# Is the fact that we're moving with a certain speed with respect to the CMB special-relativity consistent?

As a sidenote to an exercise about the aberration of CMB at the dipole level, which scope was to find the peculiar velocity we have with respect to the cosmic background (assuming the doppler effect derived from it is the only source of dipole aberration), my professor left the following question:

Is the fact that we're moving with a certain speed with respect to the CMB special-relativity consistent?

My calculations were as follows:

Apparently, the difference in the temperature of the CMB due to the dipole is $$\Delta T_{l=1}=3.372\cdot 10^{-3}K$$, and choosing our axis so that the spherical harmonic with $$m=0$$ is oriented with the dipole, it's easy to see that, at the dipole order, temperature will be written: $$T(\theta)=T_0(1+\frac{v}{c}cos\theta)$$ So I just took $$\frac{\Delta T_{l=1}}{T} = \frac{v}{c} cos\theta$$, and this leads to $$v\approx371\frac{km}{s}$$.

I don't see how this such small speed could not be consistent with special relativity, but I'm suspicious that answering "Yes it is consistent" would be too easy to be true... Is my reasoning correct? Thanks!

The actual speed with which we are moving relative to the CMB is unimportant; we could be moving at $$0.99c$$ and special relativity would still apply, since that theory only requires a velocity that is less than $$c$$. It's like the Newtonian formula for calculating kinetic energy - it works regardless of whether your speed is $$371 km/s$$ (an extremely fast speed by terrestrial standards) or $$0.001 m/s$$.