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Is it true that in all spacetimes there is some conformal time-like Killing vector $\tau^a$ in the vicinity of null geodesics?

If the above statement is true then can one argue that, for all spacetimes, an EM wave propagating along a null geodesic with wave-vector $k^a$ has a conserved quantity $k_a\tau^a=k_\tau$?

Please see @MichaelSeifert's informative answer and comments for background to Killing vectors, conserved quantities and EM waves:

https://physics.stackexchange.com/a/186483/22307

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John Eastmond
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  • Just out of curiosity, what motivated this question? – FenderLesPaul Jun 07 '15 at 16:43
  • I am wondering if it is generally true that photons propagate with constant energy with respect to conformal time. I am interested in the case of FRW cosmology where this might imply that photon energy density $\rho_{rad}\propto 1/a^3$ rather than the conventional $1/a^4$. – John Eastmond Jun 07 '15 at 16:50

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