Conformal invariance of null geodesics

[PhyPsy]

Hi, folks. I hope this is the right forum for this question. I'm not actually taking any classes, but I am doing self-study using D'Inverno's Introducing Einstein's Relativity. I have a solution, and I want someone to check it for me.

Homework Statement

Prove that the null geodesics of two conformally related metrics coincide.

Homework Equations

Conformally related metrics: \overline{g}_ab = \Omega²g_ab
Null geodesics: 0 = g_ab(dx^a/du)(dx^b/du)

The Attempt at a Solution

I define the parameter u = \frac{1}{2}\Omega². Thus \frac{du}{d\Omega} = \Omega.

Now, I use the chain rule on the null geodesics equation:
0 = \Omega²g_ab(dx^a/d\Omega)\frac{d\Omega}{du}(dx^b/d\Omega)\frac{d\Omega}{du}
0 = \Omega²g_ab(dx^a/d\Omega)(dx^b/d\Omega)(\frac{du}{d\Omega})^-2
0 = \Omega²g_ab(dx^a/d\Omega)(dx^b/d\Omega)\Omega^-2
0 = g_ab(dx^a/d\Omega)(dx^b/d\Omega), which is the null geodesics equation with the new parameter \Omega.

So is this a legitimate proof of the coinciding of null geodesics of conformally related metrics?

 

[PhyPsy]

Bump; can anyone help me?

 

[diazona]

Hmm... is u a parameter for the geodesic? If so, I don't think you can relate it to Ω like that, because the proof should work for arbitrary conformal transformations including (for example) those where Ω is a constant.

 

[Dick]

Bump; can anyone help me?

No, it's not right. 0 = gab(dxa/du)(dxb/du) isn't the geodesic equation. It's just says that it's a null curve. And those are obviously conformally invariant. Not all null curves are null geodesics. If you want to see the right way to do it look at Appendix D in Robert Wald's book General Relativity.

 

[PhyPsy]

OK, I see now that I should be using the equation:
d²x^a/ds² + \Gamma^a_bc(dx^b/ds)(dx^c/ds) = 0

Unfortunately, I am coming up with
d²x^a/ds² + {\Gamma^a_bc + (\delta^a_c\partial_b\Omega + \delta^a_b\partial_c\Omega - g^adg_bc\partial_d\Omega) / \Omega}(dx^b/ds)(dx^c/ds) = 0
as the transformation. I don't see how that is invariant unless d\Omega = 0, and I don't see any reason to make that assumption.

I think I will try looking for that Wald book you mentioned.

Update: Wow, thanks for that tip, Dick. The explanation in the Wald book really cleared things up. It starts by using the affine geodesic equation,
X^a\nabla_aX^b = 0,
and uses a relation between covariant derivatives that I did not find in the Inverno book:
\overline{\nabla}_aX_b = \nabla_aX_b + T^b_acX^c

I find how T^b_ac transforms conformally and get this equation:
X^a\nabla_aX^b = 2X^aX^b\nabla_a(ln \Omega) - g^bdg_acX^aX^c\nabla_d(ln \Omega)

Since it is a null geodesic, g_acX^aX^c = 0, so the equation reduces to:
X^a\nabla_aX^b = 2X^aX^b\nabla_a(ln \Omega)

The non-affine geodesic equation is X^a\nabla_aX^b = \lambdaX^b, so I just define \lambda = 2X^a\nabla_a(ln \Omega), and the geodesic equation is satisfied. This is why the Inverno book gave the hint that both equations need not be affinely parametized.