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Black-Hole Thermodynamics
PHYS 4315
R. S. Rubins, Fall 2009
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Quantum Fluctuations of the Vacuum
• The uncertainty principle applied to electromagnetic fields
indicates that it is impossible to find both E and B fields to be
zero at the same time.
• The quantum fluctuations of the vacuum so produced cannot
be detected by normal instruments, because they carry no
energy.
• However, they may be detected by an accelerating detector,
which provides a source of energy.
• The accelerating observer would measure a temperature of
the vacuum (the Unruh temperature), given by
TU = aħ/2πc.
Notes
i. For an acceleration of 1019 m/s2, TU ~ 1 K.
ii. TU = 0 if either ħ =0 or c = ∞, which is the classical result.
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Zeroth Law of Black-Hole Mechanics
Zeroth law
• The horizon of a stationary black hole has a uniform surface
gravity κ.
Thermodynamic analogy
• An object in thermal equilibrium with a heat reservoir has a
uniform temperature T.
Relationship between κ and T
• Analogous to the Unruh effect , Hawking showed that black
holes emit Hawking radiation at a temperature TH, given by
TH = ħκ/2πc,
where κ may be thought of as the magnitude of the
acceleration needed by a spaceship to just counteract the
gravitational acceleration just outside the event horizon. 3
Entropy of a Black Hole
• Black holes must carry entropy, because the 2nd law of
thermodynamics requires that the loss of entropy of an object
falling into a black hole must at least be compensated by the
increase of entropy of the black hole.
• The expression for the entropy of a black hole, obtained by
Beckenstein, and later confirmed by Hawking is
SBH = kAc3/4Għ,
where k is Boltzmann’s constant, A is the area of the black
hole’s horizon, and BH could stand for black hole or
Beckenstein-Hawking.
• A system of units with c=1 gives SBH = kA/4Għ, while one in
which c=1, ħ=1, k=1 and G=1 gives SBH = A/4, showing that a
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black-hole’s entropy is proportional to the area of its horizon.
First Law of Black-Hole Mechanics
1st law
dM = (κ/8π) dA + Ω dJ + Φ dQ,
where M is the mass, Ω is the angular velocity, J is the angular
momentum,Φ is the electric potential, Q is the charge, and the
constants c, ħ, k, and G are all made equal to unity.
Thermodynamic analogy
dU = T dS – P dV
Relationship between (κ/8π)dA and TdS
• Since TH = κ/2π and SBH = A/4,
(κ/8π) dA = (2πTH)(1/8π)(4dSBH) = THdSBH;
i.e. the first term is just the product of the black-hole
temperature and its change of entropy.
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Second Law of Black-Hole Mechanics
2nd law
• The area A of the horizon of a black hole is a non-decreasing
function of time; i.e. ΔA ≥ 0.
Thermodynamic analogy
• The entropy of an isolated system is a non-decreasing
function of time; i.e. ΔS ≥ 0.
Hawking radiation
• If the quantum fluctuations of the vacuum produces a particleantiparticle pair near the horizon of a black hole, and the
antiparticle drops into the hole, the particle will appear to have
come from the black hole, which loses entropy.
• This leads to a generalized 2nd law:
Δ[Soutside + (A/4)] ≥ 0.
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