Chapter 42: Problem 11

Question: A rocket ship is receding from the earth at a speed of v = 0.2 c. A light in the rocket ship appears blue to passengers on the ship. What color would is appear to an observer on the earth.

Solution: For motion directly away from the earth, we know that frequencies will be red-shifted from $\nu$ to $\nu^\prime$, and hence wavelengths will be increased by the same fraction where

$$\nu^\prime~=~\nu { \sqrt{1-{v^2\over c^2}}\over 1+{v\over c}}$$

$$\lambda^\prime~=~\lambda { 1+{v\over c}\over\sqrt{1-{v^2\over c^2}} }$$

Assuming blue light with $\lambda~=~500~{\rm nm}$in the rocket, observers on the earth will see light of wavelength $\lambda^\prime~=~600~{\rm nm}$, which is something like orange.