A Primer on Special Relativity
An inertial coordinate system is a system of space and time coordinates with respect to which (1) every material body free of external influence moves at constant speed in a straight line, and (2) if two identical material objects initially adjacent and at rest act to repell each other, they acquire equal speeds in opposite directions. Given one inertial coordinate system we can construct infinitely many others by means of arbitrary fixed translations and spatial rotations, which leave the speed of every object unchanged. Such an equivalence class of inertial coordinate systems is called an inertial reference frame. It's important to recognize that the definition of an inertial reference frame not only identifies inertial motion with straight paths of constant speed, it also establishes an operational definition of simultaneity (i.e., the synchronization of times at spatially separate events), because according to property (2) we can use identical physical objects acting against each other to synchronize clocks equidistant from their center of mass.
Given this definition of inertial reference frames, the principle of relativity asserts that for any material particle in any state of motion there exists an inertial reference frame - called the rest frame of the particle - with respect to which the particle is instantaneously at rest (i.e., the change of the spatial coordinates with respect to the time coordinate is zero). This principle is usually extended to include reciprocity, meaning that for any two systems S1 and S2 of inertial coordinates, if the spatial origin of S1 has velocity v with respect to S2, then the spatial origin of S2 has velocity -v with respect to S1. The existence of this class of reference frames, and the viability of the principles of relativity and reciprocity, are inferred from experience. Once these principles have been established, the relationship between relatively moving inertial coordinate systems can then be considered.
Let [t,x,y,z] signify a system of inertial coordinates in the rest frame of particle p, and likewise let [t',x',y',z'] signify a system of inertial coordinates in the rest frame of a particle p' moving with speed v relative to [t,x,y,z]. By means of a fixed translation we can make the origins of these two coordinate systems coincide, and by a fixed spatial rotation we can spatially align the x and x' axes. For simplicity, we will consider particles and motions confined to the x,x' axes. The question naturally arises as to how these two coordinate systems are related to each other for a given relative velocity v. Since, by definition, inertial motions are straight lines with respect to both systems, the relations between two inertial coordinate systems must be linear functions of the form
for constants A,B,C,D (for a fixed v). A stationary object in the rest frame of p' has a constant value of x', so the differential dx' = Adx + Bdt = 0 implies dx/dt = -B/A = v and hence B = -vA. The inverse of the above transformation is
Evaluating the velocity of a stationary object in the frame of p with respect to the frame of p' leads to dx'/dt' = B/D = -v (by reciprocity) and hence D = A.
Letting m2 denote the determinant AD - BC, and substituting for B and D, we have A2 + vAC = m2, so we have C = (m2 - A2)/(vA). If we define a = A/m, then the original transformation can be written in the form
and the inverse transformation has the form
If we replace v with -v these two transformations are exchanged, except for the factor m, so if we are to have spatial isotropy we must have m equal to 1. This leaves undetermined only the expression in square brackets. (Remember that the parameter a is a function of v, but it is a constant for any fixed v.) Letting k denote this quantity, we have [(a2-1)/(va)2] = k, from which we get 
, and the general transformation can be written in the form
The magnitude of the constant k (for any given v) can be absorbed by the units of space and time, so the only three distinct cases to consider are k = +1, 0, and -1. If k = -1 this transformation is simply a Euclidean rotation in the xt plane through an angle q = invtan(v). In other words, with k = -1 the above equations can be written in the form
With k = 0 the equations represent the Galilean space-time transformation, i.e., we have
The remaining case is with k = +1, which gives the Lorentzian space-time transformation
We wish to determine which of these represents the correct relation between relatively moving inertial coordinate systems. The Euclidean transformation (i.e., the case k = -1) is easy to rule out empirically, because we cannot turn around in time as we can in space. However, it isn't as easy to distinguish empirically between the Galilean and Lorentzian transformations, especially if the value of k in ordinary units of space and time is extremely close to zero. As a result, Newtonian mechanics was based for many years on the assumption that k = 0.
