- #1

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## Main Question or Discussion Point

Hey guys, I'm working on Polchinski's string book, and I have a problem. Around page 152 he uses an identity I'm not sure how to prove. Essentially he wants to compute conformal killing vector fields. So we have the eq for a CKV:

[tex]

P_{ab}=\Delta_a \xi_b+ \Delta_b\xi_a- g_{ab}\Delta_c\xi^c=0

[/tex]

And what I need to prove is that, in general

[tex]

P_{ab}P^{ab}=\Delta_a \xi_b\Delta^a \xi^b-R \xi_a\xi^a,

[/tex]

Where R is the Ricci scalar. This identity might (or might not) only work when integrated over, ie

[tex]

\int d^2x \sqrt{g}P_{ab}P^{ab} = ...

[/tex]

I suspect this might be useful to get some integration by parts.

My GR is a little rusty, so I'm not sure how to get it. I can expand all the Lie derivatives, collect some terms, etc, and I get:

[tex]

2\Delta_a \xi_b\Delta^a \xi^b+2(\xi_{b,a}-\Gamma^i_{ba}\xi_i)(\xi^a_{,b}+\Gamma^a_{jb}\xi^j)-g^{ab}\Delta_c\xi^c(\xi_{b,a}+\xi_{a,b}-2\Gamma^{i}_{ba}\xi_i)-g_{ab}\Delta_c\xi^c(\xi^b_{,a}+\xi^a_{,b}+\Gamma^b_{ia}\xi^i+\Gamma^a_{ib}\xi^i)

[/tex]

Am I on the right track? I just don't see how to get a Ricci scalar out of all those terms..For example I would need two terms like

[tex]

\Gamma^a_{bc}\Gamma^c_{de}-\Gamma^a_{ec}\Gamma^c_{db}

[/tex]

If anyone did a calculation like this before or can easily spot that it does work out, please let me know! I just need to know if I'm the right track...GR calculations seem to usually work out, but for this one I'm lacking a general sense of direction in manipulating the indices ...you need like a million contractions to get the Ricci scalar :) Thanks for any help!

[tex]

P_{ab}=\Delta_a \xi_b+ \Delta_b\xi_a- g_{ab}\Delta_c\xi^c=0

[/tex]

And what I need to prove is that, in general

[tex]

P_{ab}P^{ab}=\Delta_a \xi_b\Delta^a \xi^b-R \xi_a\xi^a,

[/tex]

Where R is the Ricci scalar. This identity might (or might not) only work when integrated over, ie

[tex]

\int d^2x \sqrt{g}P_{ab}P^{ab} = ...

[/tex]

I suspect this might be useful to get some integration by parts.

My GR is a little rusty, so I'm not sure how to get it. I can expand all the Lie derivatives, collect some terms, etc, and I get:

[tex]

2\Delta_a \xi_b\Delta^a \xi^b+2(\xi_{b,a}-\Gamma^i_{ba}\xi_i)(\xi^a_{,b}+\Gamma^a_{jb}\xi^j)-g^{ab}\Delta_c\xi^c(\xi_{b,a}+\xi_{a,b}-2\Gamma^{i}_{ba}\xi_i)-g_{ab}\Delta_c\xi^c(\xi^b_{,a}+\xi^a_{,b}+\Gamma^b_{ia}\xi^i+\Gamma^a_{ib}\xi^i)

[/tex]

Am I on the right track? I just don't see how to get a Ricci scalar out of all those terms..For example I would need two terms like

[tex]

\Gamma^a_{bc}\Gamma^c_{de}-\Gamma^a_{ec}\Gamma^c_{db}

[/tex]

If anyone did a calculation like this before or can easily spot that it does work out, please let me know! I just need to know if I'm the right track...GR calculations seem to usually work out, but for this one I'm lacking a general sense of direction in manipulating the indices ...you need like a million contractions to get the Ricci scalar :) Thanks for any help!