In Rindler's book: Relativity, Special, General and Cosmological, is stated on page 40 that the Relativity Principle (RP), when applied to just one Inertial Frame (IF), guarantees the homogeneity and isotropy of tha IF. By inertial frame Rindler means an ideal infinity extended rigid body moving freely in a world without gravity. This is distinct from an inertial coordinate system, that should be understood as an IF plus, in it, a choice of standard coordinates $x$, $y$, $z$ and $t$.
As he says, the RP concerns inertial coordinate systems: the laws of physics are invariant under a change of inertial coordinate systems.
I can't understand why this imply homogeneity and isotropy of an IF. If I suppose the existence of an special direction in some inertial reference frame, I could imagine some physical law governig the propagation of some signal (it can be light if you want, but it's not necessary), and if by measuring the velocity of this signal in two different directions and I get two different results, this would violate the isotropy of the IF and at the same time I could write the physical law in an invariant way under coordinate changes inside de IF (sure, it would depend on the special direction) and this would be in accordance with the RP as stated above. What is wrong with my reasoning?
Best Answer
I think you're right that the Principle of Relativity does not imply that the frame is isotropic. The former is a statement about physical laws - it is a kind of a meta-law. Isotropy of a frame is a statement about the inner character of the frame, i.e. about something that can be viewed as a physical body.
Consider this inertial system: a piece of crystal with preferred direction (say, beryl with hexagonal crystal structure.) I think that here the Relativity Principle and anisotropy of the frame could co-exist.
(Your idea about the universal acceleration does not work well though, because the equation holds in a frame that is not accelerating together with the bodies, and such frame does not qualify as inertial. The frame moving with the bodies does qualify as inertial, but the equation there is $\mathbf{F}'=m\mathbf{a}'$ which is isotropic.)