3.7 Does Gravity Affect Light? (Optional)
40. 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^('), the photon frequency at infty , is related to f, the photon frequency at the star's surface, by
f^(')=fe^(-G(M_(s))/(R_(s))c^(2))
(b) Show that this expression reduces to Equation 3.39 for small (M_(s))/(R_(s)). (Hint: The decrease in photon energy, hdf, as the photon moves dr away from the star is equal to the work done against gravity, F_(G)dr.)
3.7Does Gravity Affect Light?(Optional) a photon falling or rising in a gravitational field, we have assumed a small change in frequency and a con- dense that fis not small. (a) Show that f', the photon frequency at co, is related to f, the photon frequency at the star's surface,by f'=fe-GMs/Rc2 f (b) Show that this expression reduces to Equation 3.39 for small Ms/R.Hint:The decrease in photon energy h df, as the photon moves dr away from the star is equal to the work done against gravity,Fc dr.