It was not until the late 19th century that sufficiently precise experimental techniques became available to determine that the true value of k is not zero. In ordinary units it has the value (1.11)10-17 second2/meter2, which happens to equal 1/c2 where c is the speed at which electromagnetic waves propagate in vacuum. This implies that relatively moving systems of inertial coordinates are related according to the Lorentzian transformation. It follows that the constant-t surfaces of two relatively moving systems of inertial coordinates are skewed, although the skew is so slight it's almost impossible to detect in most ordinary circumstances.
Notice that for any incremental interval whose components are (dt,dx) with respect to one particular system of inertial coordinates, and (dt',dx') with respect to any other system of inertial coordinates, we have
(dt')2 - (dx')2 = (dt)2 - (dx)2
This signifies that the quantity (dt)2 - (dx)2 is invariant with respect to all inertial coordinate systems. Since there exists a system of inertial coordinates with respect to which the spatial component dx is zero, it follows that the above invariant quantity is the square of the time differential dt along a path with respect to the rest frame of the path. This particular time differential is an invariant quantity, called the proper time of the interval (usually denoted by dt to distinguish it from an arbitrary inertial time coordinate). It's easy to see that the inertial path between any two events has the maximum lapse of proper time. Any non-inertial path between those same two events will have a lesser lapse of proper time. In general, the lapse of proper time along the path of any physical entity corresponds precisely to the advance of the entity's quantum state vector.
When people first hear about special relativity they often wonder if it's necessary to think in terms of coordinate systems whose constant-t surfaces are skewed. They point out (correctly) that it's perfectly possible to construct a set of relatively moving coordinate systems that all share a common time coordinate. However, if two relatively moving systems of coordinates share a common time parameter, they cannot both be inertial coordinate systems. The definition of an inertial coordinate system already imposes a specific set of constant-t surfaces for any given time axis in order to make inertia isotropic (i.e., the same in all spatial directions). We are certainly free to think in terms of non-inertial coordinate systems, but then we must be careful to remember that inertia is not isotropic with respect to such coordinate systems.
One major shortcoming with the way in which special relativity is usually taught is that inertial coordinate systems (and frames) are not fully defined. They are typically characterized simply as coordinate systems that are not accelerated. Being unaccelerated is a necessary but not a sufficient condition for a coordinate system to be inertial, because we can have the spatial coordinates oblique to the time coordinate, relative to the inertial orientation. In such a system, all inertial motion has uniform speed in a straight line, but nevertheless the coordinates are not (in general) inertial, because although Newton's first law of motion is satisfied, his second and third laws are not (even quasi-statically). In other words, inertia is not generally isotropic with respect to oblique coordinates. For any given time axis there is a unique orientation of the spatial coordinates compatible with inertial isotropy.
Students are often told that Einstein and/or Poincare were the first to introduce operational definitions of simultaneity, but in fact there has always been an operational definition of simultaneity for inertial coordinates, because there is a unique simultaneity compatible with inertial isotropy for any given time axis. Galileo himself explained this in his "Dialogues on the Two Chief World Systems". What the discoveries of Bradley, Fizeau, Maxwell, Michelson, etc., made clear was that the propagation of electromagnetic disturbances is isotropic with respect to the same class of coordinate systems (the inertial coordinates) in terms of which mechanical inertia is isotropic. In addition, they found that the value of k is not exactly zero (coincidentally at about the same time that Planck discovered that the value of h is not exactly zero), and consequently there is an invariant speed, equal to 
, for the set of inertial coordinate systems. From this we immediately have the concept of proper time, which is identified with the phase of the quantum wave function along any given timelike path. All the familiar consequence of special relativity follow.
Another common misconception is that we cannot assert the empirical isotropy of the one-way speed of light with respect to inertial coordinates. The fact is that it's perfectly possible to demonstrate this one-way isotropy. Simply observe that two identical particles acting on each other reach any given distance from their common center of mass coincident with two pulses of light emanating from that center. Obviously this does not demonstrate the one-way speed of either the particles or the light pulses, but it does demonstrate that if we define a system of coordinates such that mechanical inertia is isotropic, then the one-way speed of light is isotropic with respect to those coordinates.