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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
r 2 r 2 ds = 1 E dt 2 + 1 E dr 2 + r 2 d 3 2 r r QR Code JIS X 0510 Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Creating QRCode In Java Using Barcode generation for Java Control to generate, create QR Code JIS X 0510 image in Java applications. (14.20) Decode QRCode In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Barcode Generator In Java Using Barcode maker for Java Control to generate, create bar code image in Java applications. For the extremal black hole in ve dimensions, the relation between mass and charge becomes m= Q 3 rE2 = 4G5 G5 (14.21) Barcode Recognizer In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Make QR In Visual C# Using Barcode encoder for .NET Control to generate, create Denso QR Bar Code image in .NET applications. where G5 is the gravitational constant in ve dimensions. The area of the horizon is
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Printing USPS PLANET Barcode In Java Using Barcode creation for Java Control to generate, create USPS Confirm Service Barcode image in Java applications. Make Bar Code In Java Using Barcode generator for Android Control to generate, create barcode image in Android applications. The Laws of Black Hole Mechanics
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Code 128B Printer In ObjectiveC Using Barcode creation for iPad Control to generate, create ANSI/AIM Code 128 image in iPad applications. Drawing Code 128 In VS .NET Using Barcode creator for Reporting Service Control to generate, create Code 128C image in Reporting Service applications. (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/grqc/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)

