How can we assume that t' (time in frame S' moving with respect to S) is independent of y,z coordinates while deriving Lorentz Tranformation?

Question

[Referring to section 2.2, derivation of lorentz tranformation, Introduction to Special Relativity by Robert Resnick].

x' = a41*x + a42*y + a43*z + a44*t

The author goes as follows: " For reasons of symmetry, we assume that t' does not depend on y and z. Otherwise, clocks placed symmetrically in the y-z plane about the x axis would appear to disagree as observed from S', which would contradict the isotropy of space"

I am new to Special Relativity and hence I am studying Robert Resnick to get insights. But I am not able to understand the meaning of above quoted lines. Kindly explain the real meaning and significance of these lines.

Answer 1

For a linear relation between the two coordinate system of the form $$t' = a_{41}x + a_{42}y + a_{43}z + a_{44}t$$ and the relative velocity being along the x-axis. If you take any four points symmetrical about the x-axis, $$P_1(x,y,z,t) = (x,1,0,t), \quad P_2(x,y,z,t) = (x,0,1,t)\\P_3(x,y,z,t) = (x,-1,0,t) \quad P_4(x,y,z,t)=(x,0,-1,t)$$
Then for a fixed time, these all lie along a unit circle with the x-axis perpendicular to the centre. Since the motion is along x-axis and the two frames are also separated only along the x and t axes. So things should be symmetric about the x-axis. This means that keeping everything else same, the $t'$ observed by the points $P_1$ and $P_3$ should be the same. (Because you can just rotate everything by 180 degrees about the x axis to get from the first point to the second). Putting this into the equation above you will find that $a_{24} = 0$. By a similar argumetn you can show that $a_{43}=0$. So your $t'$ does not depend on $y,z$ for this case.

In general for motion along a direction that has components along $y,z$ you will find $t'$ will not be independent of these.