Minkowski Space
Section Seventeen
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
Minkowski, "Space and Time", address at the 80th Assembly of German Natural Scientists and Physicians, Sept. 21, 1908
Many people feel difficulties for understanding that the Minkowski space has four dimensions. But anyone, who believes that he/she lives in a three dimensional Euclidean space in which "time flows", in effect has to admit that he/she lives in a four dimensional world. What is peculiar in Minkowski space is that time became (mathematically) on a par with spacial axes; but this is nothing but one of the consequences from the Lorentz metric (due to Einstein's definition of simultaneity and time). Thus you may put it this way: the talk of "four dimensionality" is a convenient way to treat the Lorentz metric in physics.
If you still insists on "seeing" the essential features of Minkowski space, here are a couple of pictures, one 3 dimensional (time with two dimensions of space), the other 4 dimensional (time with three dimensions of space).
Last modified, Oct. 9, 2003. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
