Black Holes in .NET framework

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Black Holes
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(b) the ergosphere is a region of zero gravitational eld (c) inside the ergosphere, spacelike geodesics rotate with the mass that is the source of the gravitational eld (d) no information can be known about the ergosphere
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CHAPTER
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Cosmology
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We now turn to the study of the dynamics of the entire universe, the science known as cosmology. The mathematical study of cosmology turns out to be relatively simple for two reasons. The rst is that gravity dominates on large scales, so we don t need to consider the local complexity that arises from the nuclear and electromagnetic forces. The second reason is that on large enough scales, the universe is to good approximation homogeneous and isotropic. By large enough scales, we are talking about the level of clusters of galaxies. We apply the terms homogeneous and isotropic to the spatial part of the metric only. By homogeneous, we mean that the geometry (i.e., the metric) is the same at any one point of the universe as it is at any other. Remember, we are talking about the universe on a large scale, so we are not considering local variations such as those in the vicinity of a black hole. An isotropic space is one without any preferred directions. If you do a rotation, the space looks the same. Therefore, we can say an isotropic space is one for which the geometry is spherically symmetric about any point. Incorporating these two characteristics into the spatial part of the metric allows us to consider spaces of constant curvature. The curvature in a space is
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Cosmology
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denoted by K . Consider an n-dimensional space Rn . A result from differential geometry known as Schur s theorem tells us that if all points in some neighborhood N about a point are isotropic, and the dimension of the space is n 3, then the curvature K is constant throughout N . In our case we are considering a globally isotropic space, and therefore K is constant everywhere. At an isotropic point in Rn , we can de ne the Riemann tensor in terms of the curvature and the metric using Rabcd = K (gac gbd gad gbc ) (12.1)
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In our case, keep in mind that we will be able to apply this result only to the spatial part of the metric. The observation that on large scales the universe is homogeneous and isotropic is embodied in a philosophical statement known as the cosmological principle.
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The Cosmological Principle
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Copernicus told us that the Earth is not the center of the solar system. This idea can be generalized to basically say that the Earth is not the center of the universe. We call this statement the cosmological principle. Basically, we are saying that the universe is the same from point to point.
A Metric Incorporating Spatial Homogeneity and Isotropy
As we mentioned earlier, the properties of homogeneity and isotropy apply only to the spatial part of the metric. Observation indicates that the universe is evolving in time and therefore we cannot extend these properties to include all of spacetime. This type of situation is described by using gaussian normal coordinates. A detailed study of gaussian normal coordinates is beyond the scope of this book, but we will take a quick look to understand why the metric can be written in the general form: ds 2 = dt 2 a 2 (t) d 2
Cosmology
Q(t2, x, y, z,)
P(t1, x, y, z,)
Fig. 12-1. A geodesic at the same spatial point, moving through time.
where d 2 is the spatial part of the metric and a (t) the scale factor, a function that implements the evolution in time of the spatial part of the metric. Note that if a(t) >0, we are describing an expanding universe. We model the universe in the following way. At a given time, it is spatially isotropic and homogeneous, but it evolves in time. Mathematically we represent this by a set of three-dimensional spacelike hypersurfaces or slices S stacked one on top of the other. An observer who sits at a xed point in space remains at that point but moves forward in time. This means that the observer moves along a geodesic that is parallel with the time coordinate. Suppose that S represents the spacelike hypersurface at some time t1 and that S is a spacelike hypersurface at a later time t2 . Let us denote two points on these slices as P and Q, and consider a geodesic that moves between the two points (see Fig. 12-1). Since the observer is sitting at the same point in space, the spatial coordinates of the points Pand Q are unchanged as we move from S to S . Therefore, the arc length of the geodesic is given by the time coordinate; i.e., t2 t1 = arc length of the geodesic. More precisely, we can write ds 2 = dt 2 Therefore, the component of the metric must be gtt = 1. To derive the form of the spatial component of the metric, we rely on our previous studies. The Schwarzschild metric had the property of spherical symmetry which is exactly what we are looking for. Let s recall the general form of the Schwarzschild metric: ds 2 = e2 (r ) dt 2 e2 (r ) dr 2 r 2 d 2 + sin2 d 2
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