It’s well known that in special relativity, observers in relative motion can disagree about the shapes and sizes of various objects. But if Alice draws any planar figure, and fires a simultaneous pulse of light from each point on it, perpendicular to the plane, any observer will agree on the figure’s shape and size.
What? Simultaneity is an observer-dependent notion! But if Alice uses *her* definition of simultaneity, she will encode her chosen planar figure in the collection of light pulses in a frame-invariant way.
What happens if Bob, flying past in a spaceship, uses *his* definition of simultaneity to pick out different events on the worldlines of the same light pulses?
Suppose A is a unit-length spacelike vector orthogonal to Alice's worldline, and C is a lightlike vector orthogonal to A. Then the worldlines for two light pulses that Alice considers to be separated by a distance x, one starting at the origin and one starting at x A, will be:
P(s) = s C
Q(u) = x A + u C
where s, u are parameters for how far we go along the respective worldlines.
At s = u = 0 we have what Alice measures:
P(0) = 0
Q(0) = x A
|Q(0)-P(0)|^2
= |x A|^2
= x^2 |A|^2
= x^2
But for *any* s, u:
|Q(u)-P(s)|^2
= |x A + (u-s) C|^2
= x^2 |A|^2 + 2(u-s) A·C + (u-s)^2 |C|^2
= x^2
So whatever events on those two worldlines Bob considers to be simultaneous, he still ends up measuring exactly the same distance between them as Alice does.
