Page:PrasadSpaceTime.djvu/2

136 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