Page:PrasadSpaceTime.djvu/2

Let us try to represent the situation graphically. Let $x,y,z$ be the rectangular co-ordinates for space and let $t$ denote time. Space and time combined always form the subject of our perception. No one has noticed a place except at a time, and a time except at a place. However, I respect the dogma that space and time have each an independent significance. I will use the word world-point for a space-point corresponding to a time-point, i.e., for a system $x,y,z,t$. The totality of all conceivable systems $x,y,z,t$ may be called the world. I could boldly chalk out four world-axes on the board. Already one of the drawn axes consists of a number of vibrating molecules and, further, makes along with the Earth a voyage in All. Thus this axis already gives us enough to reflect upon; the somewhat greater reflection connected with the axis No. 4 does not do any harm to the mathematician. In order to leave nowhere a gaping void, we imagine to ourselves that something perceptible is existent at all places and at every moment. In order to avoid using the words matter or electricity, I will use the word substance for this "some thing." Let us direct our attention towards the substantial point, existent in the world-point $x,y,z,t$, and let us imagine to ourselves that we are in a position to recognize this substantial point at any other time. The changes $dz,dy,dz$ in the space co-ordinates of this substantial point may correspond to an element of time $dt$. Thus, as the picture – so to say — of the eternal life of the substantial point, we obtain a curve in the world, i.e., a world-line whose points admit of a one-to-one correspondence with the parameter $t$ from $-\infty$ to $\infty$. The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines.

In consequence of the notions, space and time, the $x,y,z$ totality $t=0$ and its two flanks $t>O$ and $t<0$ fall asunder. If, for the sake of simplicity, we keep the zero-point of space and time fixed, then the first group of Mechanics means that, corresponding to the homogeneous linear transformations of the expression

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

into itself, we may subject the $x,y,z$-axes in $t=O$ to an arbitrary rotation round the zero-point. But the second group means that, without having to alter the mechanical laws, we may also replace $x,y,z,t$ by $x-\alpha t,\,y-\beta t,\,z-\gamma t,t$, where $\alpha ,\beta ,\gamma$ are arbitrary constants. After this, the axis of time may be given a fully arbitrary direction towards the upper half-world $t>0$. Now, what has the demand of orthogonality in space to do with this complete freedom of the axis of time upwards?

To establish the connexion, let us take a positive parameter $c$ and consider the figure