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Augbrah
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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 x^0 such that metric components \partial_0 g_{\mu \nu} = 0.
Spacetime is static := there exists a coord chart such that \partial_0 g_{\mu \nu} = 0 and g_{0i} = 0
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 V_{[\alpha}\nabla_\mu V_{v]}=0
iii) Let V be timelike Killing vector field with V_{[\alpha}\nabla_\mu V_{v]}=0. Show that this condition implies \nabla_\mu(|V|^n V_\nu)-\nabla_\nu(|V|^n V_\mu)=0 Where |V|^2=V_a V^a and n is integer which should be determined.
Killing vector field V satisfies \nabla_\mu V_\nu + \nabla_\nu V_\mu = 0
i) I have done this part. For timelike V^\mu we can find an inertial frame s.t. V^\mu = \delta^\mu_0. Then V_\mu = g_{\mu 0}. Hence \nabla_\mu V_\nu + \nabla_\nu V_\mu =...=\partial_0 g_{\mu\nu}, by expressing Christoffel symbols as partial derivatives of a metric other derivatives nicely cancel.
So we proved that timelike Killing V ⇔ \partial_0 g_{\mu\nu}=0.
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
V_{\alpha}(\nabla_\mu V_{\nu} - \nabla_\nu V_{\mu}) + \text{(other 2 cyclicly permuted pairs)}=0 Studying the first term and using same identification V_\mu = g_{\mu 0} and as before remembering that for static spacetime g_{0i} = 0 as well as \partial_0 g_{\mu \nu} = 0 we get:
V_{\alpha}(\nabla_\mu V_{\nu} - \nabla_\nu V_{\mu}) = g_{\alpha 0} (\partial_\mu g_{\nu 0} - \partial_\nu g_{\mu 0}) = 0
Same procedure for other two terms and total sum is 0.
iii) Besides product rule I did not got far. Any suggestions?
1. Homework Statement
Spacetime is stationary := there exists a coord chart with a timelike coordinate x^0 such that metric components \partial_0 g_{\mu \nu} = 0.
Spacetime is static := there exists a coord chart such that \partial_0 g_{\mu \nu} = 0 and g_{0i} = 0
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 V_{[\alpha}\nabla_\mu V_{v]}=0
iii) Let V be timelike Killing vector field with V_{[\alpha}\nabla_\mu V_{v]}=0. Show that this condition implies \nabla_\mu(|V|^n V_\nu)-\nabla_\nu(|V|^n V_\mu)=0 Where |V|^2=V_a V^a and n is integer which should be determined.
Homework Equations
Killing vector field V satisfies \nabla_\mu V_\nu + \nabla_\nu V_\mu = 0
The Attempt at a Solution
i) I have done this part. For timelike V^\mu we can find an inertial frame s.t. V^\mu = \delta^\mu_0. Then V_\mu = g_{\mu 0}. Hence \nabla_\mu V_\nu + \nabla_\nu V_\mu =...=\partial_0 g_{\mu\nu}, by expressing Christoffel symbols as partial derivatives of a metric other derivatives nicely cancel.
So we proved that timelike Killing V ⇔ \partial_0 g_{\mu\nu}=0.
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
V_{\alpha}(\nabla_\mu V_{\nu} - \nabla_\nu V_{\mu}) + \text{(other 2 cyclicly permuted pairs)}=0 Studying the first term and using same identification V_\mu = g_{\mu 0} and as before remembering that for static spacetime g_{0i} = 0 as well as \partial_0 g_{\mu \nu} = 0 we get:
V_{\alpha}(\nabla_\mu V_{\nu} - \nabla_\nu V_{\mu}) = g_{\alpha 0} (\partial_\mu g_{\nu 0} - \partial_\nu g_{\mu 0}) = 0
Same procedure for other two terms and total sum is 0.
iii) Besides product rule I did not got far. Any suggestions?