JHEP12(2007)052 hep122007052.pdf December 8, 2007 November 5, 2007 ntaining a black December 13, 2007 Gravity, Received: Accepted: Published: lidity is also tested in case of http://jhep.sissa.it/archive/papers/jhep122007052 /j Published by Institute of Publishing for SISSA Space-Time Symmetries, Black Holes, Classical Theories of The generalized second law of thermodynamics for a system co [email protected] [email protected] Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5,Department Canada, of and Astronomy, BeijingBeijing, Normal 100875, University, China E-mail: School of Physics, PekingBeijing, University, 100871, China E-mail: SISSA 2007 c ° Singularities. Hongbao Zhang Abstract: Song He The dynamicalsecond horizon law and of generalized thermodynamics hole dynamical horizon is proposedadiabatically in collapsing a thick covariant light way. shells. Its va Keywords: JHEP12(2007)052 3 5 2 1 (1.1) iabatically with the black t horizon, the notion of ivalently formulated as a quarter of the area of the hat we shall address in the zed to the dynamical hori- al horizon may also have the th the black hole dynamical ized second law of thermody- wing black hole by spherically entropy bound formulation of n or absorption of black holes estigated, where, in particular, er outside never decreases with put forth for a system including must satisfy ns are presented in the end. ′ e through the between A S , e and e A A 4 − ′ e +). Notation and conventions follow ref. [8]. A , – 1 – + ≤ , + S , − is assumed. e A ≥ ′ e A However, due to the global and teleological property of even The signature of metric takes ( collapsing thick light shells hole dynamical horizon dynamical horizon was developed and itsthe properties first were inv and secondzon laws [4 – of 7]. black Thus hole itthermal mechanics is character tempting was as to generali the conjecture eventnamics that horizon may the does, also dynamic be and applied thepresent to general paper. the dynamical In horizon. nextthe section, This we generalized is shall w second propose lawhorizon. a of covariant Then thermodynamics its validity associated issymmetrical demonstrated wi in collapse a of model thick of light a shells. gro Some discussio its two-dimensional space-like surfaces of area where 4. Discussions 3. The generalized second law of thermodynamics tested by ad black holes by Bekensteinblack [1 – hole’s 3]. event horizon It plus states thetime entropy that in of the all ordinary sum matt processes.the of It one generalized is second noteworthy that lawcovariant for entropy of the bound. thermodynamics formatio Namely, can the also entropy be flux equ 1. Introduction The generalized second law of thermodynamics was initially 2. The generalized second law of thermodynamics associated Contents 1. Introduction JHEP12(2007)052 ) a b n n The with a ′ ∇ A 2, then ab − h with the and = = a a space-time A n nt horizons of n θ a ( l b l a ∇ of one future-directed ab pace-like hyper-surface l h ted by a family of closed θ represent future-directed le, the generalized second iant way as follows: such that = abatically collapsing thick t the dynamical horizon is a ck hole dynamical horizon. l a ntified quasi-locally without competent candidate for the n θ n sion 1. Note that, in contrast to the itively, it is clear that no signal ces therein. and and a a l l ) on each leaf. b l a of the other future-directed null normal n n θ + between its apparent horizon of area – 2 – b n H a if the dominant energy condition holds for matter, l ( 2 1 A − 4 ′ + A ab g ≤ = S ab h flowing through it. a must satisfy through the black hole dynamical horizon between its appare vanishes and the expansion s : A smooth, three-dimensional, space-like sub-manifold in is assumed. ′ a S l A A black hole dynamical horizon > A ′ and ) is said to be a black hole dynamical horizon if it can be folia A ab A Now associated with the alternative boundary of the black ho Thus, roughly speaking, a black hole dynamical horizon is a s Definition In the subsequent section, its validity will be tested by adi black hole dynamical horizon M, g the expansion of the null geodesics normal can be given by outgoing and ingoing null normals,notion respectively. of See the figure event horizon,knowledge the of dynamical the horizon full cancan space-time be propagate history. ide out In of addition, thespace-like. intu dynamical All horizon of due these make toboundary the the of dynamical fact the horizon tha black become hole. a law of thermodynamics canentropy be flux naturally formulatedarea in a covar is strictly negative. If we choose the normalization of with the induced metric two-dimensional surfaces such that, on each leaf, the expan ( null normal which is foliated by closed apparent horizons, where where Figure 1: For more subtle details, please refer to ref. [5] and referen We would first like to introduce the basic definition of the bla light shells. entropy current 2. The generalized second law of thermodynamics associated JHEP12(2007)052 = 1 (3.5) (3.3) (3.2) (3.4) (3.6) (3.1) u . The energy m be chosen essentially = 2 , iabatically F a R ¶ ∂ ∂r µ − a . l, let us first compute the , 0 ¶ the non-vanishing energy 2 . ) ∂ entric light shells with a flat schild radius < ∂t tor field is time-like, i.e., (1) = 1. Thus symmetrical collapse of thick is a monotonically increasing dr F , µ , ularity. See figure 2. d but tedious calculation, one =1 F + rbitrary sphere characterized by u = | on 2 that the hyper-surface ) . a dt r 2 ( k = 0, and after the total duration of ∂t ∂u , u − b t = k + − a = a 2 k n ,n Ω 2 ∂r ∂u ′ a r d (0) = 0, and ,θ ( 2 ¶ ) ∆ F r u ∂t ∂u π mF ∂ ∂r – 3 – + 4 Schwarzschild region within light shells Minkowski region − r 2 µ , ) ) = 2 = t u dr 2(1 + ∆ ab with respect to its argument, and the null vector field r =1 − + T u ( = | 2 F l F , u θ b dt m , m r r + (1 2 2 − ∇ 0 a u a =      ¶ 2 ∇ ∂ = ∂t ab ds u µ g ) 0. u > ′ . Clearly, the energy momentum tensor satisfies the dominant ). The outgoing and ingoing null normals to these spheres can a F ) t, r dr = (1+ and ∆ are constant parameters. In addition, a l + ( m a denotes the derivative of ) ′ dt F Next, to locate the black hole dynamical horizon in this mode Specifically speaking, this model describes a family of conc collapsing thick light shells = ( a serves as a densityfinds profile that function. this Aftermomentum metric a tensor is straightforwar within a light shells solution being of given by Einstein equation with function with respect to its argument, with is a black hole dynamical horizon if and only if its normal vec 3. The generalized second law of thermodynamics tested by ad Obviously, it follows from the definition presented in secti where Here both then the corresponding expansions can be obtained as k collapse ∆ the outmost light shell finally arrives atinitial the expansion sing of the future-directedsome null value normal of to ( an a innermost light shell reaches the center at the time condition due to Minkowski interior, ending with a final black hole of Schwarz where light shells in Eddington-Finkelstein coordinate [9] Start with a model of formation of a black hole by spherically to be, respectively, JHEP12(2007)052 p t = 0 and (3.8) (3.7) t z + pparent = 0 and = r t , then the t + at a time + r p > z t r ′ − z ′ 0. reads between > ′ ′ z modynamics is satisfied s and Gauss theorem, the