## Question

Show that conformal transformations leave null geodesics invariant, that is, that the null geodesics of $g_{\mu\nu}$ are the same as those of $\omega^2g_{\mu\nu}$. (We already know that they leave null curves invariant; you have to show that the transformed curves are still geodesics.) What is the relationship between the affine parameter in the original and conformal metrics?

The answer to this is a bit feeble I think - so feeble that I forgot to post it for ten days. It relies on Carroll's assertion in section 3.4 on the properties of geodesics that from some kind of equation like his 3.58 you can always find an equation that satisfies the geodesic equation and he gives us the relationship of the affine parameter to the magic equation. It would have been nice to prove the assertion but I think that was out of scope.

It's at: Ex G1 Conformal Null Geodesics.pdf (a mere two pages).

If we write $λ=g(α)$, (3.59) becomes $$f = -(1/g) (1/g)' (1/g)^(-2) = g'/g$$ Integrate on both sides, we have $$ln g = c + \int_0^α f(x) dx$$ or equivalently, $$g = c' exp(\int_0^α f(x) dx)$$.