Show that killing vector field satisfies....

In summary, the conversation is about spacetime being stationary and static, and the relationship between these two conditions and the existence of a timelike Killing vector field. The speakers discuss the conditions for spacetime to be stationary and static, and one speaker presents their solution for part (i) and asks for suggestions for part (iii). They also discuss a potential mistake in the solution for part (ii).
  • #1
Augbrah
3
0
I'm trying to do past exam papers in GR but there are some things I don't yet feel comfortable with, so even though I can do some parts of the question I would be very happy if you could check my solution. Thank you!

1. Homework Statement

Spacetime is stationary := there exists a coord chart with a timelike coordinate [itex]x^0[/itex] such that metric components [itex]\partial_0 g_{\mu \nu} = 0[/itex].
Spacetime is static := there exists a coord chart such that [itex]\partial_0 g_{\mu \nu} = 0[/itex] and [itex]g_{0i} = 0[/itex]

i) [Done] Show that spacetime is stationary if and only if there exists a timelike Killing vector field V.
ii) [Done] Show that if spacetime is static, there exists a timelike Killing vector field satisfying [itex]V_{[\alpha}\nabla_\mu V_{v]}=0[/itex]
iii) Let V be timelike Killing vector field with [itex]V_{[\alpha}\nabla_\mu V_{v]}=0[/itex]. Show that this condition implies [tex]\nabla_\mu(|V|^n V_\nu)-\nabla_\nu(|V|^n V_\mu)=0[/tex] Where [itex]|V|^2=V_a V^a[/itex] and [itex]n[/itex] is integer which should be determined.

Homework Equations


Killing vector field V satisfies [itex]\nabla_\mu V_\nu + \nabla_\nu V_\mu = 0[/itex]

The Attempt at a Solution


i) I have done this part. For timelike [itex]V^\mu[/itex] we can find an inertial frame s.t. [itex]V^\mu = \delta^\mu_0[/itex]. Then [itex]V_\mu = g_{\mu 0}[/itex]. Hence [itex]\nabla_\mu V_\nu + \nabla_\nu V_\mu =...=\partial_0 g_{\mu\nu}[/itex], by expressing Christoffel symbols as partial derivatives of a metric other derivatives nicely cancel.

So we proved that timelike Killing V ⇔ [itex]\partial_0 g_{\mu\nu}=0[/itex].

ii) If spacetime is static, it is also stationary, so we already know from i) there must exist timelike Killing V, just need to show it satisfied the equation (right?). By expanding antisymmetrization we have
[tex]V_{\alpha}(\nabla_\mu V_{\nu} - \nabla_\nu V_{\mu}) + \text{(other 2 cyclicly permuted pairs)}=0[/tex] Studying the first term and using same identification [itex]V_\mu = g_{\mu 0}[/itex] and as before remembering that for static spacetime [itex]g_{0i} = 0[/itex] as well as [itex]\partial_0 g_{\mu \nu} = 0[/itex] we get:
[tex]V_{\alpha}(\nabla_\mu V_{\nu} - \nabla_\nu V_{\mu}) = g_{\alpha 0} (\partial_\mu g_{\nu 0} - \partial_\nu g_{\mu 0}) = 0[/tex]

Same procedure for other two terms and total sum is 0.

iii) Besides product rule I did not got far. Any suggestions?
 
Physics news on Phys.org
  • #2
You should probably how far you got with part (iii).
 
  • #3
I might be missing something but I don't think what you wrote at the end of part ii) is correct. If you put μ=i and ν=0 this is not zero but equal to [itex]g_{\alpha 0}\partial_i g_{00}[/itex] which is not necessarily zero
You really need to antisymmetrise in order to get zero identically
 
Last edited:

1. What is a killing vector field?

A killing vector field is a type of vector field in mathematics that preserves the metric of a manifold. In other words, it is a vector field which, when applied to a metric tensor, results in a zero change in the metric.

2. How is a killing vector field related to symmetries?

Killing vector fields are closely related to symmetries because they represent infinitesimal transformations that leave the geometry of a manifold unchanged. In other words, they describe the symmetries present in a manifold.

3. Why is it important to show that a killing vector field satisfies certain conditions?

It is important to show that a killing vector field satisfies certain conditions because it helps us understand the properties and behaviors of a manifold. By studying these conditions, we can gain insight into the symmetries and geometric structure of a manifold.

4. What are some common conditions that a killing vector field must satisfy?

The most common condition that a killing vector field must satisfy is the Killing equation, which states that the Lie derivative of the metric tensor with respect to the vector field must be equal to zero. Additionally, a killing vector field must also be divergence-free and have a constant norm.

5. How are killing vector fields used in physics?

Killing vector fields are used in physics to describe symmetries in the laws of nature. For example, in general relativity, killing vector fields are associated with the conservation of energy and momentum. They are also used in other areas of physics, such as fluid mechanics and quantum mechanics.

Similar threads

  • Advanced Physics Homework Help
Replies
0
Views
474
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
1K
  • Special and General Relativity
Replies
19
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
799
  • Special and General Relativity
2
Replies
62
Views
3K
  • Special and General Relativity
Replies
2
Views
846
Back
Top