Problem Set #2 1. Problem 4.15 Suppose a constant force of ...

where dz is the width of the reference frame and g 0 is the Newtonian gravitational
acceleration at the point of closest approach, O. The angular deflection is small,
so assume that the photon is initially traveling in the z-direction as it enters the
frame. (Hint: The width of the frame in the z-direction is dz, so the time for
the photon to cross the frame can be taken to be dz/c)
(b) Integrate the result you found in part (a) from α = −π/2 to +π/2 and so find the
total angular deflection of the photon as it passes through the curved spacetime
near the Sun.
(c) Your answer (which is also the answer obtained by Einstein in 1911 before he
arrived at his field equations) is only half the correct value of 1.75”’. Can you
qualitatively account for the missing factor of two?
4. Problem 17.9 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:
∆t 0
= v (
∞
= 1 − 2GM ) 1/2
∆t ∞ v 0 r 0 c 2 (4)
to show that the observer at infinity measures
(
L ∞ = L 1 − 2GM )
Rc 2 . (5)
(b) Both observers use Wien’s law
λmaxT = 0.002897755 m K, (6)
to determine the radius of the spherical blackbody. Show that
T ∞ = T
(c) Both observers use the Stefan-Boltzmann law,
√
1 − 2GM
Rc 2 . (7)
L = 4πR 2 σT 4 e , (8)
to determine the radius of the spherical blackbody. Show that
R ∞ =
R
√
1 − 2GM/Rc 2 . (9)
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.
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