Why do the primed axes tilt toward the light cone?

A boost mixes x and ct; lines of constant t′ (simultaneity in S′) and constant x′ (worldlines of rest points in S′) both lean by angle atan(β) toward the null lines ct = ±x. Light rays stay at 45° in every inertial frame. Seeing both sets of axes on one diagram makes length contraction and time dilation geometric rather than mysterious.

What does the sign of Δs² tell me?

With Δs² = (cΔt)² − (Δx)², positive values are time-like (causal order possible), zero is light-like, and negative is space-like (no causal influence). Sign conventions vary in textbooks (some use −+++ metric); this lab uses the particle-physics mostly-plus time convention stated above. Do not confuse coordinate Δt with proper time along a worldline.

Can two events be simultaneous in one frame but not another?

Yes — that is relativity of simultaneity. Horizontal lines in S are t = const; in S′ simultaneity slices are tilted. Events that share a t′ value generally have different lab times. The diagram makes that tilt explicit when you change β.

Minkowski Spacetime Diagram (Lorentz Boost)

Interactive (x, ct) spacetime diagram: a Lorentz boost at v = βc rotates the (x′, ct′) axes inward by atan(β) toward the 45° light cone — relativity of simultaneity, time dilation and length contraction become pure geometry. Click events with Shift / Alt to read the invariant interval Δs² = (cΔt)² − (Δx)² and its time-/light-/space-like classification.

About this model

An interactive (x, ct) Minkowski diagram shows a Lorentz boost at v = βc. The primed axes (x′, ct′) tilt inward by atan(β) toward the 45° light cone, so relativity of simultaneity, time dilation, and length contraction appear as pure geometry. Events can be placed with Shift/Alt modifiers; the invariant interval Δs² = (cΔt)² − (Δx)² classifies pairs as time-like, light-like, or space-like. Units are natural (c = 1 on the plot). Assumptions: 1+1 flat Minkowski spacetime, no gravity, single boost along x. You vary β to watch axes rotate and read intervals for clicked event pairs.

Who it's for: Advanced undergrad special relativity and mathematical physics courses introducing spacetime geometry.

Key terms

Minkowski diagram
spacetime diagram
Lorentz boost
invariant interval
light cone
relativity of simultaneity

How it works

Interactive Minkowski (x, ct) spacetime diagram. Slide β to boost between inertial frames S and S′ — both axes rotate inward by atan(β) toward the light cone, exposing relativity of simultaneity, length contraction and time dilation as pure geometry. Place events with Shift/Alt-click to read the invariant interval Δs² and its time- / light- / space-like classification.

Frequently asked questions

Why do the primed axes tilt toward the light cone?: A boost mixes x and ct; lines of constant t′ (simultaneity in S′) and constant x′ (worldlines of rest points in S′) both lean by angle atan(β) toward the null lines ct = ±x. Light rays stay at 45° in every inertial frame. Seeing both sets of axes on one diagram makes length contraction and time dilation geometric rather than mysterious.
What does the sign of Δs² tell me?: With Δs² = (cΔt)² − (Δx)², positive values are time-like (causal order possible), zero is light-like, and negative is space-like (no causal influence). Sign conventions vary in textbooks (some use −+++ metric); this lab uses the particle-physics mostly-plus time convention stated above. Do not confuse coordinate Δt with proper time along a worldline.
Can two events be simultaneous in one frame but not another?: Yes — that is relativity of simultaneity. Horizontal lines in S are t = const; in S′ simultaneity slices are tilted. Events that share a t′ value generally have different lab times. The diagram makes that tilt explicit when you change β.

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Minkowski Spacetime Diagram (Lorentz Boost)

About this model

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Frequently asked questions

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