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Black-Hole Thermodynamics

PHYS 4315R. 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.

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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 black-hole’s entropy is proportional to the area of its horizon.

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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 particle-

antiparticle 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.