This is a summary of the second part of Appendix A in Albert Einstein’s book . By second part, I mean the part immediately after the derivation of the Lorentz transformation. If there is no relative motion between the coordinate systems with respect to the $y$ and $z$ axis of $K$, then the $Y$ and $Z$ coordinates of the light in $K$ are equal to the coordinates of the light in $K’$, since there is no need to modify these coordinates: $Y’ = Y$ and $Z’ = Z$.

Next, the propagation of light is investigated. As mentioned at the beginning of Derivation #1, suppose there is a flash of light at the origin of $K$ at $t=0$. This light travels away from the origin in every possible direction in three dimensions. The light travels with a speed of $c$ in each direction. The distance from the origin that the light travels as a function of time $t$ is $R=ct$. This $R$ can be thought of as a radius of a sphere with a center at the origin. From geometry, the equation for a sphere as a function of position coordinates is $R^2 = X^2 + Y^2 + Z^2$, so $R = \sqrt{X^2 + Y^2 + Z^2}$. The positive root is used because distances are positive by definition. The equation for a sphere in three dimensions as a function of position coordinates can be derived with the Pythagorean Theorem. Since $R^2 = c^2t^2 = X^2 + Y^2 + Z^2$, it is clear that $0 = X^2 + Y^2 + Z^2 – c^2t^2$. This equation can be made more general with a variable $\sigma$, by multiplying both sides of the equation by $\sigma$: $0 = \sigma(X^2 + Y^2 + Z^2 – c^2t^2)$.

The same reasoning can be applied to light in the translating coordinate system $K’$ with coordinates $x’, y’, z’, t’$. This can be done because one of the assumptions of special relativity is that the speed of light is $c$ regardless of whether the coordinate system is translating with a nonzero velocity or not. Hence $R’^2 = c^2 t’^2$, $(R’)^2 = X’^2 + Y’^2 + Z’^2$, and $0 = X’^2 + Y’^2 + Z’^2 – c^2 t’^2$.

Since $0=0$ suppose that it is valid to write $X’^2 + Y’^2 + Z’^2 – c^2 t’^2 = \sigma(X^2 + Y^2 + Z^2 – c^2t^2)$.

Furthermore, suppose in $K$ it is possible for light to only move along the $x$-axis. If this is the case, then the equation of $0 = \sigma(X^2 + Y^2 + Z^2 – c^2t^2)$ reduces to $0 = \sigma(X^2 – c^2t^2)$ since the light’s position does not advance along the $y$ and $z$ axes. Similarly, for $K’$, suppose it is possible for light to only move along the $x’$-axis. If this is the case, then the equation of $0 = X’^2 + Y’^2 + Z’^2 – c^2 t’^2$ reduces to $0 = X’^2 – c^2 t’^2$ since the light’s position does not advance along the $y’$ and $z’$ axes.

Since $0=0$ suppose that it is valid to write $X’^2 – c^2 t’^2 = \sigma(X^2 – c^2t^2)$. Einstein sets $\sigma=1$ for a reason I have not been able to determine. More attention will be given to this point in the future. This is one question to which I do not have an answer.

This post brings the engaged reader up to equation (11) in Appendix A of Einstein’s book .

References

 Albert Einstein. Relativity; The Special and General Theory. Appendix A. Three Rivers Press, 1961.