## Remarks on the Lorentz Transformation and One Question

This is a summary of the second part of Appendix A in Albert Einstein’s book [1]. 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 […]