1. 1 Black-Hole Thermodynamics PHYS 4315 R. S. Rubins, Fall 2009

2. 2 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 T U = aħ/2πc. Notes i. For an acceleration of 10 19 m/s 2, T U ~ 1 K. ii. T U = 0 if either ħ =0 or c = ∞, which is the classical result.

3. 3 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 T H, given by T H = 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.

4. 4 Entropy of a Black Hole Black holes must carry entropy, because the 2 nd 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 S BH = kAc 3 /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 S BH = kA/4Għ, while one in which c=1, ħ=1, k=1 and G=1 gives S BH = A/4, showing that a black-hole’s entropy is proportional to the area of its horizon.

5. 5 First Law of Black-Hole Mechanics 1 st 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 T H = κ/2π and S BH = A/4, (κ/8π) dA = (2πT H )(1/8π)(4dS BH ) = T H dS BH ; i.e. the first term is just the product of the black-hole temperature and its change of entropy.

6. 6 Second Law of Black-Hole Mechanics 2 nd 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 2 nd law: Δ[S outside + (A/4)] ≥ 0.