Exercise:
Two events occur at the same time in an inertial frame mathcalS and are separated by a distance of km along the x axis. What is the time difference between these two events as measured in a frame mathcalS' moving with constant velocity along x and in which their spatial separation is measured as km?

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 .--. nodebelow midway same time; 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 km & Delta x' km Delta t & Delta t' First of the Lorentz transformation leads to: Delta x' gamma leftDelta x - v Delta t right gamma leftDelta t - right gamma fracDelta x'Delta x fracmm beta sqrt-fracgamma^ sqrt-fracgam^ bet v beta c bet c v The time difference between the two events in mathcalS' follows then directly from the second : Delta t' -gamma fracvDelta xc^ - frac.c mc^ -.s That means in the second frame of reference the second event took place first .micro s before it.