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The four laws of black hole mechanics. (English) Zbl 1125.83309
Summary: Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area $A$ of the event horizon and the “surface gravity” $\kappa$ of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which correspond to and in some ways transcend the four laws of thermodynamics.

##### MSC:
 83C57 Black holes 83C20 Classes of solutions of equations in general relativity
##### References:
 [1] Hawking, S. W.: Commun. math. Phys.25, 152–166 (1972). · doi:10.1007/BF01877517 [2] Hawking, S. W.: The event horizon. In: Black Holes. New York, London, Paris: Gordon and Breach 1973 (to be published). [3] Hawking, S. W., Ellis, G. F. R.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973 (to be published). [4] Carter, B.: Phys. Rev. Letters26, 331–333 (1971). · doi:10.1103/PhysRevLett.26.331 [5] Carter, B.: (Preprint, Institute of Theoretical Astronomy, Cambridge, England). [6] Carter, B.: Properties of the Kerr metric. In: Black Holes. New York, London, Paris: Gordon and Breach 1973 (to be published). [7] Smarr, L.: Phys. Rev. Letters30, 71–73 (1973). · doi:10.1103/PhysRevLett.30.71 [8] Beckenstein, J.: PhD Thesis. Princeton University, 1972. [9] Carter, B.: J. Math. Phys.10, 70–81 (1969). · Zbl 0165.58902 · doi:10.1063/1.1664763 [10] Hawking, S. W.: Commun. math. Phys.18, 301–306 (1970). · Zbl 0197.26403 · doi:10.1007/BF01649448 [11] Hawking, S. W., Hartle, J. B.: Commun. math. Phys.27, 283–290 (1972). · doi:10.1007/BF01645515 [12] Bardeen, J. M.: Astrophys. J.162, 71–95 (1970). · doi:10.1086/150635 [13] Bardeen, J. M.: Nature226, 64–65 (1970). · doi:10.1038/226064a0 [14] Christodoulou, D.: Phys. Rev. Letters25, 1596–1597 (1970). · doi:10.1103/PhysRevLett.25.1596