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Gravitational Waves in VS .NET
Gravitational Waves QR Code JIS X 0510 Decoder In .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. QRCode Generator In .NET Using Barcode creation for .NET Control to generate, create QR image in Visual Studio .NET applications. hxx < 0
Scanning QR Code In .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications. Creating Bar Code In .NET Framework Using Barcode encoder for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Fig. 133. When h xx < 0, the relative distance between particles separated along the
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Make EAN13 In Java Using Barcode creator for Java Control to generate, create EAN13 image in Java applications. Encoding DataMatrix In VB.NET Using Barcode creator for .NET framework Control to generate, create Data Matrix 2d barcode image in Visual Studio .NET applications. y hxx > 0
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displacements between the particles increase.
particles lying in the plane and show how the ring is distorted by a passing gravitational wave. In particular, imagine that the ring starts off as a perfect circle. As the wave passes, h xx will oscillate between positive, zero, and negative values, causing the relative distances between particles to change in the manner just described. A transverse wave with h xx = 0 and h xy = 0 is referred to as one with +polarization. We now examine the other polarization case, by setting h xx = 0. This time the line element is given by ds 2 = dt 2 dx 2 dy 2 dz 2 2 h xy dx dy (13.19) Consider the following transformation, which can be obtained by a rotation of /4 : dx =
dx dy 2
dy =
dx + dy 2
hxx > 0
hxx > 0
hxx = 0
hxx < 0
Fig. 136. The effect of a passing gravity wave with +polarization on a ring of particles.
The ring pulsates as the wave passes.
Gravitational Waves
Writing the line element with these coordinates, we obtain ds 2 = dt 2 1 h xy dx 2 1 + h xy dy 2 dz 2 (13.20) Now doesn t that look familiar It looks just like the line element we examined in (13.18). The behavior induced by the wave will be identical to that in the last case; however, this time everything is rotated by /4. This polarization is known as polarization. In general, a plane gravitational wave will be a superposition of these two polarizations. The Weyl Scalars
In this section we review the Weyl scalars and brie y describe their meaning. They are calculated using the spin coef cients given in (9.15) in combination with a set of equations known as the NewmanPenrose identities. In all, there are ve Weyl scalars which have the following interpretations: 0 1 2 3 4 ingoing transverse wave ingoing longitudinal wave electromagnetic radiation outgoing longitudinal wave outgoing transverse wave (13.21) In most cases of interest we shall be concerned with transverse waves, and therefore with the Weyl scalars 0 and 4 . The following NewmanPenrose identities can be used to calculate each of the Weyl scalars: 0 1 2 = D ( + ) (3 ) + ( + + 3 ) = D ( + ) ( ) + ( + ) + ( ) = + + ( + ) + 2 (13.22) (13.23) (13.24) (13.25) (13.26) = = + ( + ) ( + ) + ( ) + ( + ) (3 ) + 3 + + Gravitational Waves
Review: Petrov Types and the Optical Scalars
It is very useful to study gravitational waves using the formalism introduced in 9. First we give a quick summary of the Petrov classi cation in relation to the Weyl scalars discussed in 9. The Petrov type of a spacetime indicates the number of principal null directions that spacetime has, and how many times each null direction is repeated. We can summarize the Petrov classi cations that are primarily of interest in this chapter in relation to the Weyl scalars in the following way: Petrov Type N: There is a single principal null direction, repeated four times. If l a is aligned with the principal null direction, then 0 = 0 and 4 is the only nonzero Weyl scalar. If n a is aligned with the principal null direction, then 4 = 0 and 0 is the only nonzero Weyl scalar. Petrov Type III: There are two principal null directions, one of multiplicity one and one repeated three times. The nonzero Weyl scalars are 3 and 4 . Petrov Type II: There is one doubly repeated principal null direction and one two distinct null directions. The nonzero Weyl scalars are 2, 3 , and 4 . Petrov Type D: There are two principal null directions, each doubly repeated. In this case 2 is the only nonzero Weyl scalar. In particular, we recall three quantities de ned in terms of the spin coef cients given in (9.15). These are the optical scalars, which describe the expansion,

