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dt 2 = in .NET framework
dt 2 = Recognizing QR Code In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Draw Quick Response Code In .NET Framework Using Barcode creation for .NET Control to generate, create QR image in VS .NET applications. (13.32) QR Code ISO/IEC18004 Recognizer In Visual Studio .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in VS .NET applications. Barcode Encoder In VS .NET Using Barcode generation for .NET Control to generate, create bar code image in .NET framework applications. dz 2 =
Decode Barcode In .NET Framework Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. QR-Code Generator In Visual C# Using Barcode encoder for .NET Control to generate, create QR image in Visual Studio .NET applications. (13.33) Denso QR Bar Code Generator In .NET Using Barcode generator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Print QR Code ISO/IEC18004 In Visual Basic .NET Using Barcode drawer for .NET framework Control to generate, create QR Code image in VS .NET applications. We will now proceed to write the plane wave metric given by (13.18) in terms of U and V to nd ds 2 = dt 2 (1 h xx ) dx 2 (1 + h xx ) dy 2 dz 2 = 1 dU 2 + 2dU dV + dV 2 (1 h xx ) dx 2 4 1 dU 2 2dU dV + dV 2 (1 + h xx ) dy 2 4 Drawing Bar Code In .NET Using Barcode printer for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Draw Code 128 Code Set A In Visual Studio .NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code 128B image in .NET framework applications. = dU dV (1 h xx ) dx 2 (1 + h xx ) dy 2 Now, recalling that h xx = h xx (t z) = h xx (U ), we de ne a 2 (U ) = 1 h xx and b2 (U ) = 1 + h xx Printing Matrix 2D Barcode In .NET Framework Using Barcode drawer for .NET Control to generate, create 2D Barcode image in Visual Studio .NET applications. Generating Identcode In .NET Using Barcode maker for .NET framework Control to generate, create Identcode image in .NET applications. and we obtain the Rosen line element ds 2 = dU dV a 2 (U ) dx 2 b2 (U ) dy 2 (13.34) Printing UCC - 12 In Java Using Barcode encoder for Java Control to generate, create UPC-A Supplement 2 image in Java applications. Code 39 Extended Encoder In None Using Barcode drawer for Font Control to generate, create Code39 image in Font applications. Gravitational Waves
Code-128 Creator In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create USS Code 128 image in .NET applications. Create Code 128 In Objective-C Using Barcode generation for iPhone Control to generate, create Code 128C image in iPhone applications. We choose the following basis of one forms for this metric: 0 = with
Generating Data Matrix ECC200 In Java Using Barcode printer for Eclipse BIRT Control to generate, create Data Matrix image in BIRT applications. Print ECC200 In Objective-C Using Barcode creator for iPad Control to generate, create Data Matrix image in iPad applications. 1 (dU + dV ) , 2 Code 128A Encoder In Java Using Barcode maker for BIRT Control to generate, create Code 128 image in BIRT applications. Linear Barcode Printer In Visual C#.NET Using Barcode generator for .NET Control to generate, create Linear 1D Barcode image in VS .NET applications. 1 = 1 (dU dV ) , 2 2 = a (u) dx, 3 = b (u) dy (13.35) a b
1 0 = 0 0 0 1 0 0 0 0 0 0 1 0 0 1 An exercise (see Quiz) shows that the nonzero components of the Ricci tensor in this basis are R00 = R01 = R10 = R11 = The vacuum equations Ra b = 0 give 1 d2 b 1 d2 a + =0 a dU 2 b dU 2 1 d 1 d Letting h(U ) = a dUa2 , we see that this equation implies that b dUb2 = h(U ) and so we can write the metric in terms of the single function h. Therefore (13.34) becomes 1 d2 b 1 d2 a + a dU 2 b dU 2
ds 2 = dU dV h 2 (U ) dx 2 h 2 (U ) dy 2 We now apply the following coordinate transformation. Let u = U, v = V + x 2 aa + y 2 bb , X = ax, Y = by From the rst two equations, we obtain du = dU and V = v x 2 aa y 2 bb =v a 2 b 2 X Y a b
Inverting the last two equations for x and y gives
Gravitational Waves
1 1 X, y = Y a b a 1 dx = dX 2 X du a a x = Now we use these relations along with the de nition h(u) = a /a = b /b and obtain dV = dv 2 + (a )2 a b a b X dX X 2 du + 2 X 2 du 2 Y dY Y 2 du a a a b b (b )2 2 Y du b2 (a )2 a b X dX h(u)X 2 du + 2 X 2 du 2 Y dY + h(u)Y 2 du a a b
= dv 2 +
(b )2 2 Y du b2
We also have a dx = a
2 2 2 a 1 2 X du + dX a a b 1 X du + dX 2 b b
(a )2 2 2 a = 2 X du 2 du dX + dX 2 a a = (b )2 2 2 b Y du 2 du dY + dY 2 2 b b
b2 dy 2 = b2
Substitution of these results into ds 2 = dU dV a 2 (U ) dx 2 b2 (U ) dy 2 gives the Brinkmann metric ds 2 = h(u) Y 2 X 2 du 2 + 2 du dv dX 2 dY 2 (13.36) From here on we will drop the uppercase labels and let X x and Y y (just be aware of the relationship of these variables to the coordinates used in the original form of the plane wave metric given in (13.18)). More generally, if we de ne an arbitrary coef cient function H (u, x, y), we can write this metric as ds 2 = H (u, x, y) du 2 + 2 du dv dx 2 dy 2 (13.37) Gravitational Waves
This metric represents a pp-wave spacetime; however, it does not necessarily represent plane waves, which are a special case. We will describe the form of H in the case of plane waves below. In Example 9-5, we found the Weyl and Ricci scalars for this metric. In that problem we choose la = (1, 0, 0, 0) as the covariantly constant null vector where the coordinates are given by (u, v, x, y) . The principal null direction is along la , which means that this vector de nes the direction along which the rays of the gravity wave are coincident. The coordinate v is another null coordinate while the coordinates x, y de ne the surface of the plane wave. It is obviously true that la,b = 0. To understand the implication of this requirement, we state some of the spin coef cients here: = la,b m a m b , = la,b m a m b , = la,b m a n b We see that la,b = 0 implies that = = = 0. Therefore, as we explained in the beginning of this section, a pp-wave has no expansion, twist, or shear. In addition, the fact that = 0 tells us that the null rays de ned by la are parallel. This shows that this metric is a pp-wave spacetime. The form of the function H in the Brinkmann metric can be studied further. A generalized pp-wave is one for which H can be written in the form H = Ax 2 + By 2 + C x y + Dx + E y + F where A, B, C, D, E, and F are real valued functions of u.
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