Derivatives of Killing Vectors in Visual Studio .NET

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Derivatives of Killing Vectors
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We can differentiate Killing vectors to obtain some useful relations between Killing vectors and the components of the Einstein equation. For the Riemann tensor, we have c b X a = R a bcd X d The Ricci tensor can be related to Killing vectors via b a X b = Rac X c For the Ricci scalar, we have X a a R = 0 (8.8) (8.6)
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Constructing a Conserved Current with Killing Vectors
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Let X be a Killing vector and T be the stress-energy tensor. Let s de ne the following quantity as a current: J a = T ab X b We can compute the covariant derivative of this quantity. We have a J a = a (T ab X b ) = ( a T ab )X b + T ab ( a X b ) The stress-energy tensor is conserved; therefore, c T ab = 0 and we are left with a J a = T ab ( a X b ) Since the stress-energy tensor is symmetric, the symmetry of its indices allows us to write 1 a J a = T ab ( a X b ) = (T ab a X b + T ba b X a ) 2 = 1 ab T ( a X b + b X a ) = 0 2
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Therefore, J is a conserved current.
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1. Killing s equation is given by (a) b X a a X b = 0 (b) b X a = 0 (c) b X a + a X b = G ab (d) b X a + a X b = 0 Given a Killing vector X , the Riemann tensor satis es (a) c b X a = R a bcd X d
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(b) c b X a = R a bcd X d (c) c b X a = R a bcd X d R a bcd X b 3. Given a Killing vector X , the Ricci scalar satis es (a) X a a R = 0 (b) X a a R = R (c) X a a R = R a a
CHAPTER
Null Tetrads and the Petrov Classi cation
There is a viewpoint that the most fundamental entities that can be used to describe the structure of spacetime are light cones. After all, a light cone divides past and future, and in doing so de nes which events are or can be causally related to one another. The light cone de nes where in spacetime a particle with mass can move nothing moves faster than the speed of light (see Fig. 9-1). We begin by reviewing a few concepts you ve already seen. While they may already be familiar, they are important enough to be reviewed once again. As described in 1, we can plot events in spacetime using a spacetime diagram. One or two spatial dimensions are suppressed, allowing us to represent space and time together graphically. By de ning the speed of light c = 1, light rays move on 45 lines that de ne a cone. This light cone de nes the structure of spacetime for some event E that we have placed at the origin in the following way.
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Null Tetrads and the Petrov
Time
Future
Particle motion permitted along curves inside light cone
Elsewhere Region A Event E at the origin Space
Light rays move on paths that are the diagonals defining the cone
Only events in the past light cone of E can affect it Elsewhere Region B
The Past
Fig. 9-1. The essence of spacetime structure is well described by the light cone, shown
here with only one spatial dimension. Time is on the vertical axis. Particles with mass can move only on paths inside the light cone.
Events in the past that are causally related to E are found in the lower light cone where t < 0 in the diagram. Events that can be affected by E are inside the future light cone where t > 0. No object or particle with mass can move faster than the speed of light, so the motion of any massive particle is restricted to be inside the light cone. In the diagram we have de ned the two regions that are outside the light cone, regions A and B, as elsewhere. These regions are causally separated from each other. No event from region A can impact an event in region B because travel faster than the speed of light would be necessary for that to occur. This is all taking place in perfectly at space, where light travels on straight lines. Gravity manifests itself in the curvature of spacetime. As we will see when we examine the Schwarzschild solution in detail, gravity bends light rays. In a gravitational eld, light no longer travels on perfectly straight lines. The more curvature there is, the more pronounced is the effect. A remarkable consequence of this fact is that in strong gravitational elds, light cones begin to tip over. Inside a black hole we nd the singularity, a point where the curvature of spacetime becomes in nite. As light cones get closer to
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