### energy of photons in FLRW universe

Note that for photons $p_{\mu}p^{\mu} = 0$, unlike particles for which $p_{\mu}p^{\mu} = m^2c^2$

we know that $p_t = \frac{E}{c}$, $p^r = \frac{dr}{d\lambda}$, where $\lambda$ is an affine parameter. Also, $p^{\theta} = mc\frac{d\theta}{d\lambda} = 0$. And $p_{\phi} = L$

calling $\frac{GM}{c^2} = \mu$

say $u = \frac{1}{r}$ implying that $du = -\frac{dr}{r^2}$
and let’s call $\frac{E}{Lc}= \frac{1}{b}$

differentiating this equation, we get

we need to solve this perturbatively i.e $u = u_0 +\epsilon u_1$ where $\epsilon = \frac{3\mu}{b}$

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