Page:BatemanTime.djvu/12

in order that an observer P who is at a point $\left(x_{1},y_{1},z_{1}\right)$ at time $t_{1}$ may be in a position to record the effect of a disturbance which issued from a point $\left(x_{2},y_{2},z_{2}\right)$ at time $t_{2}$.

It should be remarked that the conditions given above with regard to the possibility of P seeing Q at the given times are necessary, but not sufficient. This accounts for the fact that the transformations for which the condition

$\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}=c^{2}\left(t_{1}-t_{2}\right)^{2}$

is invariant are limited to a certain group.

It is clear that P is able to see Q if there is a flow of energy from P to Q. If $\left(s_{x},s_{y},s_{z}\right)$ denotes the direction in which energy is flowing from Q at time $t_{2}$, and we regard the differences $x_{1}-x_{2}$, etc., as small, so that

$x_{1}-x_{2}=dx,\ y_{1}-y_{2}=dy,\ z_{1}-z_{2}=dz,\ t_{1}-t_{2}=dt$

we may consider transformations such that the equations

${\frac {dx}{s_{x}}}={\frac {dy}{s_{y}}}={\frac {dz}{s_{z}}}$ $dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}=0$

are invariant. Since the flow of energy depends upon the state of the electromagnetic field, the formulæ of transformation will depend upon the character of the electromagnetic field, but it can be shown that if the above equations are invariant, the fundamental equations which describe the sequence of electromagnetic disturbances are also invariant.^[1] The description of any series of phenomena is thus qualitatively the same in the two series of coordinates.

It is interesting to compare the result just obtained with some ideas with regard to the way in which experience is interpreted by the mind.

If we suppose that some physical process taking place

1. These are not the only transformations which may be used to transform a particular electromagnetic field into another, there appear to be many different types of transformation which are suitable for this purpose.