You make several of those:
This is only true for the non-relativistic case.
Well, it can be fiddled with so that it isn't (as mentioned). Drop it down to a 10 degree blackbody, and they will not be a bad assumption at all. It just simplifies calculations with a 10,000 degree star with 1/10,000 energy difference. (simple numbers)
This assumes an isotropic distribution of velocities, which is certainly not the case if you view a star moving away at high speed.
There is after you subtract out the star's motion. (My problem was Galilean Transforming the speed to do so, rather than Lorentz transforming; see below)
And this is the big one: Time is dilated, but also length is contracted. So you cannot convert velocities like that. And the definition of thermal velocity is always relative to the system and makes no sense if you view a moving system from the outside.
Only in the direction of motion. The thermal motions in all other directions are not affected. Since thermal motion is random, there will be plenty in the non-length contracted directions of motion to work with.
You can only do that if you're in the inertial system of the star. And then there are no relativistic effects because nothing is moving fast and thus time dilation plays no role.
Ok, I think this might be what I did wrong.
I think I just did a simple Galilean Transformation with the implied velocity conversions to subtract out the motion of the star from the observed motion of the atoms (observed velocity of atoms subtract velocity of star to get the thermal motion).
Or am I doing it wrong now, and did it right earlier? (
, just making sure)
This is obviously wrong, as the effects have to be the same no matter which inertial system I am observing them from, save for the appropriate Lorentz transforms.
Hence my asking questions where I messed up.
I realized it was wrong, I just didn't know where.
The big flaw in your reasoning is that you try to look at thermal effects form the point of view of a moving observer. This cannot succeed, as the description of temperature makes the assumption, that the whole system does not move in the inertial system you look at it. What you have to do is to look at the system itself where no relativity is needed look at what comes out and then do the relativistic conversion on the output. In your case this is just the Doppler shift of the usual black-body radiation of the star.
Well... All observers are moving, as there is no stationary point. The view from the star that it is stationary and everything else is moving is just as valid as everything else is stationary and the star is moving. So to illustrate my point of the apparent inconsistency, we just assume the observer is stationary, and changed the location of the observers from away from the star to on the star.
Anyways, correct me if I'm wrong:
If an A0 star is moving such that it is 1% time dilated, after Lorentz addition (in this case, subtraction) of the appropriate velocities, you find the star's thermal motion is no different.
Thus the will emit the standard 10,000 K blackbody temperature, which will then be Doppler shifted to varying wavelengths depending on the motion of the star.
Likewise, an observer on the star will have to perform the correct Lorentz addition to derive the appropriate thermal velocities, and apparent paradox solved.