Killing vectors in KS coordinates

In summary, the conversation discusses the investigation of whether the given vector with components is a Killing vector of the KS space-time with line element. The vector is shown to satisfy the constraints and is an isometry, making it a Killing vector. The conversation also mentions using GRT to handle the constraints, but it is found to be unnecessary as the proof can be done by inspection.
  • #1
pervect
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I want to prove (or disprove) that the vector with components
[tex]\xi^u = \frac{v}{2 r_s}\hspace{5 mm} \xi^v = \frac{u}{2 r_s}[/tex]

is a Killing vector of the KS space-time with line element

[tex]\frac{4 r_s^3}{r} e^{-\frac{r}{r_s}} \left( du^2 -dv^2\right) + r^2 \left( d\theta^2 + sin^2 \theta d\phi^2\right) [/tex]

Here r is an implicit function of u,v, which is handled by a constraint equation.

However, I seem to be having a hard time getting GRT to handle the contraints. I try

[tex]\left( \frac{r}{r_s} - 1 \right) e^{\frac{r}{r_s}} = u^2 - v^2[/tex] directly, but it doesn't simplify.

I try to feed it the following
[tex]

{\frac {\partial }{\partial u}}r \left( u,v \right) =2\,{{\it r\_s}}^{
2}u \left( r \left( u,v \right) \right) ^{-1} \left( {e^{{\frac {r
\left( u,v \right) }{{\it r\_s}}}}} \right) ^{-1}

[/tex]

but it complains about "illegal use of object as a name", I can't see what it's objecting to.

So a) -does this look like the right expression for the Killing Vector? And b) - how does one successfully get the constraints into GrTensor?
 
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  • #2
Don't need any complicated algebra, it's true almost by inspection. Just show directly that the Lie derivative of gμν with respect to ξμ is zero. The coordinate transformation is u → u + εv, v → v + εu. Then u2 - v2 → u2 - v2 so it obeys the constraint, and du2 - dv2 → du2 - dv2 so it's an isometry, and that's really all there is to it!
 
  • #3
I set r_s to 1, and renamed r to rr,and finally got GRT to crunch through it,but your way is both much easier and more insightful. Thanks!
 

FAQ: Killing vectors in KS coordinates

1. What are Killing vectors in KS coordinates?

Killing vectors in KS coordinates refer to a set of vector fields in the Kerr-Schild coordinates that satisfy the Killing equation. These vectors represent the symmetries of the spacetime manifold and play a crucial role in understanding the properties of the Kerr black hole.

2. How are Killing vectors in KS coordinates related to the Kerr metric?

The Kerr metric, which describes the geometry of the Kerr black hole, can be expressed in terms of the Killing vectors in KS coordinates. These vectors provide a coordinate basis for the metric and allow for a simpler representation of the spacetime geometry.

3. What is the significance of Killing vectors in KS coordinates?

Killing vectors in KS coordinates have several important applications, such as calculating the conserved quantities of a particle orbiting a Kerr black hole and finding the symmetries of the spacetime. They also play a crucial role in understanding the geodesic motion of particles in the Kerr metric.

4. How are Killing vectors in KS coordinates used in the study of black holes?

Killing vectors in KS coordinates are used extensively in the study of black holes, particularly the Kerr black hole. They help in understanding the properties of the black hole, such as its event horizon, ergosphere, and the singularity at its center.

5. Can Killing vectors in KS coordinates be extended to other spacetimes?

Yes, Killing vectors in KS coordinates can be extended to other spacetimes, as long as the spacetime has a Kerr-Schild metric. However, the properties and applications of these vectors may differ depending on the specific spacetime geometry.

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