# MIT 8 821 - Lecture 23: Black Hole Thermodynamics (5 pages)

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## Lecture 23: Black Hole Thermodynamics

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Lecture Notes

- Pages:
- 5
- School:
- Massachusetts Institute of Technology
- Course:
- 8 821 - String Theory and Holographic Duality

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8 821 F2008 Lecture 23 Black Hole Thermodynamics Lecturer McGreevy Scribe Tongyan Lin December 2 2008 In today s lecture we ll discuss the laws of black hole thermodynamics and how AdS black holes are related to finite temperature CFTs and Koushik will give a related presentation 1 Laws of Thermodynamics Recall from last time that for a black hole Area Entropy T 1 where was the surface gravity The near horizon metric is ds2 2 2 dt2 d 2 2 2 d 2 d 2 2 when we go to Euclidean time it If has periodicity 2 then the euclidean geometry is regular Recall the canonical ensemble thermal partition function is Zth tre H T 3 where e H T propagates the system with imaginary time t 1 iT Thermal equilibrium is equivalent to periodic euclidean time with period 1 T so we identify with temperature T The laws of stationary black hole thermodynamics analogous to the usual laws of thermodynamics are 0th thermal equilibrium is constant over the event horizon This means temperature is constant in space and time Thus stationary black holes are in thermal equilibrium with constant temperature John thinks the proof of the 0th law doesn t depend on the shape of the black hole as long as its a stationary solution 1 1st conservation of energy dE dM dJ dQ dA P dV 8 G 4 dJ is the change in rotational energy dQ is the electrical energy and 8 G dA T dS is heat exchange This law relates the change in the energy or equivalently mass to changes in various properties of the black hole The last term describing mechanical work P dV isn t present for black holes but IS for black branes 2nd entropy increases This is the area theorem for a black hole we proved last lecture A Proof of the exact relation between S and A in a later lecture A 0 since S 4 G 3rd absolute zero entropy or rather T cannot taken to zero in a finite number of steps This doesn t mean that S T 0 0 but it does probably mean at T 0 there is a minimum in entropy These laws follow from Einstein s Equation the energy condition we discussed

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