Hyperbolic Rotations and Invariant Proper Time

In special relativity, the quantity

is the analogue of squared distance in Euclidean space. Lorentz transformations preserve this “Minkowski norm”, so curves of constant are invariant.

Hyperbola Preservation Under Lorentz Boosts

Move the slider to change the rapidity η. The hyperbolae t² − x² = 1 and t² − x² = 4 stay fixed, while the primed axes rotate around them.

xtt'x'

Lorentz boosts are hyperbolic rotations: they tilt the coordinate axes while leaving the Minkowski norm t² − x² (and hence these hyperbolae) invariant.

These hyperbolae are the sets where is constant. As you change the rapidity (and therefore the velocity ), the primed axes rotate, but the hyperbolae stay fixed: the geometry of spacetime is the real invariant object.