Isotropy of the universe in different reference frames

Suppose that we put Bob and Alice into intergalactic space. If they look around they will see the light from distant galaxies shifted according to the Hubble law. More importantly, the light is (on average) isotropic.

Now suppose we accelerate Bob to e.g. $$\beta = 0.999$$. He should see the light in forward angles shifted blue and in backward angles shifted more towards red. In other words, the universe isn't isotropic to Bob anymore.

Then again, if we accelerate both of them to $$\beta$$, the relative situation is identical, but now both 'should' see anisotropic universe. According to this reasoning, there exists a special reference frame where the universe is isotropic. This, of course, isn't what we measure.

What is the solution to this issue?

My take: Suppose the motion is along a common $$x$$ axis. At $$t=0$$ (according to Alice) two distant galaxies at $$x = a$$ and $$x = -a$$ shoot out a signal. Then Bob sees these events occuring at $$t_{\pm} = \pm \gamma \, \frac{\beta}{c} a$$. In other words, the backward signal was emitted when this galaxy was much younger and therefore (Hubble law) moving more slowly (giving a small redshift). But as Bob is moving away from it, it redshifts further. Also the forward galaxy is much older and is originally redshifted a lot. But as Bob is moving towards it, it blueshifts.

It would be great is there was some explanation without using the whole machinery of GTR. Thanks!

• According to this reasoning, there exists a special reference frame where the universe is isotropic I failed to see the reasoning. Ability to choose a reference frame does not guarantee that a random cosmological solution of GR could become isotropic in any frame. You can do it only if the solution was isotropic in some special frame to begin with. – A.V.S. Nov 9 '18 at 3:05