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__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]

^{2}g

_{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]

^{2}g

_{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]

^{2}g

_{ab}(dx

^{a}/d[itex]\Omega[/itex])(dx

^{b}/d[itex]\Omega[/itex])([itex]\frac{du}{d\Omega}[/itex])

^{-2}

0 = [itex]\Omega[/itex]

^{2}g

_{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?