(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the general equation for gravitational redshift [tex]\frac{\Delta v}{v} = - \frac{GM}{c^2}(\frac{1}{r_1} - \frac{1}{r_2})[/tex] reduces to [tex]\frac{\Delta v}{v} = \frac{gH}{c^2}[/tex] near the surface of the earth.

2. Relevant equations

The two above, plus Newton's Law of Universal Gravitation

[tex]F = \frac{GMm}{r^2}[/tex]

3. The attempt at a solution

Begin with Newton's Law and solve for GM.

[tex]GM = \frac{F r^2}{m}[/tex]

Expand F to mg, and cancel m.

[tex]GM = g r^2[/tex]

Plug this into the gravitational redshift equation.

[tex]\frac{\Delta v}{v} = - \frac{g r^2}{c^2}(\frac{1}{r_1} - \frac{1}{r_2})[/tex]

r_1 is the distance to the observer, which I assume to be a distant star or similar ([tex]r_1 >> r[/tex]). r_2 is the distance from the Earth's surface to the light source ([tex]r_2 = r[/tex])

Multiplying by r,

[tex]\frac{\Delta v}{v} = - \frac{g r}{c^2}(\frac{r}{r_1} - \frac{r}{r_2})[/tex]

[tex]\frac{r}{r_1} = 0[/tex] and [tex]\frac{r}{r_2} = 1[/tex]

[tex]\frac{\Delta v}{v} = - \frac{g r}{c^2}(-1)[/tex]

[tex]\frac{\Delta v}{v} = \frac{g H}{c^2}[/tex]

Is what I have done logical and valid? Please advise.

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# Homework Help: Gravitational Redshift Near the Earth

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