I'll just repeat some of the philosophical observations about SR that I've posted a bunch of times before in this forum:
* A theory of physics is a set of statements that can be used to predict the results of experiments. (Actually we have to change that to "...predict the probabilities of the logically possible results of experiments" or something like that to make sure that quantum mechanics qualifies as a theory, but we don't have to worry about that here).
* No mathematical model makes any predictions about the real world, and therefore a mathematical model can't define a theory of physics.
* SR is defined by a set of postulates that tell us how to interpret the mathematics of Minkowski space as predictions about the results of experiments. (I have listed a few of those postulates below).
* SR is a theory of space, time and motion, and it's also a framework in which we can define theories of matter.
* To define a theory of matter in the framework of SR, we either add matter "manually" (e.g. by associating a mass and other properties with a curve that represents the motion of a particle), or by adding the principle of least action as an additional postulate, and let each Lagrangian define a theory of matter.
* Both classical and quantum theories of matter can be defined in this framework.
* A (global) coordinate system is just a function from Minkowski space into \mathbb R^4. It's a function that assigns four numbers (coordinates) to each event.
* The set underlying Minkowski space is \mathbb R^4, but we have some freedom to choose what structure to define on that set. It doesn't matter if we define Minkowski space as a vector space or as a smooth manifold. We get the same predictions about the results of experiments anyway, assuming that we state our axioms correctly.
* There's nothing that forces us to associate a particular coordinate system with a physical observer that isn't accelerating, but it's
conventional to let the path that represents his motion be his time axis, with the scale defined by what his clock measures, and to let his spatial axes be such that any two events that have the same time coordinate are simultaneous according to the standard definition of simultaneity.
* This is the standard definition of simultaneity: If a radar device equipped with a clock emits a signal when the clock displays -T and detects the reflected signal when the clock displays T, then the reflection event is simultaneous with the event where the clock displays 0.
* The coordinate systems defined in this way are the (global) inertial frames. The word "global" is usually omitted, so I'll do that from now on.
* There's a bijective correspondence between the set of inertial frames and the set of isometries of the metric (the Poincaré group).
* Note that the inertial frames that we (choose to) associate with two physical observers with different velocities are "slicing" space-time into 3-dimensional hyperplanes that represent "space, at a specific time" in different ways. Your slices aren't parallel to mine unless we have the same velocity.
* This is the cause of "length contraction". The motion of the endpoints of a rod are represented by two straight lines in Minkowski space. Its length in your frame is the (proper) length of a straight line that connects the world lines of the endpoints and lies entirely in the hyperplane that you think of as "space, at that time". But the length of the rod in
my frame is different, because the hyperplane that I'm using to represent "space, at that time" isn't parallel to the one you're using.
Some of the postulates that define the theory:
1. Physical events are represented by points in Minkowski space. (Note that this implies that the time evolution (the motion) of a classical system is represented by a set of curves in Minkowski space. It also makes it natural to define a "classical particle" as a physical system whose motion can be represented by
one such curve).
2. A clock measures the proper time of the curve in Minkowski space that represents its motion. (A clock is never a point particle, but it's impossible to state these axioms without making a few idealizations).
Unfortunately, I haven't seen a complete list of the postulates needed. I've been trying to write down a complete list myself, but it's more difficult than it seems.
If you're specifically interested in the differences between inertial frames in SR and pre-relativistic physics, I suggest that you focus on the properties of transition functions, i.e. functions of the form x\circ y^{-1}, where x and y are both inertial frames. Such a function represents a change of coordinates from one inertial frame to another. They have the following properties in both SR and pre-relativistic physics:
a) They are smooth functions from \mathbb R^4 into \mathbb R^4.
b) They take straight lines to straight lines.
In pre-relativistic physics, they also have the property
c) Each transition function maps each hyperplane of simultaneous events onto itself.
In SR, this is changed to
c') Each transition function maps a light-cone onto itself.
D'oh...I didn't intend to make this post this long.
