Black-Hole Thermodynamics

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Black-Hole Thermodynamics. PHYS 4315 R. S. Rubins, Fall 2009. 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.

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### Black-Hole Thermodynamics

PHYS 4315

R. S. Rubins, Fall 2009

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.

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.

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 givesSBH = kA/4Għ, while one in which c=1, ħ=1, k=1 and G=1 givesSBH = A/4, showing that a 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

• SinceTH = κ/2πandSBH = 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.

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.