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: [itex]\overline{g}[/itex]_ab = [itex]\Omega[/itex]²g_ab
Null geodesics: 0 = g_ab(dx^a/du)(dx^b/du)

The Attempt at a Solution

I define the parameter u = [itex]\frac{1}{2}[/itex][itex]\Omega[/itex]². Thus [itex]\frac{du}{d\Omega}[/itex] = [itex]\Omega[/itex].

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

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 + [itex]\Gamma[/itex]^a_bc(dx^b/ds)(dx^c/ds) = 0

Unfortunately, I am coming up with
d²x^a/ds² + {[itex]\Gamma[/itex]^a_bc + ([itex]\delta[/itex]^a_c[itex]\partial[/itex]_b[itex]\Omega[/itex] + [itex]\delta[/itex]^a_b[itex]\partial[/itex]_c[itex]\Omega[/itex] - g^adg_bc[itex]\partial[/itex]_d[itex]\Omega[/itex]) / [itex]\Omega[/itex]}(dx^b/ds)(dx^c/ds) = 0
as the transformation. I don't see how that is invariant unless d[itex]\Omega[/itex] = 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[itex]\nabla[/itex]_aX^b = 0,
and uses a relation between covariant derivatives that I did not find in the Inverno book:
[itex]\overline{\nabla}[/itex]_aX_b = [itex]\nabla[/itex]_aX_b + T^b_acX^c

I find how T^b_ac transforms conformally and get this equation:
X^a[itex]\nabla[/itex]_aX^b = 2X^aX^b[itex]\nabla[/itex]_a(ln [itex]\Omega[/itex]) - g^bdg_acX^aX^c[itex]\nabla[/itex]_d(ln [itex]\Omega[/itex])

Since it is a null geodesic, g_acX^aX^c = 0, so the equation reduces to:
X^a[itex]\nabla[/itex]_aX^b = 2X^aX^b[itex]\nabla[/itex]_a(ln [itex]\Omega[/itex])

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