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Let's consider Lorentz Transformation (if to choose system of units where light speed = 1)

T = (t - vx )
X = (-vt + x)


where as usual = 1 / ( 1 - v 2 )1/2
v - is a speed of inertial system K which moves along axis x of the system "at rest" k; x,t are measured in k;
X,T are measured in K.

The function

= t 2 - x 2 .


has notable property:

=


where = T 2 - X 2 . In other words, points T(X) representing the same event will form contour of function or solution of equation:

T 2 - X 2 = const


For example, if to choose an event (t,x) which happened at the time t at origin of the system "at rest" x=A,
and then consider this event in all inertial systems which move along axis x,
then the graph T(X) of such events will form "hyperbola" with minimum at X=A and asymptotes T=X and T=-X.




Using substitution method, it can be checked that Lorentz Transformation does not chage a shape of group of events which belong one line. In other words, line t(x) = ax + b transforms to line also.

So, events which lie on axis x happen simultaneously (like events D and C ), but at certain v this line transforms to a line D'C' with neither of its events happened simultaneously because they have different ordinates.

Moreover, events lying on line D'' C'' have different time order when they observed in system where they presented by line D' C'. For example, event D'' happened "before" event C'', but the same event D' happened "after" event C'.





More details about Fig. 1.

Events A happened after event B in system k. They both happened at origin x=0.
When we imagine sequence of inertial systems with decreasing speed v,
then point A moves along the right branch of hyperbola 1;
point B moves along hyperbola 2.

For example, if v < 0 then, event A viewed in system K, will be represented by point A'.
The same way, it is easy to verify that event D moves along the down branch of hyperbola D, D'.