Consider a spherical blackbody of constant temperature and mass M whose surface lies at radial coordinate r=R. An observer located at the surface of the sphere and a distant observer both measure the blackbody radiation given off by the sphere.
(a) If the observer at the surface of the sphere measures the luminosity of the blackbody to be L, use the gravitational time dilation formula, Eq. (17.13), to show that the observer at infinity measures
L∞=L(1−Rc22GM).(17.31)(b) Both observers use Wien's law, Eq. (3.15), to determine the blackbody's temperature. Show that
T∞=T1−Rc22GM.(17.32)(c) Both observers use the Stefan–Boltzmann law, Eq. (3.17), to determine the radius of the spherical blackbody. Show that
R∞=1−2GM/Rc2R.(17.33)
Thus, using the Stefan–Boltzmann law without including the effects of general relativity will lead to an overestimate of the size of a compact blackbody.