The big idea
Two ways to "rotate"
In ordinary Euclidean geometry, rotating a point keeps its distance from the origin fixed. Points trace out a circle: x² + y² stays constant.
In hyperbolic geometry, there's an analogous motion — a hyperbolic rotation (or boost) — but it keeps a different quantity fixed. Points slide along a hyperbola: x² − y² stays constant. That single sign change is the seed of all of special relativity.
Circular rotation
y' = x·sin θ + y·cos θ
invariant: x² + y² = r²
Mixes x and y with trig functions. Preserves Euclidean distance.
Hyperbolic rotation
t' = x·sinh φ + t·cosh φ
invariant: t² − x² = const
Mixes space and time with hyperbolic functions. Preserves the spacetime interval.
The rapidity φ plays the role of the rotation angle θ. As φ grows, the boost gets stronger — and corresponds to a faster relative velocity.
Interactive figure 1
Watch a hyperbolic rotation
Drag the rapidity φ slider (or press Play). A point glides along the orange hyperbola — never crossing the red light cone — while the grid shears. This shearing is the Lorentz transformation.
The shaded orange wedge is the boost "angle" θ. Unlike a circular angle, its area equals the rapidity φ — the genuine hyperbolic-angle measure of the boost. The arcs on the axes show its visible Euclidean tilt, arctan β, which grows toward 45° (the light cone) but never reaches it.
Notice the invariant box stays locked at a constant value no matter how far you push φ. The geometry bends, but the interval is rigid.
The physics
Why this is special relativity
Einstein's two postulates — the laws of physics are the same for all inertial observers, and the speed of light c is the same for everyone — force spacetime to have a hyperbolic geometry rather than a Euclidean one.
The Lorentz boost is a hyperbolic rotation
Writing velocity as β = v/c and defining rapidity by φ = artanh β, the familiar Lorentz transformation becomes literally the hyperbolic rotation above:
ct' = γ(ct − βx) = ct·cosh φ − x·sinh φ
x' = γ(x − βct) = x·cosh φ − ct·sinh φ
Velocities add by adding rapidities
Boosts compose like rotations: stack two boosts and the rapidities simply add, φ = φ₁ + φ₂. Converting back to velocity reproduces Einstein's velocity-addition formula — and explains why you can never reach c.
Time dilation & length contraction
Because the t′ and x′ axes tilt toward the light cone (you saw it in figure 1), moving clocks run slow and moving rulers shrink — both by the factor γ = cosh φ.
Interactive figure 2
The invariant spacetime interval
Different observers disagree about when and where an event happens — but they all agree on the spacetime interval separating two events:
A hyperbolic rotation changes the individual coordinates ct and x, yet leaves s² untouched. Drag below to boost the same event between reference frames and watch the interval hold steady.
Time-like s² > 0
Events that can be cause and effect. A real observer can travel between them; they share a "before/after" everyone agrees on.
Space-like s² < 0
Too far apart in space to influence each other. Their time order depends on the observer.
Light-like s² = 0 — the boundary, traced by light itself. The light cone is the one structure every hyperbolic rotation leaves fixed.
In one breath
Putting it together
- Euclidean rotation preserves x² + y² and traces circles. Hyperbolic rotation preserves t² − x² and traces hyperbolas.
- A Lorentz boost — changing to a moving observer's frame — is exactly a hyperbolic rotation by the rapidity φ = artanh(v/c).
- Because the boost is a hyperbolic rotation, the spacetime interval s² = (ct)² − x² is invariant: all observers measure the same value.
- γ = cosh φ and β = tanh φ tie the geometry directly to time dilation, length contraction, and the cosmic speed limit c.