 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
asp net display barcode Null Tetrads and the Petrov in .NET framework
Null Tetrads and the Petrov Scan QRCode In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. QRCode Generation In .NET Using Barcode creator for .NET Control to generate, create Quick Response Code image in .NET applications. In the context of gravitational radiation, we have the following interpretations: QR Code 2d Barcode Reader In .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Generation In VS .NET Using Barcode encoder for .NET Control to generate, create barcode image in VS .NET applications. 0 1 3 4 Barcode Decoder In .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. QR Code ISO/IEC18004 Drawer In C# Using Barcode creator for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET applications. transverse wave component in the n a direction longitudinal wave component in the n a direction transverse wave component in the l a direction longitudinal wave component in the l a direction Painting Denso QR Bar Code In .NET Using Barcode creator for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications. Create Denso QR Bar Code In Visual Basic .NET Using Barcode generator for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications. Finally, we mention the physical meaning of some of the spin coef cients. First we consider a congruence of null rays de ned by l a : =0 Re ( ) Im ( )   l a is tangent to a null congruence expansion of a null congruence (rays are diverging) twist of a null congruence shear of a null congruence Print EAN128 In Visual Studio .NET Using Barcode printer for VS .NET Control to generate, create USS128 image in Visual Studio .NET applications. Paint Code39 In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create USS Code 39 image in .NET framework applications. For a congruence de ned by n a , these de nitions hold for , , and . EXAMPLE 95 Consider the Brinkmann metric that describes plane gravitational waves: ds 2 = H (u, x, y) du 2 + 2 du dv dx 2 dy 2 Find the nonzero Weyl scalars and components of the Ricci tensor, and interpret them. SOLUTION 95 One can calculate these quantities by de ning a basis of orthonormal one forms and proceeding with the usual methods, and then using (9.22) and related equations to calculate the desired quantities directly. However, we will take a new approach and use the NewmanPenrose identities to calculate the Ricci and Weyl scalars directly from the null tetrad. The coordinates are (u, v, x, y) where u and v are null coordinates. Specifically, u = t x and v = t + x. The vector v de nes the direction of propagation of the wave. Therefore, it is convenient to take l a = (0, 1, 0, 0) = v We can raise and lower the indices using the components of the metric tensor, so let s begin by identifying them in the coordinate basis. We recall that we can Generate Bar Code In .NET Framework Using Barcode creation for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. Creating Uniform Symbology Specification Code 93 In .NET Using Barcode creation for Visual Studio .NET Control to generate, create Uniform Symbology Specification Code 93 image in Visual Studio .NET applications. write the metric as
Barcode Decoder In C# Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. Bar Code Reader In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Null Tetrads and the Petrov
Recognizing Barcode In .NET Using Barcode Control SDK for ASP.NET Control to generate, create, read, scan barcode image in ASP.NET applications. UPC  13 Drawer In Java Using Barcode encoder for Eclipse BIRT Control to generate, create GS1  13 image in Eclipse BIRT applications. ds 2 = gab dx a dx b Therefore we can just read off the desired terms. Note that the cross term is given by 2 du dv = guv du dv + gvu dv du and so we take guv = gvu = 1. The other terms can be read off immediately and so ordering the coordinates as [u, v, x, y], we have H 1 (gab ) = 0 0 1 0 0 0 0 0 0 1 0 0 0 1 Decoding Barcode In VB.NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET framework applications. DataMatrix Maker In Java Using Barcode creator for Java Control to generate, create ECC200 image in Java applications. The inverse of this matrix gives the components of the metric tensor in the coordinate basis with raised indices: 0 1 = 0 0 1 H 0 0 0 0 1 0 0 0 0 1 Encoding Code 3 Of 9 In None Using Barcode generator for Software Control to generate, create Code 39 image in Software applications. Barcode Scanner In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. g ab
With this information in hand, we can calculate the various components of the null vectors. We start with l a = (0, 1, 0, 0). Lowering the index, we nd la = gabl b = gav l v The only nonzero term in the metric of this form is guv = 1, and so we conclude that lu = guv l v = 1 la = (1, 0, 0, 0) (9.25) To nd the components of the rest of the tetrad, we can apply (9.8) and (9.9). To avoid ipping back through the pages, let s restate the identity for gab here: gab = la n b + lb n a m a m b m b m a Null Tetrads and the Petrov
Once we have all the components with lowered indices, we can raise indices with g ab to obtain all the other terms we need. Since there are not many terms in the metric, this procedure will be relatively painless. Since we also have lu = 1 as the only nonzero component of the rst null vector, we can actually guess as to what all the terms are without considering every single case. Starting from the top, we have guu = H = 2lu n u 2m u m u guv = 1 = lu n v + lv n u m u m v m u m v = lu n v m u m v m u m v gvu = 1 = lv n u + lu n v m v m u m v m u gxx = 1 = 2l x n x 2m x m x = 2m x m x g yy = 1 = 2l y n y 2m y m y = 2m y m y We are free to make some assumptions, since all we have to do is come up with a null tetrad that gives us back the metric. We take m u = m v = m u = m v = 0 and assume that the x component of m is real. Then from the rst and second equations, since lu = 1, we obtain nu = 1 H 2 and nv = 1 Now n is a null vector so we set n x = n y = 0. The fourth equation yields 1 mx = mx = 2 These terms are equal to each other since they are complex conjugates and this is the real part. The nal equation gives us the complex part of m. If we choose 1 1 m y = i 2 then the fth equation will be satis ed if m y = i 2 , which is what we expect anyway since it s the complex conjugate. Anyway, all together we have la = (1, 0, 0, 0) , na = 1 H, 1, 0, 0 2 1 m a = (0, 0, 1, i) 2 1 m a = (0, 0, 1, i) , 2 (9.26)

