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code 128 generator vb.net Black Holes in General Relativity in Java
Black Holes in General Relativity QR Code 2d Barcode Scanner In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Draw Denso QR Bar Code In Java Using Barcode generation for Java Control to generate, create QR Code ISO/IEC18004 image in Java applications. 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 spacetime to the matterenergy content as follows: 1 R g R + g = 8 G4 T 2 (14.1) Denso QR Bar Code Reader In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Barcode Creator In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. CHAPTER 14 Black Holes
Scanning Barcode In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. QR Code JIS X 0510 Generator In Visual C#.NET Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. This equation contains the following elements: Make QR Code In Visual Studio .NET Using Barcode generator for ASP.NET Control to generate, create QR image in ASP.NET applications. Create QR Code 2d Barcode In Visual Studio .NET Using Barcode maker for .NET framework Control to generate, create Quick Response Code image in Visual Studio .NET applications. R is the Ricci tensor. In a moment we will see that it is related to the curvature of spacetime 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 spacetime. 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 spacetime dimensions, because the form of the gravitational constant depends on the number of spacetime dimensions. T is the energymomentum 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) Denso QR Bar Code Creator In Visual Basic .NET Using Barcode creator for Visual Studio .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. Creating 2D Barcode In Java Using Barcode printer for Java Control to generate, create Matrix 2D Barcode image in Java applications. When studying the gravitational eld outside of the source, the energymomentum tensor can be set to 0 and we study the vacuum eld equations. T = 0 in a region of empty spacetime where no matter or energy is present. The equations are 1 R g R = 0 2 (14.4) UPCA Supplement 5 Creator In Java Using Barcode maker for Java Control to generate, create UPC Symbol image in Java applications. ECC200 Maker In Java Using Barcode drawer for Java Control to generate, create Data Matrix image in Java applications. The vacuum eld equations describe the structure of spacetime 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 spacetime 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) USS Codabar Maker In Java Using Barcode creation for Java Control to generate, create USD4 image in Java applications. Matrix 2D Barcode Generator In C# Using Barcode generation for .NET framework Control to generate, create Matrix Barcode image in VS .NET applications. String Theory Demysti ed
Bar Code Encoder In .NET Framework Using Barcode encoder for VS .NET Control to generate, create bar code image in .NET framework applications. EAN13 Maker In VS .NET Using Barcode creation for .NET Control to generate, create GS1  13 image in .NET applications. 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 spacetime 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 Barcode Generation In VB.NET Using Barcode drawer for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. Bar Code Decoder In .NET Framework Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. 2 1 Generate EAN13 In ObjectiveC Using Barcode generator for iPad Control to generate, create EAN13 image in iPad applications. EAN / UCC  13 Generator In None Using Barcode encoder for Online Control to generate, create EAN13 image in Online applications. (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 spacetime the surface of a sphere which divides spacetime 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 spacetime 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 Ddimensional spacetime is given by

