In many relativity illustrations, it is mentioned that someone in a spaceship believes he is standing still and the observers are moving and vice versa. The example of a light-beam clock slowing down for an observer but not for the passenger of the ship is often how relativity is explained.
But if the ship is in orbit about another body, the I think the passengers can measure this speed. Does the mirror clock argument still work?
Well Yes, but actually no. You know that Earth is orbiting the Sun, because you can observe relative motion of Sun to the other (fixed) stars on the sky. But let Say now, that we don't have any other Stars or reference objects. And let's introduce 2D polar coordinates. Let say, that we are on the Sun, and we believe We are stationary. We can then expres position of Earth as:
$$ \textbf{r}=r_0(\cos \phi, \sin \phi)=r_0(\cos \omega t, \sin \omega t). $$
But we can se, that if we set the coordinate system to the system of Earth we se $\textbf{r}_{Sun-reference}=-\textbf{r}_{Earth-reference}$, ergo: $$ \textbf{r}=r_0(\cos \phi, -\sin \phi)=r_0(\cos \omega t, -\sin \omega t) $$.
We can observe, that we don't have any special effect in the motion, which will show us, which object is actually orbiting around other. But it can be seen, that the rotation changes sign (clockwise to counterclockwise and vice versa.
Bear in mind, that this is only explanation, why you can't know if you are moving or not, if you observe orbiting. I would be happy to see any comments if this is really valid explanation if this case, or should I've been using Lorentz transformations.
