law of thermodynamics has to be generalized to include the entropy of matter in exterior space-time. Then, the total entropy of the radiating black hole will become S = SBH + Sext and, since the Hawking radiation is thermal, therefore Sext will increase, so that eventually S is always a non-decreasing function of time.

Mini-black holes are very rare, and it is even possible if they do not exist at all in the real universe because the big bang could not have produced such fluctuations, but they represent a major theoretical advance towards a better understanding of the link between quantum theory and gravity.

Thus, for mini-black holes whose mass is about 10^15 g, will evaporate in about 10^10 years which is on a timescale shorter than the age of the universe. It is also possible that some of them evaporate now and give rise to a huge burst of high energy radiation. But nothing similar has ever been observed till now. Therefore, such an observational constraint limits the density of mini-black holes to be less than about 100/(lightyear)^3.

Now talking about the entropy of black holes, the black hole entropy is given by

where kB is Boltzmann’s constant. On simplifying the above formula we will get S = 10^77 kB (M/Mo )² for a Schwarzschild black hole. Whereas the entropy of a non-collapsed star like the Sun is approximately 1058 kB, which means the black holes are huge entropy reservoirs.

So by Hawking radiation, the event horizon area of a black hole decreases, which is in violation of the Second Law of black hole thermodynamics. Therefore, black hole’s second law of thermodynamics has to be generalized to include the entropy of matter in exterior space-time. Then, the total entropy of the radiating black hole will become S = SBH + Sext and, since the Hawking radiation is thermal, therefore Sext will increase, so that eventually S is always a non-decreasing function of time.

Mini-black holes are very rare, and it is even possible if they do not exist at all in the real universe because the big bang could not have produced such fluctuations, but they represent a major theoretical advance towards a better understanding of the link between quantum theory and gravity.

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