4amr in Visual Studio .NET

Drawing QR Code ISO/IEC18004 in Visual Studio .NET 4amr

2 4amr
Recognizing QR Code 2d Barcode In .NET
Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in VS .NET applications.
QR Code Generation In .NET Framework
Using Barcode maker for Visual Studio .NET Control to generate, create Quick Response Code image in .NET applications.
sin2
Denso QR Bar Code Scanner In VS .NET
Using Barcode scanner for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications.
Bar Code Generator In .NET
Using Barcode generation for .NET framework Control to generate, create bar code image in Visual Studio .NET applications.
dt d sin2 d 2
Scanning Bar Code In Visual Studio .NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications.
Quick Response Code Maker In C#
Using Barcode creator for VS .NET Control to generate, create QR image in VS .NET applications.
dr 2
QR Code ISO/IEC18004 Maker In .NET
Using Barcode generator for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications.
Paint Quick Response Code In Visual Basic .NET
Using Barcode generation for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications.
d 2 (11.9)
DataMatrix Generation In .NET Framework
Using Barcode printer for VS .NET Control to generate, create Data Matrix image in .NET framework applications.
Encode EAN13 In Visual Studio .NET
Using Barcode printer for Visual Studio .NET Control to generate, create EAN 13 image in VS .NET applications.
r 2 + a2 +
Paint GS1 DataBar-14 In Visual Studio .NET
Using Barcode creator for VS .NET Control to generate, create GS1 DataBar-14 image in Visual Studio .NET applications.
UPC - E0 Drawer In Visual Studio .NET
Using Barcode printer for .NET framework Control to generate, create UCC - 12 image in VS .NET applications.
2a 2 mr sin2
Data Matrix Recognizer In None
Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications.
Barcode Creation In None
Using Barcode creation for Office Word Control to generate, create barcode image in Microsoft Word applications.
We have written the metric in Boyer-Lindquist coordinates. The components of the metric tensor are given by gtt = 1 2mr , gt = g t = 2mar sin2
Encoding EAN 13 In Java
Using Barcode creation for Java Control to generate, create EAN13 image in Java applications.
USS Code 128 Generator In Visual C#
Using Barcode encoder for Visual Studio .NET Control to generate, create Code-128 image in Visual Studio .NET applications.
grr = , g = , g = r 2 + a 2 +
Barcode Maker In None
Using Barcode generator for Online Control to generate, create bar code image in Online applications.
Data Matrix 2d Barcode Generator In Visual Basic .NET
Using Barcode maker for Visual Studio .NET Control to generate, create ECC200 image in .NET applications.
Black Holes
GTIN - 128 Creator In Java
Using Barcode creation for BIRT Control to generate, create GTIN - 128 image in BIRT applications.
UPC Symbol Generator In None
Using Barcode generation for Font Control to generate, create UPC-A Supplement 5 image in Font applications.
sin2 (11.10)
2ma 2r sin2
Note that the metric components are independent of t and . This implies that two suitable killing vectors for the spacetime are t and . To invert this complicated metric, we rst make the observation that the off-diagonal terms involve only gt . Therefore, we can invert the terms grr and g using grr grr = 1 grr = , g g = 1 g = 1 (11.11)
To nd the other components, we form the matrix gtt g t gt g = 2 2mar sin 1 2mr 2mar sin2 r 2 + a2 + 2ma 2r sin2 sin2
(11.12) This matrix can be inverted with a great deal of tedious algebra, or using computer (the path we choose), which gives g =
r 2 + a2
a 2 sin2
g t =
2mar
g =
a 2 sin2
(11.13)
The fact that there are mixed components of the metric tensor leads to some interesting results. For example, we can consider the four momentum of a particle with components pt , pr , p , p . Notice that p t = g ta pa = g tt pt + g t p p = g a pa = g t pt + g p and so a particle will have a nonzero p = g t pt even when p = 0. It is possible to simplify matters a bit and still get to the essential features of the Kerr metric. Let s consider the equatorial plane, which is a plane that cuts
Black Holes
right through the equator of a sphere. If we imagine the sphere being the earth or some other rotating body, the plane is perpendicular to spin axis. In the case of a black hole, we can also imagine a plane through the center of the black hole and perpendicular to the spin axis. In this case, = 0, which means that cos = 1 and d = 0. Looking at the metric in (11.9) together with the de nitions of and , we see that the metric is greatly simpli ed. We can write ds 2 = 1 2m r dt 2 + 4ma 1 dtd r 1 2m + r dr 2
a2 r2
1+
a 2 2ma 2 2 2 r d + 3 r2 r
(11.14)
With this simpli ed metric, some features of the spacetime about a spinning black hole jump out at you rather quickly. First let s note that the time coordinate used in the metric, t, is the time recorded by a distant observer as it was for the Schwarzschild metric. With this in mind, let s follow the same procedure used with the Schwarzschild metric and note where terms go to zero or blow up. The rst thing to note about this metric is that the coef cient gtt is the same as that we saw in the Schwarzschild metric (10.33). Let s set it to zero to get 2m =0 r
Solving for r , this term goes to zero at rs = 2m. This is the same value we found in the Schwarzschild case, but since the present metric is more complicated we are going to nd other values of r where interesting things are happening. So we call this the static limit. More on this in a moment. For now, let s turn to the grr term. It is here that we see the rst impact of rotation. In the Schwarzschild case we were interested in seeing where this term blew up. We are in this case as well, and notice that now grr depends on the angular momentum per unit mass a. We have grr = 1 1
Copyright © OnBarcode.com . All rights reserved.