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Black Holes in VS .NET
Black Holes Reading Denso QR Bar Code In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. QR Code Creation In .NET Framework Using Barcode creation for .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. r r=2m
Denso QR Bar Code Decoder In .NET Framework Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET framework applications. Barcode Maker In Visual Studio .NET Using Barcode creation for VS .NET Control to generate, create bar code image in VS .NET applications. Fig. 112. Using EddingtonFinkelstein coordinates (v, r ) removes the coordinate singularity at r = 2m. As r gets smaller, light cones tip over. For r < 2m, all geodesics directed toward the future head toward r = 0. Bar Code Decoder In Visual Studio .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. Print QR Code ISO/IEC18004 In Visual C# Using Barcode printer for Visual Studio .NET Control to generate, create Quick Response Code image in .NET framework applications. Kruskal Coordinates
Paint Quick Response Code In .NET Framework Using Barcode generator for ASP.NET Control to generate, create QR image in ASP.NET applications. QR Generation In Visual Basic .NET Using Barcode creation for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. KruskalSzekeres coordinates allow us to extend the Schwarzschild geometry into the region r < 2m. Two new coordinates which we label u and v are introduced. They are related to the Schwarzschild coordinates t, r in the following ways, depending on whether r < 2m or r > 2m: r > 2m: u = er/4m v = er/4m t r 1 cosh 2m 4m t r 1 sinh 2m 4m (11.5) Make Bar Code In VS .NET Using Barcode generator for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications. EAN 128 Drawer In Visual Studio .NET Using Barcode creator for .NET Control to generate, create UCC128 image in VS .NET applications. r < 2m: u = er/4m 1 v = er/4m 1 r t sinh 2m 4m t r cosh 2m 4m (11.6) GS1 RSS Printer In Visual Studio .NET Using Barcode creation for Visual Studio .NET Control to generate, create GS1 DataBar image in .NET applications. Generate Rationalized Codabar In VS .NET Using Barcode maker for Visual Studio .NET Control to generate, create Codabar image in .NET framework applications. Black Holes
Code 128 Code Set C Encoder In Java Using Barcode printer for Android Control to generate, create Code 128B image in Android applications. Draw EAN / UCC  13 In None Using Barcode generation for Online Control to generate, create GTIN  13 image in Online applications. The Kruskal form of the metric is given by ds 2 = 32m 3 r/2m e du 2 dv 2 + r 2 d 2 + sin2 d 2 r (11.7) Print EAN / UCC  13 In Java Using Barcode drawer for BIRT reports Control to generate, create UCC  12 image in BIRT applications. ANSI/AIM Code 128 Creation In .NET Framework Using Barcode generation for ASP.NET Control to generate, create ANSI/AIM Code 128 image in ASP.NET applications. These coordinates are illustrated in Fig. 113. These coordinates exhibit the following features: UPC Symbol Drawer In Java Using Barcode generation for Android Control to generate, create GTIN  12 image in Android applications. Recognizing USS Code 128 In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. The outside world is labeled by O is the region r 2m, which corresponds to u v. The line u = v corresponds to the Schwarzschild coordinate t while u = v corresponds to t . The region inside the event horizon, which is r < 2m, corresponds to v > u. In Kruskal coordinates, light cones are 45 . ANSI/AIM Code 128 Encoder In None Using Barcode creator for Software Control to generate, create Code128 image in Software applications. Bar Code Generator In None Using Barcode printer for Font Control to generate, create barcode image in Font applications. We also have the following relationships: u2 v 2 = r 1 er/2m 2m and t v = tanh u 4m (11.8) The coordinate singularity at r = 2m corresponds to u 2 v 2 = 0. The real, curvature singularity r = 0 is a hyperbola that maps to v 2 u2 = 1 Once again we can examine the paths of light rays by studying ds 2 = 0. For the Kruskal metric, we have ds 2 = 0 = This immediately leads to du dv 32m 3 r/2m e du 2 dv 2 r
In Kruskal coordinates massive bodies move inside light cones and have slope dv du
r=0 v I O' A r = 2m, t = + r = const O u
Black Holes
I' r = 2m, t = r=0
Fig. 113. An illustration of Kruskal coordinates. Regions O and O are outside the event
horizon and so correspond to r > 2m. Regions I and I correspond to regions r < 2m. The hyperbola r = const is some constant radius outside of r = 2m; it could, for example, represent the surface of a star. which tells us that the velocity of light is 1 everywhere. Therefore there is no boundary to light propagation in these coordinates. Furthermore, u serves as a global radial marker. v serves as a global time marker. The metric is equivalent to the Schwarzschild solution but does not correspond to at spherical coordinates at large distances. There is no coordinate singularity at r = 2m. It still has the real singularity at r = 0. In Fig. 113, the dashed line indicated by A represents a light ray traveling radially inward. The slope is 1 and in Kruskal coordinates it must hit the singularity at r = 0. The Kerr Black Hole
Observation of astronomical objects like the Earth, Sun, or a neutron star reveals one fact: most (if not all) of them rotate. While the Schwarzschild solution still works as a description of the spacetime around a slowly rotating object, to accurately describe a spinning black hole we need a new solution. Such a solution is given by the Kerr metric. The Kerr metric reveals some interesting new phenomena that are wholly unexpected. For example, we will nd that an object that is placed near a spinning black hole cannot avoid rotating along with the black hole no matter what state of motion we give to the object. Put a rocket ship there. Fire the most powerful Black Holes
engines that can be constructed so that the rocket ship will move in a direction opposite to that in which the black hole is rotating. But the engines cannot help no matter what we do the rocket ship will be carried along in the direction of rotation. In fact, we will see below the rotation even effects light! We re also going to see that a rotating black hole has two event horizons. In between these event horizons is a region called the ergosphere, and we will see that it is there where the effects of rotation are felt. It is also possible to extract energy from a Kerr black hole using a method known as the Penrose process. Let s get started examining the Kerr black hole by making some de nitions. As you know a spinning object is characterized by its angular momentum. When describing a black hole, physicists and astronomers give the angular momentum the label J and are usually concerned with angular momentum per unit mass. This is given by a = J/M, and if we are using the gravitational mass for M then the units of a are given in meters. With the Kerr metric, the effects of spinning on the spacetime around the black hole will be seen by the presence of angular momentum in the metric along with mixed or cross terms. These are terms of the form dt d that indicate a change in angle with time a rotation. The Kerr metric is a bit complicated, and so we are simply going to state what it is. To simplify notation, we make the following de nitions: = r 2 2mr + a 2 = r 2 + a 2 cos2 where, as de ned above, a is the angular momentum per unit mass. With these de nitions the metric describing the spacetime around a rotating black hole is ds = 1

