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A photon of frequency omega_(0) is emitted from the surface of a star whose mass is M and radius R. Find the gravirational shift of frequency Delta omega//omega_(0) of the photon at a very great distance from the star. |
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Answer» Solution :We shall only consider stars which are not too compact so that the graviational field at their surface is weak: `(gammaM)/(c^(2)R)lt lt 1` We shall also clarify the PROBLEM by making clear the MEANING of the (slightly changed) NOTATION. SUPPOSE the photon is emitted by some atom whose total relativistic energies(including the rest mass) are `E_(1)&E_(2)` with `E_(1) lt E_(2)`. These energies are defined in the absence of gravitational field and we have `omega_(0) = (E_(2)-E_(1))/(cancelh)` as the frequency at infinity of the photon that is emitted in `2rarr1` transition. On the surface of the star, the energies have the values `E'_(2) = E_(2)-(E_(2))/(c^(2)).(gammaM)/(R ) = E_(2) (1-(gammaM)/(c^(2)R))` `E'_(1) = E_(1)(1-(gammaM)/(c^(2)R))` Thus, from `cancelh omega = E'_(2) - E'_(2)` we get `omega = omega_(0)(1-(gammaM)/(c^(2)R))` Here `omega` is the frequency of the photon emitted in the transition `2rarr1` when the atom is on the surface of the star. In shown that the frequency of spectral lines emitted by atoms on the surface of some star is less than the frequency of lines emitted by atoms here on earth (where the gravitational effect is quite small). Finally `(Delta omega)/(omega_(0)) =- (gammaM)/(c^(2)R)`. The answer given in the book is incorrect is general through it agrees with the above result for `(gammaM)/(c^(2)R) lt lt1`. |
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