Gravitational Waves in VS .NET

Drawer QR Code 2d barcode in VS .NET Gravitational Waves

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.
QR-Code 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. 13-3. When h xx < 0, the relative distance between particles separated along the
Decode Bar Code In VS .NET
Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications.
QR Code ISO/IEC18004 Printer In Visual C#
Using Barcode generator for Visual Studio .NET Control to generate, create QR Code image in .NET framework applications.
y-axis increases.
Encode QR Code JIS X 0510 In .NET Framework
Using Barcode encoder for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications.
Create QR Code JIS X 0510 In VB.NET
Using Barcode maker for .NET framework Control to generate, create QR-Code image in .NET framework applications.
distance is given by ds 2 = (1 h xx ) dx 2 First we consider h xx < 0. The form of the line element shown here indicates that the relative distance between the two particles will decrease. This behavior is shown in Fig. 13-4. On the other hand, when h xx > 0, the line element becomes more positive and therefore the relative distances between particles will increase. This is shown in Fig. 13-5. The behavior of the particles discussed in these special cases allows us to extrapolate to a more general situation. It is common to consider a ring of
Code 128 Code Set C Drawer In .NET
Using Barcode maker for .NET Control to generate, create Code 128 Code Set B image in .NET applications.
Barcode Drawer In Visual Studio .NET
Using Barcode printer for .NET framework Control to generate, create barcode image in .NET framework applications.
y hxx < 0
GS1-128 Maker In VS .NET
Using Barcode maker for Visual Studio .NET Control to generate, create EAN 128 image in .NET framework applications.
Draw USPS POSTal Numeric Encoding Technique Barcode In VS .NET
Using Barcode generator for Visual Studio .NET Control to generate, create Postnet image in .NET framework applications.
Fig. 13-4. Particles separated along the x-axis. When h xx < 0, the relative physical
Data Matrix Scanner In Visual Basic .NET
Using Barcode decoder for .NET Control to read, scan read, scan image in VS .NET applications.
Making Code 39 In Objective-C
Using Barcode drawer for iPad Control to generate, create ANSI/AIM Code 39 image in iPad applications.
displacements between the particles decrease.
Generate Barcode In None
Using Barcode creation for Font Control to generate, create barcode image in Font applications.
Creating Bar Code In None
Using Barcode maker for Excel Control to generate, create bar code image in Office Excel applications.
Gravitational Waves
Make EAN-13 In Java
Using Barcode creator for Java Control to generate, create EAN-13 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
Data Matrix 2d Barcode Recognizer In VS .NET
Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications.
GTIN - 13 Generator In .NET
Using Barcode encoder for ASP.NET Control to generate, create EAN13 image in ASP.NET applications.
Fig. 13-5. Particles separated along the x-axis with h xx > 0. The relative physical
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. 13-6. 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 Newman-Penrose 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 Newman-Penrose 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,
Copyright © OnBarcode.com . All rights reserved.