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liron
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Homework Statement
A distant observer is at rest relative to a spherical mass and at a distance where the effects of gravity are negligible. The distant observer sends a photon radially towards the mass. At the distant observer, the photon's frequency is f. What is the momentum relative to the distant observer of the photon when it is distance r from the mass? Assume that r > the radius of the mass.
Homework Equations
h = Planck's constant
f = frequency of the photon at some distance from the mass relative to the distant observer's frame of reference
fobs = frequency of the photon at the distant observer's position relative to the distant observer
v = velocity of the photon at some distance from the mass relative to the distant observer's frame of reference
l = wavelength of the photon relative to the distant observer's frame of reference
r = radial distance from the mass to the photon
c = the speed of light in free space
p = momentum of the photon relative to the distant observer's frame of reference
Rs = Schwarzschild radius of the point mass = 2GM/c2 where G = gravitational constant and M = mass of the spherical mass.
The Attempt at a Solution
Here are two attempts with two different answers. They make assumptions which may not be correct.
Attempt 1
When the distant observer sends out the photon, it has a momentum of -hfobs/c. If the photon were to hit the mass and its energy totally absorbed by the mass and converted into kinetic energy, then the momentum of the mass would be -hfobs/c due to conservation of momentum. Thus the momentum of the photon relative to the distant observer would be -hfobs/c just prior to the collision, and it would be -hfobs/c at all times regardless of its distance r to the radial mass.
Attempt 2
p = -h/l
l = v/f
v = c(1-Rs/r) - from the Schwarzschild metric
f = fobs
therefore p = -hfobs/c(1-Rs/r)
The reason that f(r) = fobs is that an observer at r will see a blueshifted photon (f'(r) = blueshifted(fobs) ) but their clock is slower so they'll see more cycles per their second. The distant observer will see fewer cycles per their second so that the frequency at r relative to the distant observer = f = redshifted(f') = redshifted(blueshifted(fobs)) = fobs.