Interactive · Frame Flip

Spacetime, two ways at once.

A Minkowski diagram is a portrait of spacetime. The black axes belong to the lab. Drag β and the teal axes — the primed frame’s — scissor inward toward the light cone. Every event has two coordinate names; the diagram shows you how to read them off.

Boost between frames

Relative velocity v / c β = 0.50 γ = 1.155

sweeps β through 0 → +0.9 → 0 → −0.9 → 0 (8 s loop)

Show line of simultaneity through each event Show hyperbolae of constant interval

Pick a third event

Slide event C around. Watch its (x, t) in the lab frame and its (x′, t′) in the primed frame both update.

x 2.00 t 0.00

event	(x, t) lab	(x′, t′) primed
A	(-1.50, 0.00)	(-1.73, 0.87)
B	(1.50, 0.00)	(1.73, -0.87)
C	(2.00, 0.00)	(2.31, -1.15)

What you’re looking at

Lab axes — orthogonal: x horizontal, ct vertical.
Primed axes — boosted by β. As β grows, both axes tilt toward the light cone (they never cross it).
Light cone — 45° lines. Invariant: every observer sees light at exactly c.
Lines of simultaneity in the primed frame — events on a single dashed line all share the same $t^{'}$ , but generally different $t$ .

The Lorentz transform, applied

x^{'} = γ (x - β c t), c t^{'} = γ (c t - β x)

Pick event C at ( $x = 2.00$ , $t = 0.00$ ). With $γ = 1.155$ at $β = 0.50$ :

x^{'} = 1.155 (2.00 - 0.50 \cdot 0.00) = 2.309

t^{'} = 1.155 (0.00 - 0.50 \cdot 2.00) = - 1.155

Two events, simultaneous in the lab, are not simultaneous in the primed frame.

Events A and B were placed at $(- 1.5, 0)$ and $(1.5, 0)$ — same lab time, separated only in space. In the primed frame their times are:

t_{A}^{'} = - β γ (- 1.5) / c = 0.866, t_{B}^{'} = - β γ (1.5) / c = - 0.866

At any nonzero β, $t_{A}^{'} \neq = t_{B}^{'}$ . Simultaneity is frame-dependent.

The spacetime interval, however, is invariant: $(Δ s)^{2} = (Δ t)^{2} - (Δ x)^{2} = - 9.00$ in every frame. That’s the bedrock — frames disagree on x and t separately, but agree on the combination $t^{2} - x^{2}$ .