Light traveling through a uniform gravitational potential suffers a shift in its frequency. Photons have no rest mass, but we can assign them an effective gravitational mass based on their energy (m = E/c^2). Use conservation of energy and this effective mass to derive the gravitational redshift equation for a photon traveling from the ground to a height H. (This type of semi-classical approach works to first order, but beyond that we need a full general relativistic treatment.)
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First, we know that the energy of a photon is given by E = hf, where h is Planck's constant and f is the frequency of the light. Show more…
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In deriving expressions for the change in frequency of a photon falling or rising in a gravitational field, we have assumed a small change in frequency and a constant photon mass of $h f / c^{2}$. Suppose that a star is so dense that $\Delta f$ is not small. (a) Show that $f^{\prime},$ the photon frequency at $\infty,$ is related to $f,$ the photon frequency at the star's surface, by $$f^{\prime}=f e^{-G M_{\mathrm{s}} / R_{\mathrm{s}} c^{2}}$$ (b) Show that this expression reduces to Equation 3.39 for small $M_{\mathrm{s}} / R_{\mathrm{s}}$. (Hint: The decrease in photon energy, $h d f,$ as the photon moves $d r$ away from the star is equal to the work done against gravity, $F_{\mathrm{G}}$ dr. )
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