Calculating Killing vectors of Schwarzschild metric

In summary, the conversation discusses the use of Killing vector fields in solving exercise 7.10(e) on pages 175-176 of Robert Scott's student's manual to Schutz's textbook. These vector fields are associated with invariance for rotations about the three spatial Cartesian axes and can be transformed into spherical coordinates. The person also asks for clarification on how to find certain components of the vector fields, which is later answered.
  • #1
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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:
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  • #2
Ok, I believe I found my answer: ##S^x=z , S^z=-x## and ##T^x=0, T^z=-y##.
 
  • #3
There's typo in the text, it should be ##T^y=z , T^z=-y##.
 

What is the Schwarzschild metric?

The Schwarzschild metric is a solution to Einstein's field equations in general relativity that describes the curvature of spacetime around a non-rotating, uncharged, spherically symmetric mass. It is commonly used to describe the geometry of black holes.

What are Killing vectors?

Killing vectors are vector fields in a spacetime that preserve the metric, meaning that they do not change the distance between two points. In other words, they represent symmetries in the geometry of the spacetime.

Why is it important to calculate Killing vectors of the Schwarzschild metric?

Calculating Killing vectors of the Schwarzschild metric allows us to understand the symmetries of spacetime around a black hole, which can provide insight into the behavior of particles and light in its vicinity. It also helps in solving various problems in general relativity and in developing new theories.

How do you calculate Killing vectors of the Schwarzschild metric?

The process involves finding the vector fields that satisfy the Killing equation, which is a set of differential equations that must be satisfied for a vector field to be a Killing vector. In the case of the Schwarzschild metric, the Killing equation reduces to a set of 4 equations that can be solved to find the Killing vectors.

What are some applications of calculating Killing vectors of the Schwarzschild metric?

One application is in studying the behavior of test particles and light in the vicinity of a black hole. The Killing vectors can also be used to find conserved quantities, such as energy and angular momentum, for particles moving in the black hole's gravitational field. Additionally, the symmetries provided by the Killing vectors can help in finding exact solutions to Einstein's equations for more complex systems.

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