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CHAPTER 14 Black Holes
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Here, GD is the gravitational constant in D dimensions. In string theory, it depends on the volume of compacti ed space V, the string-coupling constant gs, and the string length scale ls through a 10-dimensional gravitational constant: GD = Now de ne: r h = 1 H r
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Then, the Schwarzschild metric in D dimensions can be written as
ds 2 = hdt 2 + h 1dr 2 + r 2 d 2 2 D
(14.12)
Charged Black Holes
In the introduction we noted that classically a black hole can be completely characterized by its mass, charge, and angular momentum. So in relativity theory there aren t too many choices available to study more complicated black holes. We could have A static black hole of mass m (Schwarzschild). A static black hole with mass m and electric charge Q. A rotating black hole. A static black hole with electric charge is called a Reissner-Nordstr m black hole, while a rotating black hole is called a Kerr black hole. Real astrophysical black holes are best described by Kerr black holes. Stars rotate so when they collapse to a black hole conservation of angular momentum dictates that the black hole will rotate as well. If the rotation is very slow, the Schwarzschild solution would be a good approximation. Real astrophysical black holes, as far as we know, don t carry electrical charge. However, this example is simpler than the Kerr case and it has some advantages which simplify calculations in string theory, so if you re interested in string theory you should familiarize yourself with charged black holes. It is interesting to note that if you add a
String Theory Demysti ed
charge Q to a static black hole in string theory, you can arrive at an exotic black hole that is supersymmetric. First we de ne: = 1 2mG4 Q 2G4 + 2 r r (14.13)
Notice that we are basically extending the Schwarzschild solution by adding a Coulomb-type term. The metric for a static, charged black hole is given by ds 2 = dt 2 + 1dr 2 + r 2 d 2 This metric has two coordinate singularities which are given by r = MG4 ( MG4 )2 Q 2 G4 The two horizons are denoted by r+ is the outer horizon. r is the inner horizon. The outer horizon is the event horizon the point of no return when approaching the black hole. Now, before stating our next result, we need to talk a little bit about singularities. Stephen Hawking and Roger Penrose did a great deal of work on singularities in the context of classical general relativity. They found out some interesting results about singularities. If a singularity is present in space-time without a horizon, it is called a naked singularity. This is because the horizon, like clothing, keeps you from seeing what s behind the veil. In this case the veil is provided by the fact that light and hence no information can escape from beyond the horizon. The singularity is essentially shut off from the rest of the universe. Hawking and Penrose conjectured that classical physics does not permit the existence of naked singularities. Charged black holes are related to this concept in the following way. A charged black hole with a mass m is limited in the amount of charge Q that it can carry. It avoids having a naked singularity only if m G4 Q (14.16) (14.15) (14.14)
CHAPTER 14 Black Holes
When the maximum charge per mass is allowed, we obtain a special type of charged black hole which is called an extremal black hole. If you search the literature you will nd this term used over and over extremal black holes are an active area of study. The condition for having an extremal black hole is m G4 = Q (14.17)
In this case, the radii of the inner and outer horizons are equal. There is only the event horizon whose location is determined to be rE = r = mG4 (14.18)
Extremal black holes are important because they have unbroken supersymmetry. The metric of an extremal black hole assumes the form r r ds = 1 E dt 2 + 1 E dr 2 + r 2 d 2 r r
2 2 2
(14.19)
The key breakthrough for string theory and black holes involved the derivation of black hole entropy from the microscopic states for an extremal black hole in D = 5 dimensions. In that case the metric can be written as
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