2301-20 Spacetime Diagrams

Here are some tools for making spacetime diagrams, i.e. plots of events where “x” runs from left to right and “t” runs up the page.

event[{x_,t_},string_]:=Tooltip[{Gray,Disk[{x,t},.03]},string]
blueline[{x1_,t1_},{x2_,t2_},string_] := Tooltip[{Blue,Line[{{x1,t1},{x2,t2}}]},string]
redline[{x1_,t1_},{x2_,t2_},string_] := Tooltip[{Red,Line[{{x1,t1},{x2,t2}}]},string]
photon[{x1_,t1_},{x2_,t2_},string_] := Tooltip[{Dashed,Green,Line[{{x1,t1},{x2,t2}}]},string]

Alan+Beth meet

Here we choose parameters and label some events in the story of Alan and Beth and the firecrackers.
In this telling of the story, Alan has assistants Andy and Arnold, who stand at +L and -L.
According to Alan, here is how the various events are arranged in space and time:

AlanWorld=
{
event[{0,0},"Alan meets Beth"],
event[{L,0},"Front Flash"],
event[{-L,0},"Rear Flash"],
redline[{0,-1/2},{0,2},"Alan worldline"],
redline[{L,-1/2},{L,2},"Andy worldline"],
redline[{-L,-1/2},{-L,2},"Arnold worldline"],
blueline[{-β/2,-1/2},{2β,2},"Beth worldline"]
};
params={L->1,γ->1/Sqrt[1-β^2],β->1/3}
Graphics[AlanWorld/.params]

And here is Beth’s version of events, which we get quickly with one Lorentz transformation.

$Post=MatrixForm
Λ= {{γ,-β γ},{-β γ,γ}}
Graphics[GeometricTransformation[AlanWorld,Λ]//. params]

Or here is a version where one can continuously transform.

Manipulate[Graphics[GeometricTransformation[AlanWorld/.params,{{Cosh[θ],Sinh[θ]},{Sinh[θ],Cosh[θ]}}],PlotRange->GoldenRatio],{{θ,0},-1,1}]

Here are some events and lines to add on to that diagram:
-- event where Alan sees the light
-- photon lines for the front and back flashes
-- events where Beth sees the flashes
-- lines of simultaneity for Alan (tA=0) and for Beth (tB=0)

Hwk07 Problem: Make two spacetime diagrams showing the events and worldlines from both points of view for the   “Twice Simultaneous” problem (Morin 11.39), making your own choice for “v”.