Exercise:
Two events occur at the same place in a certain inertial frame and are separated by a time erval of s. What is the spatial separation between these two events in an inertial frame in which the events are separated by a time erval of s?

Solution:
center tikzpicture scope foreach y in ... drawthin colorblack!!white y--y; foreach x in ... drawthin colorblack!!white x--x; drawthick colorgreen!!black-stealth -.--. nodeabove x; drawthick colorgreen!!black-stealth -.--. noderight ct; drawthick colorred .--. nodeabove same place; filldrawcolorblue . circle .cm; filldrawcolorblue . circle .cm; scope scopexshiftcm defbta. defgam. foreach y in ... drawthin colorblack!!white -.*bta+y--*bta+y; foreach x in ... drawthin colorblack!!white -.*bta+x--*bta+x; drawthick colorgreen!!black-stealth -.-.*bta--*bta nodeabove x'; drawthick colorgreen!!black-stealth -.*bta-.--*bta noderight ct'; drawthin dashed colorgreen!!black-stealth -.--. nodeabove x; drawthin dashed colorgreen!!black-stealth -.--. noderight ct; drawcolorred dashed .--.; drawcolorred dashed .--.; filldrawcolorblue . circle .cm; filldrawcolorblue . circle .cm; scope tikzpicture center The s of the Lorentz transformation are: Delta x' gamma Delta x - v Delta t Delta t' gamma leftDelta t - fracv Delta xc^ right Given data of the exercise are: Delta x & Delta x' Delta t s & Delta t' s Second of the Lorentz transformation leads to: Delta t' gamma leftDelta t - fracv Delta xc^ right gamma leftDelta t - right gamma fracDelta t'Delta t fracss . beta sqrt-fracgamma^ sqrt-frac.^ . v beta c .c v The spatial difference between the two events is in the second frame of reference mathcalS' is therefore: Delta x' gamma Delta x - v Delta t -gamma v Delta t -. . ncc s -.em The minus sign means that the second event is located before the first event on the x'-axis of system mathcalS' as can be seen in the sketch above.