# Photon on null geodesic

If given an FRW metric

$ds^2 = -dt^2 + a^2(t)[dx^2+dy^2+dz^2]$

and for the trajectory followed by a photon (null geodesic; $ds^2=0$) with affine parameter $\lambda$, know that

$g_{\mu\nu}\,\frac{dx^{\mu}}{d\lambda}\,\frac{dx^{\nu}}{d\lambda} = 0.$

How does one find

$\frac{dt}{d\lambda} = \frac{w0}{a(t)}$?

Already found the nonzero Christoffel coeffs and the remaining geodesic equations,

$0 = \frac{d^2x^i}{d\lambda^2} + \Gamma^i_{ti}\,\frac{dt}{d\lambda}\,\frac{dx^i}{d\lambda}$ and $0 = \frac{d^2t}{d\lambda^2} + \Gamma^t_{ii}\,\frac{dx^i}{d\lambda}\,\frac{dx^i}{d\lambda}$

with $\Gamma^i_{ti} = \frac{1}{a(t)}\,\frac{da(t)}{dt}$ and $\Gamma^t_{ii} = a(t)\,\frac{da(t)}{dt}$

I expect it to be pretty simple (yet not seeing it) and that the null geodesic in the $x$-direction come from something like (?):

$ds^2 = -(dt^2)+a^2(t)\,dx^2 = 0 \rightarrow \frac{dt}{a(t)} = \pm dx$

I also believe that $dx/d\lambda = c$ if the affine parameter is related by $\lambda = t$.

This is where I am confused and need it to move on to look at the cosmological redshift and derive the ratio of emitted and observed energies of a photon at $t1$ and $t2$.

Anything to straighten me out would be great!

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I'll use dots for derivative with respect to affine parameter. The FRW metric has Killing vectors $\partial_x, \partial_y, \partial_z$ each of which leads to a conservation equation: \begin{align} c_x &= \dot x\cdot\partial_x = a^2\dot x \\ c_y &= \dot y\cdot\partial_y = a^2\dot y \\ c_z &= \dot z\cdot\partial_z = a^2\dot z \end{align} which implies $$a^4(\dot x^2 + \dot y^2 + \dot z^2) = w_0^2, \qquad w_0^2 \equiv c_x^2 +c_y^2 +c_z^2$$ The null geodesic equation can, as you basically point out, be written $$\dot t^2 = a^2 (\dot x^2 + \dot y^2 + \dot z^2)$$ multiplying both sides by $a^2$ and using the conservation condition derived above gives $$a^2\dot t^2 = w_0^2$$ which is essentially what you were looking for.
so does the 1st implied equation then imply that $(a^2\,\dot{x})\,\dot{x}$ is a conserved quantity? I know that $V_{\mu}\,U^{\mu}$ is conserved... Where did these two implied equations come from? The 2nd, doesn't it say that the quantity on the right is conserved along any of the three geodesics? But the first...? –  nate Mar 30 '13 at 0:45
Sorry there was an error in the first implied equation, it should have $a^4$ on the left and the $0^2$ should have read $w_0^2$. I fixed them; sorry again I was being sloppy in the algebra. –  joshphysics Mar 30 '13 at 1:00