Postulates of SR without inertial frames?

[WannabeNewton]

As a member of what? The Poincare group in the case of SR? I don't see how that would work. A global coordinate chart, for this space - time is simply the usual pair $(\mathbb{R}^{4},\phi )$ and the coordinate system is the 4 - tuple of functions given by $\phi(p) = (x^{0}(p),...,x^{3}(p)),p\in \mathbb{R}^{4}$. If this coordinate system happened to be set up by an inertial observer and if we have another global coordinate chart $(\mathbb{R}^{4},\phi')$ then whether or not the associated coordinate system was set up by another inertial observer can be determined by the transition map.

On the other hand, each $g\in G$, with $G$ being the Poincare group, has associated a right translation $\chi _g$. The killing fields of $(\mathbb{R}^{4},\eta _{ab})$, that is the vector fields $\xi ^{a}$ such that $\mathcal{L}_{\xi }\eta _{ab} = 0$, then correspond to the resulting right invariant vector fields on $G$. So the elements of the Poincare group are intimately related to the geometrical symmetries of Minkowski space - time. I'm not seeing how we can make coordinate systems be members of this.

 

[Fredrik]

Yes, the Poincaré group in the case of SR. You don't have to define the Poincaré group in terms of killing fields or anything else that involves differential geometry. It's perfectly adequate to define it as the set of maps $x\mapsto \Lambda x+a:\mathbb R^4\to\mathbb R^4$ such that $\Lambda$ is linear and $\Lambda^T\eta\Lambda=\eta$. With this definition, it's just a subgroup of the permutation group of $\mathbb R^4$. Since the underlying set of spacetime is $\mathbb R^4$, any smooth permutation of $\mathbb R^4$ can be thought of as a coordinate system.

The Poincaré group can also be defined as the group of isometries of the Minkowski metric, i.e. as the set of all diffeomorphisms $\phi:\mathbb R^4\to\mathbb R^4$ such that $\phi^*g=g$, where $\phi^*$ is the pullback function associated with $\phi$. This definition is equivalent to the simple one above. I prefer it over that stuff involving killing fields, but maybe that's just because I understand this approach much better.