Page:Grundgleichungen (Minkowski).djvu/58

 in direction of B*C*.. Now by $$OA'/B*D*$$ is to be understood the ratio of the two vectors in question.

It is clear that this proposition at once shows the covariant character with respect to a -group.

Let us now ask how the space-time filament of F behaves when the material point F* has a uniform translatory motion, i.e., the principal line of the filament of F* is a line. Let us take the space time null-point in this, and by means of a -transformation, we can take this axis as the t-axis. Let x, y, z, t, denote the point B, let $$\tau^{*}$$ denote the proper time of B*, reckoned from O. Our proposition leads to the equations

and

where

and

In consideration of (27), the three equations (25) are of the same form as the equations for the motion of a material point subjected to attraction from a fixed centre according to the Newtonian Law, only that instead of the time t the proper time $$\tau$$ of the material point occurs. The fourth equation (26) gives then the connection between proper time and the time for the material point.

Now for different values of $$\tau$$ the orbit of the space-point x, y, z is an ellipse with the semi-major axis a and the eccentricity e. Let E denote the excentric anomaly, T the increment of the proper time for a complete description of the orbit, finally $$n\mathsf{T}=2\pi$$, so that from a properly chosen initial point r, we have the Kepler-equation

If we now change the unit of time, and denote the velocity of light by c, then from (28), we obtain