Question

3) A. What is Hawking radiation? Briefly explain how virtual particle pair creation contributes to this process.

B. What is entropy? Consider an object sliding along the floor and coming to rest because of friction. Explain how the entropy of the Universe is increasing during this process.

Answer 1

HAWKING RADIATION

electromagnetic radiation which, according to theory, should be emitted by a black hole. The radiation is due to the black hole capturing one of a particle-antiparticle pair created spontaneously near to the event horizon.

how virtual particle pair creation contributes to this process.

Hawking undertook the task of applying quantum mechanics to black hole dynamics. While his formulation is beyond the scope of this web page, a slightly quantitative and highly qualitative examination of the problem can yield a very good picture of what Hawking discovered. Hawking first attempted to examine the space-time outside the black hole using quantum field theory, which has a very different picture of empty space than the classical definition. His first step was to consider what happens when any field (for example, the electromagnetic field) is quantized in the space-time exterior to a black hole. The quantum mechanical description of a vacuum is space seething with virtual particles and antiparticles whose presence cannot be detected directly. The exact description involves two Hermitian operators, the annihilation and creation operators. Again, the exact formulation is beyond the scope of our treatment of the subject. Nevertheless, a good qualitative interpretation can be obtained by considering the energy-time form of the Heisenberg uncertainty principle:

which states that the product of the uncertainty in the energy and the uncertainty in the time is on the order of h-bar. If we restrict ourselves to very short time scales, the uncertainty in the energy can be very large. In the intermediate stage of virtual pair production, conservation of energy appears to have been violated. This is where the uncertainty principle comes to the rescue, as the particles are created and annihilated on such short time scales as to agree with the uncertainty relation above. When we take delta-t to be large, energy is conserved as we could never have observed the particles in the intermediate stage.

At first glance, this process of virtual particle creation may seem a little phony. With that in mind, we can consider the more tangible case of electric field particle creation. This process can actually be observed by applying a strong electric field across a capacitor in a vacuum:

The quantum mechanical description of the vacuum allows for the creation of the particle/antiparticle pairs, and the electric field tends to separate the charges. If the field is strong enough, the particles tunnel through the quantum barrier and materialize as real particles. The field necessary to accomplish this feat is achieved when the work done to separated the charges by a Compton wavelength equals the energy necessary to create the particles:

It should be noted that conservation of energy is not violated, as the energy it took to create the particles would be precisely equal to the decrease in the energy of the weakened electric field.

So far we have not introduced the possibility of a charged black hole, and will only consider an uncharged, non-rotating black hole. The logical next question is, if there is no electric field outside the black hole, how can these virtual particle/antiparticle pairs be of any consequence to to black holes? The answer is that if a virtual particle pair is created very near to the event horizon, one of the particles could fall the black hole while one escapes to infinity:

To an outside observer, it looks as if the black hole as radiated a particle. Indeed, to conserve energy, the energy it took to create the particle and shoot it off to infinity must have come from the black hole. The particle that did not escape must have had the vexing property of negative energy. This is what happens when a system messes around with particles that classically were not supposed to be created. When looking at the entire system, the total energy of the black hole has decreased and therefore, by Einstein's famous relation, the mass must have decreased. This is the process by which black holes radiate, which is now known as Hawking radiation.

DEFINITION OF ENTROPY

The entropy of an object is a measure of the amount of energy which is unavailable to do work. Entropy is also a measure of the number of possible arrangements the atoms in a system can have. In this sense, entropy is a measure of uncertainty or randomness

OBJECT SLIDING ON THE TABLE

irreversible processes are those which involve dissipative forces. When a book sliding across a table comes to rest due to friction, we describe the frictional force as a dissipative force. While the book was sliding, the average velocity of most molecules was directed the same way. The book had the macroscopic type of kinetic energy we discuss in classical mechanics. As the microscopic form of kinetic and potential energy that we have called thermal energy. The energy has been book slowed and this macroscopic kinetic energy began to disappear, it was replaced by the dissipated, or spread out, not just in a spatial sense (some of the energy is now in the table) but also in velocity space. Whereas at first the distribution of molecular velocities was centered on the velocity of the book, by the time sliding stops, the distribution of velocities is broader and the center shifts to zero. The broadening of the velocity distribution is associated with more accessible microscopic states, and thus with an increase in entropy.

ENTROPY OF UNIVERSE EVER INCREASING

if an isolated system was not in equilibrium,

then it evolved in such a way as to maximize the entropy. In probabilistic terms, the most

likely macroscopic thermodynamic state is the one with the most accessible microscopic

states. Thus a system that is not in thermodynamic equilibrium will display an increase

in its entropy as it evolves towards an equilibrium state. This evolution can be

accomplished in isolation so the entropy increase will not be compensated for by an

entropy decrease elsewhere in the universe. Since evolution towards equilibrium results

in a net increase in entropy of the universe, it is an irreversible process.