Killing Equation: Why Set g_μν,ρ V^ρ = 0?

[kent davidge]

When deriving the Killing Equation, one ends up with a term $g_{\mu \nu, \rho} V^\rho$ where $V$ is the Killing Vector. This terms is often set to zero. Why?
I see no motivation for demanding that the derivative of the metric coefficients vanish.

 

[martinbn]

Can you give more details?

 

[kent davidge]

Can you give more details?

What kind of details? I tried to put everything into the opening post.

 

[Orodruin]

What kind of details? I tried to put everything into the opening post.

You did not though. You said ”This term is usually set to zero. Why?” without any reference to a text actually doing so or reciting an actual derivation doing so.

 

[stevendaryl]

Are you sure that they didn't write $g_{\mu \nu;\rho} V^\rho$? The covariant derivative (indicated by semicolon, rather than comma) of $g$ is zero.

 

[kent davidge]

Are you sure that they didn't write $g_{\mu \nu;\rho} V^\rho$?

It's me. I'm deriving it. However I get $\mathcal L g _{\sigma \rho} = \nabla_\sigma V_\rho + \nabla_\rho V_\sigma + 2 V_\mu \Gamma^\mu{}_{\sigma \rho} + g_{\sigma \rho, \kappa} V^\kappa$.

 

[Orodruin]

It's me. I'm deriving it. However I get $\mathcal L g _{\sigma \rho} = \nabla_\sigma V_\rho + \nabla_\rho V_\sigma + 2 V_\mu \Gamma^\mu{}_{\sigma \rho} + g_{\sigma \rho, \kappa} V^\kappa$.

So why are you withholding the rest of your derivation?

 

[kent davidge]

So why are you withholding the rest of your derivation?

because I was not wanting to bore you with it. but never mind, i found my fault. i was equating $g_{\mu \nu} V^\nu{}_{, \ \rho} = V_{\mu, \ \rho}$ right away.

 

[PeterDonis]

because I was not wanting to bore you with it

As you see, this is not a good strategy. When you ask a question about something you did, you need to include all of what you did. You can't assume that just the part you think you need to include is relevant, because if you already knew what was relevant and what was not, you wouldn't have made a mistake and you wouldn't have needed to post a question in the first place. Please keep that in mind for the future.

Since you say you have found your mistake, this thread is closed.