code 128 generator vb.net r 2 r 2 ds = 1 E dt 2 + 1 E dr 2 + r 2 d 3 2 r r in Java

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r 2 r 2 ds = 1 E dt 2 + 1 E dr 2 + r 2 d 3 2 r r
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(14.20)
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For the extremal black hole in ve dimensions, the relation between mass and charge becomes m= Q 3 rE2 = 4G5 G5 (14.21)
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where G5 is the gravitational constant in ve dimensions. The area of the horizon is
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3 A = 2 2rE
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(14.22)
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The Laws of Black Hole Mechanics
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In the early 1970s, James Bardeen, Brandon Carter, and Stephen Hawking found that there are laws governing black hole mechanics which correspond very closely to the laws of thermodynamics.1 The zeroth law states that the surface gravity at the horizon of a stationary black hole is constant. The rst law relates the mass m, horizon area A, angular momentum J, and charge Q of a black hole as follows: dm =
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dA + dJ + dQ 8
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(14.23)
This law is analogous to the law relating energy and entropy. We will see this more precisely in a moment. The second law of black hole mechanics tells us that the area of the event horizon does not decrease with time. This is quanti ed by writing: dA 0 (14.24)
This is directly analogous to the second law of thermodynamics which tells us that the entropy of a closed system is a nondecreasing function of time. A consequence of Eq. (14.24) is that if black holes of areas A1 and A2 coalesce to form a new black hole with area A3 then the following relationship must hold: A3 > A1 + A2 As you probably recall, an analogous relationship holds for entropy. Finally, we arrive at the third law of black hole mechanics. This law states that it is impossible to reduce the surface gravity to 0. The correspondence between the laws of black hole mechanics and thermodynamics is more than analogy. We can go so far as to say that the analogy is taken to be real and exact. That is, the area of the horizon A is the entropy S of the black hole and the surface gravity is proportional to the temperature of the black hole. We can express the entropy of the black hole in terms of mass or area. In terms of mass the entropy of a black holes is proportional to the mass of the black
1 Bardeen, J.M., B. Carter, and S.W. Hawking, The four laws of black hole mechanics, Comm. Math Phys. vol. 31, (2), 1973, 161 170.
CHAPTER 14 Black Holes
hole squared. In terms of area, the entropy is 1/4 of the area of the horizon in units of Planck length: S= A 4 2p (14.25)
Computing the Temperature of a Black Hole
Let us compute the temperature in the case of a Schwarzschild black hole. In the Quiz you will get a chance to try your luck nding the temperature of a charged black hole. We follow a procedure outlined in a note published by P.R. Silva.2 We proceed as follows. We perform a Wick rotation t i and write the Schwarzschild metric as 2G M 2G M ds 2 = 1 4 d 2 + 1 4 dr 2 + r 2 d 2 r r Now set 2G4 M Rd = 1 r 2G4 M dR = 1 r
1/ 2 1
(14.26)
1/ 2
and integrate. We take the limits of integration to be
: 0 2 : 0 r : 2G4 m r r
This gives us two relations: 2 R = ( 2G4 m ) 1/ 2 (r 2G4 m )1/ 2 R = 2( 2G4 m )1/ 2 (r 2G4 m )1/ 2 (14.27) (14.28)
Available on the arXiv at http://arxiv.org/ftp/gr-qc/papers/0605/0605051.pdf.
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Dividing Eq. (14.27) by Eq. (14.28) we obtain 2 =
4G4 m
= 8 G4 m The used here is the same one used in thermodynamics, and so we obtain the following expression for the temperature of a Schwarzschild black hole: T= 1 8 mG4 (14.29)
With the temperature in hand, we can proceed ahead to obtain an expression for the entropy. Recalling that the rst law of thermodynamics states that dE = TdS (think tedious), using E = mc 2 (but taking c = 1 ) we obtain the rst law of black hole mechanics for a static, uncharged black hole: dm = Tds Hence, mdm = Integrating we nd m2 1 S = 2 8 G4 Hence, the entropy of a Schwarzschild black hole is given by S = 4 G4 m 2 (14.31) 1 dS 8 G4 (14.30)
This con rms our earlier claim, that the entropy is proportional to the mass squared. Before proceeding, let s quickly refresh our memories. What is entropy anyway Suppose that we have a density of states n( E ) for some microscopic system. The entropy is S = kB ln[n( E )] where k B is Boltzman s constant. (14.32)
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