If the Four Laws of Black Hole Physics looked familiar, it’s because they sound just like the Four Laws of Thermodynamics, which are:
|The Four Laws of Thermodynamics|
|0||The temperature T of a system in thermal equilibrium has the same value everywhere in the system.|
|1||The change in energy of a system is proportional to the temperature times the change in entropy.dE = T dS|
|2||The total entropy of a system can only increase, never decrease.|
|3||It is impossible to lower the temperature T of a system to zero through any physical process.|
There seems to be a direct correspondence between the properties of a classical thermodynamic system, and the properties of a black hole, shown in the table below
|Thermodynamic system||Black hole|
|temperature||surface gravity at horizon|
|energy||black hole mass|
|entropy||area of horizon|
A black hole spacetime seems to behave like a thermodynamic system. How could this be true? This is spacetime geometry, after all, not a cylinder of gas or a pot of liquid. The importance of this apparent thermodynamic behavior of black holes was made undeniable when black hole radiation was discovered by Hawking.
Black hole radiation, known as Hawking radiation, comes about because relativistic quantum field theory is invariant under Lorentz transformations, but not under general coordinate transformations. In flat spacetime, two observers moving at a constant velocity relative to one another will agree on what constitutes a vacuum state, but if one observer is accelerating relative to the other, then the vacuum states defined by the two observers will differ.
This idea, when extended to the spacetime of a black hole, leads to the conclusion that to an observer who stays at a fixed distance from a black hole event horizon, the black hole appears to radiate particles with a thermal spectrum with temperature (in units with GN=c=1) T=1/8pMkB, where kB is Boltzmann’s constant and M is the black hole mass.
Since plane waves and Fourier transforms are at the heart of relativistic quantum field theory, this effect can be illustrated using a classical plane wave, without even appealing to quantum operators. Consider a simple monochromatic plane wave in two spacetime dimensions with the form
An observer travelling in the x-direction with constant velocity b perceives this plane wave as being monochromatic but the frequency w is Doppler-shifted:
An observer travelling in the x-direction with constant acceleration does not perceive this plane wave as being monochromatic. The accelerated observer sees a complicated waveform:
This wave as perceived by the accelerated observer is a superposition of monochromatic waves of frequency n with a distribution function f(v) that, as shown below
appears to be a thermal distribution with temperature T=a/2pkB. The result from this simple example matches Hawking’s black hole result if the acceleration is related to the black hole mass by a=1/4M. And indeed, the acceleration at the event horizon of a black hole of mass M does satisfy a=1/4M. Why does this work so well? Because an observer held at a fixed distance from the event horizon of a black hole sees a coordinate system that is almost identical to that of an observer undergoing constant acceleration in flat spacetime.
But don’t be misled by this to think that the full black hole radiation calculation is as simple. We’ve neglected to mention the details because they are very complicated and involve the global causal structure of a black hole spacetime.
Conservation of energy still applies to this system as a whole, so if an observer at a fixed distance sees a hot bath of particles being radiated by the black hole, then the black hole must be losing mass by an appropriate amount. Hence a black hole can decrease in area, through Hawking radiation, through quantum processes.
But if the area is like entropy, and the area will decrease, doesn’t that mean that, in violation of the Second Law of Thermodynamics, the entropy of a black hole can then decrease?
No — because the radiated particles also carry entropy, and the total entropy of the black hole and radiation always increases.
Where does the entropy come from?
One of the great achievements of quantum mechanics in the 20th century was explaining the microscopic basis of the thermodynamic behavior of macroscopic systems that were understood in the 19th century. The quantum revolution began when Planck tried to explain the thermal behavior of light, and came up with the concept of a quantum of light. The thermodynamic properties of gases are now well understood in terms of the quantized energy states of their constituent atoms and molecules.