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Black Holes in .NET framework
Black Holes Read Quick Response Code In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Encoding Quick Response Code In .NET Framework Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. (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 Decode QR Code 2d Barcode In Visual Studio .NET Using Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Drawer In .NET Framework Using Barcode creation for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. CHAPTER
Bar Code Decoder In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Generating QRCode In C#.NET Using Barcode generation for VS .NET Control to generate, create QR Code 2d barcode image in .NET applications. Cosmology
Print QRCode In VS .NET Using Barcode maker for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. Quick Response Code Generator In VB.NET Using Barcode generator for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications. 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 GS1 DataBar Limited Generator In VS .NET Using Barcode maker for .NET framework Control to generate, create DataBar image in .NET framework applications. Print ECC200 In .NET Using Barcode generator for VS .NET Control to generate, create DataMatrix image in VS .NET applications. Copyright 2006 by The McGrawHill Companies, Inc. Click here for terms of use.
Drawing European Article Number 13 In .NET Using Barcode creation for .NET framework Control to generate, create European Article Number 13 image in Visual Studio .NET applications. Generate Identcode In .NET Framework Using Barcode printer for .NET Control to generate, create Identcode image in .NET applications. Cosmology
Encoding Barcode In None Using Barcode generator for Office Word Control to generate, create barcode image in Word applications. Bar Code Maker In ObjectiveC Using Barcode printer for iPad Control to generate, create barcode image in iPad applications. denoted by K . Consider an ndimensional 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) Bar Code Creation In VS .NET Using Barcode maker for Reporting Service Control to generate, create bar code image in Reporting Service applications. Code 128 Code Set B Creator In None Using Barcode drawer for Microsoft Word Control to generate, create Code 128 Code Set A image in Microsoft Word applications. 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. Reading Data Matrix 2d Barcode In Visual C#.NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Paint EAN 128 In Java Using Barcode drawer for Android Control to generate, create EAN 128 image in Android applications. The Cosmological Principle
Make UCC  12 In Visual C#.NET Using Barcode encoder for VS .NET Control to generate, create EAN 128 image in VS .NET applications. Data Matrix ECC200 Reader In VB.NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. 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. 121. 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 threedimensional 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. 121). 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

