Page:De Raum Zeit Minkowski 011.jpg

— 2 —

at a particular time, or has observed a time except at a particular place. Yet I still respect the dogma that time and space have independent existences each. I will call a space-point at a time-point, i.e., a system of values x, y, z, t, as a world-point. The manifold of all possible value systems of x, y, z, t, shall be denoted as the world. I boldly could draw four world-axes with a chalk upon a table. Even one axis drawn consists of nothing but quickly vibrating molecules, and besides, takes part in all the journeys of the earth in the universe; and therefore gives us plenty occasions for abstractions. The greater abstraction connected with the number of 4 does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to say either matter or electricity, we shall simply use the word substance for this something. We direct our attention to the substantial point located at world-point x, y, z, t, and suppose that we are in a position to recognize this substantial point at any other time. Let dt be the time element corresponding to the changes dx, dy, dz of space coordinates of this substantial point. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point), a curve in the world, the world-line, whose points can unambiguously be connected to the parameter t from $+\infty$ to $-\infty$. The whole world appears to be resolved in such world-lines, and I may just anticipate, that according to my opinion the physical laws would find their most perfect expression as mutual relations among these world-lines.

By concepts of time and space, the x, y, z manifold $t=0$ and its two sides $t<0$ and $t>0$ fall apart. If, for the sake of simplicity, we keep the null-point of time and space fixed, then the first mentioned group of mechanics signifies that at $t=0$ we can give the x, y, z-axes an arbitrary rotation about the null-point, corresponding to the homogeneous linear transformation of the expression

$x^{2}+y^{2}+z^{2}\,$

in itself. Yet the second group denotes that – also without changing the expression for the mechanical laws – we can substitute

x, y, z, t by $x-\alpha t,\,y-\beta t,\,z-\gamma t,t$

with any constants $\alpha ,\beta ,\gamma$. According to this we can give the time-axis any possible direction in the upper half of the world $t>0$. Now, what has the demand of orthogonality