# Homework Help: Gravitational Redshift Near the Earth

1. Oct 14, 2008

### mattst88

1. The problem statement, all variables and given/known data

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

2. Relevant equations

The two above, plus Newton's Law of Universal Gravitation
$$F = \frac{GMm}{r^2}$$

3. The attempt at a solution

Begin with Newton's Law and solve for GM.

$$GM = \frac{F r^2}{m}$$

Expand F to mg, and cancel m.

$$GM = g r^2$$

Plug this into the gravitational redshift equation.

$$\frac{\Delta v}{v} = - \frac{g r^2}{c^2}(\frac{1}{r_1} - \frac{1}{r_2})$$

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

Multiplying by r,

$$\frac{\Delta v}{v} = - \frac{g r}{c^2}(\frac{r}{r_1} - \frac{r}{r_2})$$

$$\frac{r}{r_1} = 0$$ and $$\frac{r}{r_2} = 1$$

$$\frac{\Delta v}{v} = - \frac{g r}{c^2}(-1)$$
$$\frac{\Delta v}{v} = \frac{g H}{c^2}$$