Why don't stars stop moving? In my astro class we are learning about red/blue-shifting and I was wondering how to related to stars. So if a star is moving in some direction, the light leaving the direction of motion would be blueshifted, while the light coming out the back of the star would be red-shifted. The problem is then, doesn't the light leaving in the direction of motion carry more momentum due to its lower wavelength. So we have an unbalanced situation. So shouldn't the star slow or even stop over time?
My intuitive guess is that it has something to do with the two different relative frames, one from the star and one from an external observer. That or something do with with the changing mass of the star? Like it really is slowing down but it is losing mass at the same time so it balances out.
Not really expecting an answer but just wondering where my thinking is wrong.
 A: First off, we know the star can't actually slow down, because we can go to the initial reference frame of the star. In this frame, the star is stationary and emits radiation equally in all directions, so it can't start accelerating. So it doesn't accelerate in the original frame either.
In the original frame, it is true that the radiation emitted carries net momentum, so the momentum of the star goes down. But momentum depends on both mass and velocity, and it's in fact the mass of the star that goes down, not the velocity. Specifically, the mass goes down because $E = mc^2$ and the star is losing energy to the radiation. In fact, an argument much like this was one of Einstein's arguments for $E = mc^2$.
A: The acceleration (or proper acceleration) due to the proper forces on the star must be calculated in the rest frame of the star, in which there is no Doppler shift. Otherwise you are only looking at a mangled form of the Newtonian equations, which is confusing at best and likely to be actually incorrect.
Incidentally, the idea that mass increases is a part of the mangling of Newtonian equations. It was superseded when Einstein produced general relativity in 1915 showing how to formulate the laws of physics using tensor quantities (including vectors and scalars), which are the same for all observers. Modern treatments use only invariant mass, or rest mass. It is a great pity that relativistic mass has not been banished from low grade accounts.
