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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

f

_{obs}= 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/c

^{2}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 -hf

_{obs}/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 -hf

_{obs}/c due to conservation of momentum. Thus the momentum of the photon relative to the distant observer would be -hf

_{obs}/c just prior to the collision, and it would be -hf

_{obs}/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 = f

_{obs}

therefore p = -hf

_{obs}/c(1-Rs/r)

The reason that f(r) = f

_{obs}is that an observer at r will see a blueshifted photon (f'(r) = blueshifted(f

_{obs}) ) 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(f

_{obs})) = f

_{obs}.