ミンコフスキー空間

最近アインシュタインの特例相対理論を読みながら４次元の空間を考えています。
Basically, I was wondering why the four-dimensional view of space and time is superior to the old one, that of a three-dimensional space and separate single dimension for time. My initial intuition was that there should be a way to express the proximity of events that happen either in the same place at different times or at the same time in different places. With separate "worlds" used for space and time, it is difficult to express such proximity within a single framework. In other words, events that happen at the same position in space are not comparable to those which happen at the same time. This makes it difficult to see whether spatial coordinates or temporal coordinates are more important to experimental results.
Einstein notes that, in the Galilean world, time is considered independent of space and absolute. In other words, given that t' is the time in reference frame K' and t is the time in frame K, t' = t. So time is independent of the K' motion with respect to K. In a relativistic world, the time depends on the motion of K', as:
t' = (t-(v/c**2)x)/sqrt(1-(v/c**2))
If the time lapse between two events, dt, is zero in K, dt' may still be non-zero. Thus, just as spatial coordinates are mapped via the Lorentz transform, so time is also. In addition, a mapping of the time function enables it to be written such that the absolute distance in Minkowski space is equivalent in K and K'.
For 3-d space,
x**2 + y**2 + z**2 - c**2t**2 = x'**2 + y'**2 + z'**2 - c**2t'**2 holds.
Substituting sqrt(-1)*ct as w, we obtain:
x**2 + y**2 + z**2 + w**2 = x'**2 + y'**2 + z'**2 + w'**2
So distances are preserved by the Lorentz transform, and the speed of light remains constant.