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Black Holes in General Relativity
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The existence of black holes is predicted by Einstein s theory of general relativity. Readers interested in a detailed description of black holes in this context may want to consult Relativity Demysti ed. The Einstein eld equations are a set of differential equations which relate the curvature of space-time to the matter-energy content as follows: 1 R g R + g = 8 G4 T 2 (14.1)
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CHAPTER 14 Black Holes
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This equation contains the following elements:
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R is the Ricci tensor. In a moment we will see that it is related to the curvature of space-time through the metric. It can be calculated from the Riemann curvature tensor using R = R . g is the metric tensor which describes the geometry of space-time. R is the Ricci scalar which is computed by contraction of the Ricci tensor. is the cosmological constant. G4 is Newton s gravitational constant. It has been noted that this is the gravitational constant in four space-time dimensions, because the form of the gravitational constant depends on the number of space-time dimensions. T is the energy-momentum tensor. The Riemann curvature tensor is Ra bgd = g a bd a bd + e bd a eg e bg a ed where the Christoffel symbols are given in terms of the metric tensor as 1 = ( g + g g ) 2 (14.3) (14.2)
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When studying the gravitational eld outside of the source, the energy-momentum tensor can be set to 0 and we study the vacuum eld equations. T = 0 in a region of empty space-time where no matter or energy is present. The equations are 1 R g R = 0 2 (14.4)
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The vacuum eld equations describe the structure of space-time outside of a massive body. We use this form of the equations when studying a black hole, because all of the mass is concentrated at a single point at the center called the singularity. We can use the vacuum eld equations to characterize the structure of the space-time outside this region. As an aside, note that perturbative string theory adds corrections to the vacuum eld equations. These corrections are of the order O[( R )n ]. If we took the rstorder correction from string theory, Eq. (14.4) would be modi ed as follows: 1 R g R + O ( R ) = 0 2 (14.5)
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We will ignore that here, we only mention it for information purposes. Continuing, the simplest case we can imagine is a black hole of mass m which is static (i.e., nonrotating) and spherically symmetric. The metric which describes the space-time outside a black hole of this form is called the Schwarzschild metric. The full solution to the vacuum eld equations used to arrive at this metric can be found in Chap. 10 of Relativity Demysti ed. We simply state the metric here: 2mG4 2 2mG4 2 2 2 ds = 1 + dt + 1 + dr + r d r r
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(14.6)
where d 2 = d 2 + sin 2 d 2. The point rH = 2G4 m is called the horizon. This appears to be a singular point because setting r = 2G4 m causes the coef cient of dr to blow up. It can be shown, however, that this is not a real singularity this singular behavior is just an artifact of the coordinate system. To see this we can calculate a scalar which is an invariant, which gives us insight into the true nature of the horizon. One such invariant is R
R
2 12rH = 6 r
(14.7)
This expression tells us that there is a true singularity at r = 0. Although rH = 2G4 m is not a singularity, it is still an important location. This location as we have already indicated denotes the event horizon. This is a boundary, in the case of (3 + 1)-dimensional space-time the surface of a sphere which divides space-time into the external world and a point of no return. Nothing that crosses the event horizon can ever return to the rest of the universe, not even light. This is why black holes are black, because light cannot escape from inside the horizon. It will be of interest to study black holes in arbitrary space-time dimension D. With that in mind, before moving on to our next black hole let s de ne some basic quantities. The rst item to note is the volume of a unit sphere in d dimensions. This is given by d = 2 ( d +1)/ 2 d + 1 2 (14.8)
where is the gamma factorial function. The radius of the horizon in D-dimensional space-time is given by
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