Hawking Radiation - Emission Process

Emission Process

Hawking radiation is required by the Unruh effect and the equivalence principle applied to black hole horizons. Close to the event horizon of a black hole, a local observer must accelerate to keep from falling in. An accelerating observer sees a thermal bath of particles that pop out of the local acceleration horizon, turn around, and free-fall back in. The condition of local thermal equilibrium implies that the consistent extension of this local thermal bath has a finite temperature at infinity, which implies that some of these particles emitted by the horizon are not reabsorbed and become outgoing Hawking radiation.

A Schwarzschild black hole has a metric

The black hole is the background spacetime for a quantum field theory.

The field theory is defined by a local path integral, so if the boundary conditions at the horizon are determined, the state of the field outside will be specified. To find the appropriate boundary conditions, consider a stationary observer just outside the horizon at position . The local metric to lowest order is

which is Rindler in terms of and . The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration diverges as .

The horizon is not a special boundary, and objects can fall in. So the local observer should feel accelerated in ordinary Minkowski space by the principle of equivalence. The near-horizon observer must see the field excited at a local inverse temperature

this is the Unruh effect.

The gravitational redshift is by the square root of the time component of the metric. So for the field theory state to consistently extend, there must be a thermal background everywhere with the local temperature redshift-matched to the near horizon temperature:

The inverse temperature redshifted to r' at infinity is

and is the near-horizon position, near, so this is really

So a field theory defined on a black hole background is in a thermal state whose temperature at infinity is

This can be expressed more cleanly in terms of the surface gravity of the black hole; this is the parameter that determines the acceleration of a near-horizon observer. In natural units, the temperature is

where is the surface gravity of the horizon. So a black hole can only be in equilibrium with a gas of radiation at a finite temperature. Since radiation incident on the black hole is absorbed, the black hole must emit an equal amount to maintain detailed balance. The black hole acts as a perfect blackbody radiating at this temperature.

In SI units, the radiation from a Schwarzschild black hole is black-body radiation with temperature

where is the reduced Planck constant, c is the speed of light, k_B is the Boltzmann constant, G is the gravitational constant, and M is the mass of the black hole.

From the black hole temperature, it is straightforward to calculate the black hole entropy. The change in entropy when a quantity of heat dQ is added is

The heat energy that enters serves to increase the total mass, so

.

The radius of a black hole is twice its mass in natural units, so the entropy of a black hole is proportional to its surface area:

Assuming that a small black hole has zero entropy, the integration constant is zero. Forming a black hole is the most efficient way to compress mass into a region, and this entropy is also a bound on the information content of any sphere in space time. The form of the result strongly suggests that the physical description of a gravitating theory can be somehow encoded onto a bounding surface.