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Black Holes in .NET
CHAPTER Scanning QR Code In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Encode QR Code In .NET Using Barcode generation for .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications. Black Holes
Read QR Code In VS .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Paint Barcode In .NET Framework Using Barcode generator for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. A black hole is a region of spacetime where gravity is so strong that not even light can escape. In nature, it is believed that black holes form at the end of a stars lifetime, when a massive star runs out of fuel and ends its life in collapse. We shall begin our study of black holes by taking a closer look at the Schwarzschild solution. As we will see, according to classical general relativity, a black hole is completely characterized by just three parameters. These are Barcode Recognizer In VS .NET Using Barcode reader for .NET Control to read, scan read, scan image in .NET framework applications. Paint Denso QR Bar Code In C# Using Barcode generator for VS .NET Control to generate, create QR Code image in VS .NET applications. Mass Charge Angular momentum
Print Quick Response Code In .NET Using Barcode creation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Denso QR Bar Code Maker In VB.NET Using Barcode encoder for VS .NET Control to generate, create QR Code image in .NET applications. This characterization results in three general classes of black holes that are studied: EAN13 Creator In Visual Studio .NET Using Barcode creator for Visual Studio .NET Control to generate, create European Article Number 13 image in .NET framework applications. Create Data Matrix In .NET Using Barcode creator for VS .NET Control to generate, create ECC200 image in Visual Studio .NET applications. Static black holes with no charge, described by the Schwarzschild solution Black holes with electrical charge, described by the ReissnerNordstr m solution Rotating black holes, described by the Kerr solution Print GTIN  128 In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create UCC128 image in Visual Studio .NET applications. Identcode Creation In .NET Framework Using Barcode drawer for .NET Control to generate, create Identcode image in .NET framework applications. Copyright 2006 by The McGrawHill Companies, Inc. Click here for terms of use.
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Print Code 128 In None Using Barcode creation for Microsoft Word Control to generate, create Code 128 Code Set A image in Microsoft Word applications. UPC Symbol Generator In Java Using Barcode maker for Java Control to generate, create UPCA Supplement 5 image in Java applications. In this chapter we will consider two cases: the Schwarzschild and Kerr black holes. To begin, let s review the problem of coordinate singularities and see how to remove the singularity from the Schwarzschild metric. Code128 Reader In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. Generate Code 3 Of 9 In Visual Basic .NET Using Barcode generation for .NET Control to generate, create Code 39 Extended image in .NET framework applications. Redshift in a Gravitational Field
Draw Bar Code In None Using Barcode printer for Font Control to generate, create barcode image in Font applications. Matrix Barcode Creation In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create Matrix Barcode image in ASP.NET applications. When studying black holes you will often see an in nite redshift being discussed. Let s take a moment to see what happens to light as it is emitted upward in a gravitational eld, that is from an observer located at some inner radius ri to an observer positioned at some outer radius ro . The key to seeing what happens to the light is to see how time passes for each observer. In other words we are interested in the period of the light wave as seen by each observer. Recall that the proper time is the time a given observer measures on his or her own clock. For the Schwarzschild metric, for a stationary observer the proper time relates to the time as measured by a distant observer via the relationship 2m dt r d = It is a simple matter to calculate a redshift factor by comparing the proper time for observers located at two different values of r . This is best illustrated by an example. EXAMPLE 111 Consider two xed observers located nearby a Schwarzschild black hole. One observer, located at r1 = 3m, emits a pulse of ultraviolet light to a second observer located at r2 = 8m. Show that the second observer nds that the light has been redshifted to orange. SOLUTION 111 To nd the redshift factor, we simply calculate the ratio d 2 /d 1 where d i =
2m dt ri
Black Holes
Let s denote the redshift factor by . Then we have d 2 = = d 1 1 1 1 1
2m r2 2m r1 2m 8m 2m 3m
dt dt =
1 1 1 4 2 3 2m r2 2m r1 3 4 1 3
1 1 9 3 = 4 2 We can take the wavelength of ultraviolet light to be 1 400 nm. The wavelength that the second observer sees is then 2 = 1 = 1.5 400 nm = 600 nm This is in the orange region of the spectrum, which roughly runs from 542 nm to 620 nm. Coordinate Singularities
Let s step back for a moment and review the distinction between coordinate and curvature singularities. First recall the Schwarzschild metric given in (10.33): ds 2 = 1 2m r dt 2 dr 2 r 2 (d 2 + sin2 d 2 ) (1 2m/r ) It s pretty clear that the Schwarzschild metric exhibits unusual behavior at r = 2m. For r > 2m, gtt > 0 and grr < 0. However, notice that for r < 2m, the signs of these components of the metric reverse. This means that a world line along the taxis has ds 2 < 0 and so describes a spacelike curve. Meanwhile, a world line along the r axis has ds 2 > 0 and so describes a timelike curve. The time and space character of the coordinates has reversed. This indicates that a massive particle inside the Schwarzschild radius could not remain stationary at a constant value of r. Now let s take a direct look at r = 2m. Considering gtt rst, we see that at r = 2m gtt = 1 2m =1 1=0 2m

