I decided to re-visit special relativity and spacetime. All went well until I reached the subject of time-like and space-like events and their relation to causality. Time-like Events The residual spacetime value between two time-like events is defined as: ∆S = √(c∆t)^ - ∆x^2, where c∆t > ∆x. Thus, ∆S is real and defines as series of hyperbolic curves in the past and future light-cones of an Event (0,0). Events in these regions can have a causal relationship with Event (0,0). Proper time is given by ∆S/c = ∆t = t√(1 - (V^2/c^2)) Any attempt to determine proper distance in these regions, results in an imaginary value (i.e. there is no proper distance for space-like separated events). So far, so good. Space-like Events This is where my understanding breaks down. The residual spacetime value between two space-like events is defined as: ∆S = √(c∆t)^2 - ∆x^2, where c∆t < ∆x. Thus, ∆S is imaginary. This makes sense, as space-like events are in the "elsewhere regions" of Event (0,0) and thus cannot have a causal relationship with it. However, The texts that I've read , magically reverse the signs within the square root, such that: ∆S = √∆x^2 - (c∆t)^2 and ∆S thus becomes real. I think this mathematical manipulation is akin to rotating the elsewhere regions by ninety degrees to where the past and future lights cones used to be. Once this is done: ∆S = ∆x√(1 - (c2/V2)), which is defined as the "proper distance" between Event (0,0) and events in the elsewhere regions. Because of this mathematical manipulation, even though V > c, the proper distance is a real value. In fact if V < c, proper distance becomes imaginary. So my questions are: 1) How can the spacetime relationship between ct and x be flipped at will to make things work-out right? 2) How can there be a "proper distance" between Event (0,0) and events in the elsewhere regions, if they are not causally related? 3) Shouldn't we just define the "elsewhere regions" as imaginary/not causally related and ignore the concept of "proper distance"?