Life of a photon in gravity? I have a simple thought experiment and I am not sure about the answer:
What would happen to a photon that is emitted by an excited hydrogen in an otherwise empty universe?
Would gravity of the atom pull the photon eventually back? How would the frequency of the photon change on the way?
Let's assume this happens in a non-expanding and flat universe. Is there an obvious answer in general relativity to this question?
edit: Assuming a flat universe besides curvature induced by the gravity of atom and photon
 A: I'm assuming you mean the spacetime would be flat except for the gravitational field of the hydrogen atom. Then to good approximation the spacetime outside the atom is the Schwarzschild spacetime with mass parameter $M = m_H$ i.e. the mass of a hydrogen atom. We will assume the photon is emitted approximately one-half Bohr radius $r_H$ from the centre of the atom.
The Schwarzschild radius of a hydrogen atom $R_s$ is about $2.5\times 10^{-54}$ metres, which is much shorter than $r_H = 2.5 \times 10^{-11}$ metres. Therefore the photon is never pulled back.
Its frequency is redshifted (very slightly) by the atom's gravitational field. Specifically, the frequency at infinity $f_\infty$ is given by
$$ f_\infty = \frac{f_0}{z_\infty+1}$$
where $f_0$ is the frequency measured at emission and the redshift factor to infinity $z_\infty$ is given in this spacetime by
$$z_\infty = \frac{1}{\sqrt{1 - \frac{R_s}{r_h}}} - 1 = \frac{1}{\sqrt{1 - 10^{-43}}} - 1.$$
This number is zero to within machine epsilon for double precision floats. So for all practical purposes the photon is unaffected.
