| From: Greg L. ® | 13/01/2001
1:01:19 |
| Subject: Black Hole Thermodynamics | post id:
205437 |
| A lot of people recently have
been asking questions relating to Black Holes and in particular, Hawking
Radiation. I've come across an interesting section in Clifford Will's
article 'The Renaissance of General Relativity' that discusses this
phenomenon in some detail. As our usual relativity and BH man (Chris)
seems to be on holidays, I will reproduce the section here for everyone's
info. By 1972, enough was known about both the specific Schwartzchild and Kerr Black Holes, and about horizons in general, that it became possible for James Bardeen, Brandon Carter, and Stephen Hawking to condify many of their properties into a set of laws that were very suggestive. The four laws of BH mechanics were: 1) The zeroth law: In a stationery situation, or equilibrium, the surface gravity k of a black hole is constant over the horizon. 2) The first law: In a transformation from one state to a nearby state, the energy of the system changes by delta E = (c^2/G)(kdeltaA/8pi) + W where A is the surface area of the horizon and W is the total of any work done in changing the rotation of the BH, and any work done on any matter that may be outside the BH. 3) The second law: During any process in an isolated system, the total area of all horizons is non-decreasing. 4) The third law: It is impossible for any finite sequence of steps to bring the surface gravity to the value zero (for the Kerr solution with the critical rotation rate - a naked singularity-k is zero.) These laws are in exact paralell with the laws of thermodynamics, with k playing the role of temperature, and the area playing the role of entropy. In 1972, Jacob Berkenstein suggested that, because of quantum mechanics, this was more than just a coincidence, and that black holes could have real entropy. His argument was based on the notion that black holes have 'no hair.' When a particle falls down a BH, information (hair) is lost, but, according to the 'information theory' approach to thermodynamics, the missing information is equivalent to entropy. | |
| From: Greg L. ® | 13/01/2001
1:16:27 |
| Subject: re: Black Hole Thermodynamics | post id:
205443 |
| Therefore,
there must be an entropy associated with the BH. Berkenstein went on to
show that if entropy was proportional to the area, as suggested by the
four laws, the proportionality constant could be estimated by noticing
that the Heisenberg uncertainty principle applied to the injection of a
particle into a BH provided a minimum amount by which the area of the BH
could be increased during any injection. By equating this to the entropy gained by the hole upon loss of information about the injected particle, he arrived at a value that was surprisingly close to the correct proportionality constant. By 1974, Hawking had applied the mathematics of quantum field theory to BH spacetimes, and proved that BH's could evaporate by the creation of particle-antiparticle pairs from the vacuum near the horizon. The emitted particles were found to have a thermal, or blackbody spectrum with a precise temperature given by T=hk/(2pi)(k)(c) where h is Planck's constant and k is Boltzmann's constant. The associated entropy was indeed proportional to the area of the horizon, and Hawking's calculation gave the precise proportionality constant. The entropy S was given by S={(1/4)(kc^3)/(G)(h)}A. For a Shwartschild BH, the area and hence the entropy are proportional to M^2, and the temperature is inversely proportional to M. Because the BH can emit particles with a thermal spectrum at a definite temperature, and can of course absorb such particles, it can be in thermal equilibrium with external systems. Thus the four laws of BH mechanics are truely thermodynamic laws. It is now possible to state, for example, a generalised second law of thermodynamics: In any process in a closed system the total entropy of matter and black holes can never decrease. This merging of the laws of gravitation, quantum mechanics and thermodynamics is one of the stunning achievements of modern theoretical physics. | |
| From: Greg L. ® | 13/01/2001
1:27:31 |
| Subject: re: Black Hole Thermodynamics | post id:
205448 |
| This emission or
evaporation process will ultimately cause a BH to decay away entirely.
However, for most astrophysical purposes, it is unimportant. The reason is
that the temperature of a solar mass BH is less than a microkelvin,
and its lifetime against the evaporation is greater than 10^70
years. So for all intents and purposes, BH's formed by stellar
collapse do not evaporate. However, Hawking and others suggested the
possibility that 'mini' BH's were created in the big bang. For mini BH's
of a mass of about 10^12 kg, the lifetime will be about 20 billion years,
so they would be in the final stages of evaporation today. Because their
masses would now be very small, their temperatures and evaporation rates
would be very high, so they would emit a final burst of high-energy gamma
rays and elementary particles that would be detectable. Such bursts have
not been detected to date, a result that sets a stringent upper limit on
the possible number of such mini BH's. From The Renaissance of General Relativity by Clifford Will, The New Physics, Cambridge University Press, 1989. | |