If the Four Laws of Black Hole Physics looked
familiar, it's because they sound just like the Four Laws of Thermodynamics,
|The Four Laws of Thermodynamics
T of a system in thermal equilibrium has the same value
everywhere in the system.
in energy of a system is proportional to the temperature
times the change in entropy.
dE = T dS
entropy of a system can only increase,
||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
||black hole mass
||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
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 traveling in the x-direction with constant
velocity b perceives this plane wave as being monochromatic but
the frequency w is Doppler-shifted:
An observer traveling in the x-direction with
constant acceleration a 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 area is like entropy, and the area can
decrease, doesn't that mean that the entropy of a black hole can therefore
decrease, in violation of the Second Law of thermodynamics? 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.
So what is the microscopic physics that underlies
the thermodynamic properties of black holes? String theory suggests
an answer that we will explain in the next section.