The Lorentz-transform calculation above uses a definition of extended-simultaneity (i.e. of when and where events occur at which you were not present) that might be referred to as the co-moving or "tangent free-float-frame" definition. This definition is naturally extrapolated to events in gravitationally-curved spacetimes, and to accelerated observers, through use of a radar-time/distance definition that (unlike the tangent free-float-frame definition for accelerated frames) assigns a unique time and position to any event.