# Inertial Frames in Special Relativity

According to the postulates of special relativity, all inertial frames are equal in all respects. Then how does it follow from this, that the space is isotropic and homogenous for an inertial frame and time is homogenous. Or does the former follow from the latter?

Or are they two independent postulates about inertial frames? Can isotropy and homogeneity of space and homogeneity of time for an inertial frame follow from a perhaps even more fundamental postulate in physics?

Let us start from classical physics.

The fundamental invariance postulate in classical physics is that all physical laws describing an isolated physical system in an inertial reference frame are invariant in form under the action of Galileian (Lie) group. That group is made of $4$ (Lie) subgroups.

(1) spatial translations, invariance under space translations is also known as spatial homogeneity in a given inertial reference frame. It implies that the total momentum of the system is conserved along its the evolution.

(2) spatial rotations, invariance under space translations is also known as spatial isotropy in a given inertial reference frame. It implies that the total angular momentum of the system is conserved along its the evolution.

(3) temporal translations, invariance under time translations is also known as temporal homogeneity in a given inertial reference frame. It implies that the total energy of the system is conserved along its the evolution.

(4) subgroup of pure Galileian transformations ($t\to t'=t$, ${\bf x} \to {\bf x}' = {\bf x} + t{\bf v}$). This last type of invariance also implies physical equivalence of all inertial reference frames. It implies the existence of a relation connecting the total momentum of the system and the motion of the center of mass.

In special relativistic physics, passing to deal with Poincaré group in place of Galileo group, the situation is almost identical. It is certainly identical regarding (1), (2) and (3) as the mentioned groups of transformations are also subgroups of Poincaré group. The only relevant difference concerns (4). Here the subgroup of pure Galileian transformation is replaced with the set of pure Lorentz transformations: the boosts. Invariance under boost implies again physical equivalence of all inertial reference frames. The associated conservation law is a theorem analogous to the classical one, with the fundamental difference that now the total mass of the system includes the contribution of energies of the parts of the system in accordance with $m=E/c^2$.

• Thank you for your answer. I would be grateful if you could add what happens after relativistic corrections as the wikipedia article for Lorentz group is bit difficult for me to decode at this moment ? – Isomorphic Mar 5 '14 at 16:04
• Passing to Poincaré group the situation is almost identical. The only relevant difference is relative to (4). Here the subgroup of pure Galileian transformation is replaced with the set of pure Lorentizian transformations (the boosts). Invariance under boosts implies again physical equivalence of all inertial reference frames. The associated conservation law is a theorem analogous to the classical one, with the fundamental difference that now the total mass of the system includes the contribution of energies of the parts of the system in accordance with $m=E/c^2$. – Valter Moretti Mar 5 '14 at 16:20

There is. Have a look at http://en.wikipedia.org/wiki/Noether%27s_theorem. Isotropie is "coupled" with conservation of angular momentum, homogenity of space to conservation of momentum and homogenity of time to conservation of energy.