Calculating Killing vectors of Schwarzschild metric

  • #1
MathematicalPhysicist
Gold Member
4,300
205
I am trying to understand the solution to exercise 7.10(e) on pages 175-176 of Robert Scott's student's manual to Schutz's textbook.

He writes the following:
Table 7.1 rows four, five and six, lists the three Killing vector fields associated with invariance for rotations about the three spatial Cartesian axes.
Because Schwarzschild also has spherical symmetry it enjoys the same Killing vector fields.
We can transform these into the spherical coordinates of (ii) using relations in Appendix B giving:
$$\vec{Q}=\vec{e_t}$$
$$\vec{R}=\vec{e_\phi}$$
$$\vec{S}=\bigg(\frac{\partial \theta}{\partial x} S^x +\frac{\partial \theta}{\partial z} S^z\bigg)\vec{e_\theta}+\bigg(\frac{\partial \phi}{\partial x} S^x+\frac{\partial \phi}{\partial z} S^z\bigg)\vec{e_\phi}=\cos \phi \vec{e_\theta}-\cot \theta \sin \phi \vec{e_\phi}$$
$$\vec{T}=\bigg(\frac{\partial \theta}{\partial x} T^x +\frac{\partial \theta}{\partial z} T^z\bigg)\vec{e_\theta}+\bigg(\frac{\partial \phi}{\partial x} T^x+\frac{\partial \phi}{\partial z} T^z\bigg)\vec{e_\phi}=\sin \phi \vec{e_\theta}-\cot \theta \cos \phi \vec{e_\phi}$$
I don't understand how to find ##S^x, S^z## or ##T^x,T^z## from the metric or from the cartesian representation of the rotation vectors?
The derivatives are calculated with spherical coordinates which I understand how to achieve them.

Any help?
 
Last edited:

Answers and Replies

  • #2
MathematicalPhysicist
Gold Member
4,300
205
Ok, I believe I found my answer: ##S^x=z , S^z=-x## and ##T^x=0, T^z=-y##.
 
  • #3
MathematicalPhysicist
Gold Member
4,300
205
There's typo in the text, it should be ##T^y=z , T^z=-y##.
 

Related Threads on Calculating Killing vectors of Schwarzschild metric

Replies
4
Views
1K
  • Last Post
Replies
20
Views
11K
Replies
3
Views
700
Replies
7
Views
3K
  • Last Post
Replies
3
Views
868
  • Last Post
Replies
23
Views
1K
  • Last Post
Replies
10
Views
6K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
4K
Top