Momentum of a photon heading towards a spherical mass

In summary, when a distant observer sends a photon radially towards a spherical mass at a distance where the effects of gravity are negligible, the photon's momentum relative to the distant observer is -hfobs/c(1-Rs/r), where fobs is the frequency of the photon at the observer's position and r is the radial distance from the mass to the photon. This assumes that the photon exchanges momentum with the gravitational field before hitting the mass.
  • #1
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.
 
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  • #2
liron said:
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.
The photon interacts with the gravitational field (or curvature of spacetime if you prefer the GR view) before it hits the sphere. There is no reason to assume that this interaction does not exchange momentum (and it does).

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.
Well, you should get the same result for an observer at the place of the photon.

I'm not sure if the momentum of a photon far away is such a well-defined concept, but I guess your attempt 2 result is the most meaningful value.
 

Related to Momentum of a photon heading towards a spherical mass

1. What is the momentum of a photon heading towards a spherical mass?

The momentum of a photon heading towards a spherical mass can be calculated using the equation p = h/λ, where p is the momentum, h is Planck's constant, and λ is the wavelength of the photon. This momentum can also be expressed in terms of its energy, E = hc/λ, where c is the speed of light.

2. How does the mass of the spherical mass affect the momentum of the photon?

The mass of the spherical mass does not affect the momentum of the photon. According to the theory of relativity, the momentum of a photon is solely determined by its energy and direction, and is independent of the mass of the object it is approaching.

3. What is the relationship between the momentum of the photon and its frequency?

The momentum of a photon is directly proportional to its frequency. This relationship is described by the equation p = hf/c, where p is the momentum, h is Planck's constant, f is the frequency, and c is the speed of light.

4. How does the momentum of a photon change as it approaches the spherical mass?

As a photon approaches a spherical mass, its momentum remains constant. However, the direction of the photon's momentum will change due to the gravitational pull of the mass. This results in a curved path of the photon as it passes by the mass.

5. Can the momentum of a photon be affected by the gravitational field of the spherical mass?

Yes, the momentum of a photon can be affected by the gravitational field of a spherical mass. This is due to the phenomenon known as gravitational lensing, where the path of a photon is bent by the curvature of space caused by the mass of an object.

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