A spaceship starts falling under gravity with an acceleration $g$ as measured by an observer Barry at rest on Earth. At the instant that the ship starts to fall, an astronaut Harry at the base of the rocket ship sends a light signal of frequency $w$ vertically upward to another astronaut Sally a distance $h$ above.

Barry argues that the light signal reaching Sally ought to be Doppler shifted toward the blue. This Doppler shift $Δw$ is given by $\frac{Δw}{w_{Doppler}} = \frac{Δu}{c} $

where $Δu$ is the velocity of the rocket ship after a time $Δt = \frac{h}{c}$

What I want to know is how that Doppler shift equation came about mathematically? The text said they used the formula for the low velocity approximation to the relativistic Doppler shift which is

$w' = w \sqrt(\frac{c+v}{c-v}) $

But I just don't see how that happened. Maybe I'm missing something...

A spaceship starts falling under gravity with an acceleration $g$ as measured by an observer Barry at rest on Earth. At the instant that the ship starts to fall, an astronaut Harry at the base of the rocket ship sends a light signal of frequency $w$ vertically upward to another astronaut Sally a distance $h$ above.

Barry argues that the light signal reaching Sally ought to be Doppler shifted toward the blue. This Doppler shift $Δw$ is given by $\frac{Δw}{w_{Doppler}} = \frac{Δu}{c} $

where $Δu$ is the velocity of the rocket ship after a time $Δt = \frac{h}{c}$

What I want to know is how that Doppler shift equation came about mathematically? The text said they used the formula for the low velocity approximation to the relativistic Doppler shift which is

$w' = w \sqrt{\frac{c+v}{c-v}} $

But I just don't see how that happened. Maybe I'm missing something...