Assume there is a stationary observer O and a moving observer O' at v in collinear relative motion. When origined with O, O' emits a spherical light pulse. Now, the light pulse is described by O as x^2 + y^2 + z^2 = (ct)^2 and by O' as x'^2 + y^2 + z^2= (ct')^2. By considering only the x-axis points of the light sphere, it is the case that x'^2= (ct')^2. Thus, ct' = ± x'. Here is my question. What are the equations strictly from the coordinates and proper time of O to describe ct' = ± x'. This means, what are the x points in O and what are the times in O for the light sphere of O'.