A spacetime diagram with the rest frame in the background grid. Boost to a moving frame and watch axes shear toward the light cone. Time and space mix; events change order. Light rays (45°) are invariant — every observer sees them at the same angle. Lorentz invariance is the geometric content of special relativity.
why time and space mix
A spacetime point ("event") has coordinates (x, t) in one frame and (x', t') in a frame moving at velocity v. Einstein's two postulates (1905):
1. The laws of physics are the same in all inertial frames
2. The speed of light c is the same in all inertial frames
imply that the coordinates must transform as:
x' = γ(x − vt), t' = γ(t − vx/c²)
where γ = 1/√(1−v²/c²) is the Lorentz factor.
Time and space mix: a "purely temporal" displacement in one frame has spatial components in another. Conversely, two events that are simultaneous in one frame can be ordered differently in another.
Spacetime interval: s² = c²t² − x² is invariant. Different observers disagree on x and t separately, but they all agree on s². This is the deepest content of special relativity — events have a geometric relationship that transcends individual frames.
Three consequences:
1. Time dilation: Δt' = γ Δt. Moving clocks tick slow.
2. Length contraction: Δx' = Δx/γ. Moving rulers shrink.
3. Relativity of simultaneity: two spacelike-separated events have observer-dependent ordering.
On the diagram: the orange line at 45° is a light ray. The blue and gold axes are the t and x axes of the moving frame. Notice they shear toward the light cone but never cross it. As β → 1, the boosted frame's axes collapse onto the light line, and the rest frame view becomes singular.