Does the Relativity Principle of Special Relativity imply homogeneity and isotropy of all the reference frames?

Question

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?

Answer 1

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.)

Answer 2

I don't have Rindler with me since I am on holidays but I don't believe he means PR inplies homogeneity and isotropy, and you yourself say "''gives that this frame should be homogeneous and isotropic." So I'm almost certain that the venerable Prof Rindler would be saying that it's the other way around in many cases: for example, the derivation of the Lorentz boost from group postulates assumes that an inertial observer's measurement of distances and times does not depend on where in space they are and in which direction they are going. Even if there is matter around (a "little bit", so that its stress energy tensor can be left out of our calculations unless we want to be really accurate, e.g. for synchronising the GPS system), one would expect that its presence would not affect how an IR measures their co-ordinates, even if some physics is affected by its presence (e.g. electromagnetism, fluid mechanics).

You are quite right: the physical laws must be clearly independent from human constructs like co-ordinates, but this is different from the notions of homogeneity and isotropy : in your example of physical anisotropy, your covariance principle simply means that the direction of physical anisotropy must of course be the same whatever co-ordinate system we may reckon it from.

In your example of a constant physical force in a constant direction (a word of advice: don't write that symbol $c$ to stand for anything other than the Lorentz-invariant $c$ when thinking about relativity, unless you're a real sucker for punishment and like to get yourself confused :) ) even the measurement of time and distance would depend on position and direction: if everything is feeling an apparent force relative to an observer, that observer's accelerometers would be showing an acceleration and the observer would not be inertial: his/her plight would be equivalent to the constantly accelerating elevator / rocketschip studied in GR and would likely find it easiest to do his/her calculations in the venerable Prof. Rindler's own co-ordinates.

Answer 3

Gravitation chapter 27 provides a little better insight into this statement. The selection of a inertial reference frame comes up in the discussion of homogeneity. Homogeneity in the most general sense means that the universe is "the same" everywhere at any given moment of time (e.g. some set of parameters or governing equations or whatever is the same at all points in space for a given moment of time).

This is easily understood in non-relativistic theories, but is ill-defined in relativistic theories. Unless space-time is flat, there are no global inertial frames in general relativity. This means one has to select an inertial frame from which the can define a 3-dimensional space-like hyper-surface.

The events on the hyper-surface have a local Lorentz frame whose surface of simultaneity coincides to the hyper-surface. The 4-velocity of the Lorentz frames are orthogonal to the hyper-surface.

Isotropy is a statement that an observer can not distinguish a preferred spatial direction from any other. This means that all world lines are orthogonal to the hyper-surface. This is important because the universe can not be made to look isotropic to all observers, only for ones that are moving as part of the "cosmological fluid". This is interpreted to mean that "co-moving observers" share the same hyper-surface and agree the universe is the same everywhere with no preferred direction. Observers who are not co-moving will not see the universe as isotropic, for instance, an observer moving in one direction near the speed of light will see a highly blue-shifted universe in front of them and a highly red-shifted universe behind him, which is clearly not isotropic.

The Special Principle of Relativity then is a statement related to observers on hyper-surfaces. It tells us that co-moving observers will see the same physical universe, but we can smoothly transform from one co-moving hyper-surface to another. Every observer will discover they are at rest relative to the their hyper-surface of homogeneity and they can measure their velocity relative to that hyper-surface. As soon as they change velocity in some direction (accelerate in some direction), there will be a definite direction identified, breaking isotropy.

So in to answer the question, if you are choosing a preferred direction, you are accelerating away from your original hyper-surface. It is General Relativity that tells us you can still formulate your physical laws to be invariant under coordinate changes. The choosing of direction, and associating it with acceleration, means you are in a non-inertial reference frame, which do not abide by the principles of special relativity.