Laws of black hole mechanics
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The four laws of black hole mechanics are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking and James Bardeen.
Contents
Statement of the Laws
The Zeroth Law
The horizon has constant surface gravity for a stationary black hole [\kappa].The First Law
We have
- [dM = \fracdA+\Omega dJ+\Phi dQ],
The Second Law
The horizon area is, assuming the weak energy condition, a non-decreasing function of time,
- [dA \geq 0]
The Third Law
It is not possible to form a black hole with vanishing surface gravity. [\kappa]=0 is not possible to achieve.Discussion of the Laws
The Zeroth Law
The zeroth law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature. T constant for thermal equilbrium for a normal system is analogous to [\kappa] constant over the horizon of a stationary black holeThe First Law
The left hand side, [\, dM], is the change in mass/energy. Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and electromagnetism. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right hand side the term [ \, T dS].The Second Law
The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the entropy of a closed system is a non-decreasing function of time, suggesting a link between entropy and the area of a black hole horizon. However,this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalised second law introduced as total entropy = black hole entropy + outside entropyThe Third Law
Extremal black holes have vanishing surface gravity. Stating that [\kappa] cannot go to zero is analogous to the third law of thermodynamics which, in its weak formulation, states that it is impossible to reach absolute zero temperature in a physical process. The strong version of the third law of thermodynamics, which states that as the temperature approaches zero, the entropy also approaches zero, does not have an analogue for black holes.Interpretation of the Laws
The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the no hair theorem, infinite entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at temperature
- [T_H = \frac].
- [S_ = \frac].
References
- J. M. Bardeen, B. Carter and S. W. Hawking, "The four laws of black hole mechanics", Commun. Math. Phys. 31, 161 (1973).
- J. D. Bekenstein, "Black holes and entropy", Phys. Rev. D 7, 2333 (1973).
- S. W. Hawking, "Black hole explosions?", Nature 248, 30 (1974).
- S. W. Hawking, "Particle creation by black holes", Commun. Math. Phys. 43, 199 (1975).
- S. W. Hawking and G. F. R. Ellis, "The large-scale structure of space-time", Cambridge University Press (1973).
- S. W. Hawking, "The Nature of Space and Time", (1994) [link]
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