Two events in S are separated by a distance D=x2 - x1 and a time T = t2 - t1 .
a) Use the Lorenrz transformation to show that in frame S' , which is moving with speed v relative to S, the time separation is t2 - t1 = ã(T - vD/c2 ).
b) Show that the events can be simultaneous in frame S' only if D is greater than cT.
c) If one of the events is the cause of the other, the separation D must be less than cT since D/c is the smallest time that a signal can take to travel from x1 to x2 in frame S. Show that if D is less than cT, t2' is greater than t1' in all refence frames. d) Suppose that a signal could be sent with speed c'>c so that in frame S the cause precedes the effect by the time T=D/c'. Show that there is then a reference frame moving with speed v less than c in which the effect precedes the cause.