If given an FRW metric
ds2=−dt2+a2(t)[dx2+dy2+dz2]
and for the trajectory followed by a photon (null geodesic; ds2=0) with affine parameter λ, know that
gμνdxμdλdxνdλ=0.
How does one find
dtdλ=w0a(t)?
Already found the nonzero Christoffel coeffs and the remaining geodesic equations,
0=d2xidλ2+Γitidtdλdxidλ and 0=d2tdλ2+Γtiidxidλdxidλ
with Γiti=1a(t)da(t)dt and Γtii=a(t)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 (?):
ds2=−(dt2)+a2(t)dx2=0→dta(t)=±dx
I also believe that dx/dλ=c if the affine parameter is related by λ=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!
Answer
I'll use dots for derivative with respect to affine parameter. The FRW metric has Killing vectors ∂x,∂y,∂z each of which leads to a conservation equation: cx=˙x⋅∂x=a2˙xcy=˙y⋅∂y=a2˙ycz=˙z⋅∂z=a2˙z
Unsolicited Advice (which has helped me enormously in research by the way)
When solving for geodesics, learn to love conserved quantities, and therefore Killing vectors!!!
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