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.
R. S. Rubins, Fall 2009
TU = aħ/2πc.
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.
Relationship between κ and T
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.
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.
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.
dU = T dS – P dV
Relationship between (κ/8π)dA and TdS
(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.
Δ[Soutside + (A/4)] ≥ 0.