Length Contraction - Geometrical Representation

Geometrical Representation

See also: Minkowski diagram

The Lorentz transformation geometrically corresponds to a rotation in four-dimensional spacetime, and it can be illustrated by a Minkowski diagram: If a rod at rest in S' is given, then its endpoints are located upon the ct' axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x') positions of the endpoints are O and B, thus the proper length is given by OB. But in S the simultaneous (parallel to the axis of x) positions are O and A, thus the contracted length is given by OA. On the other hand, if another rod is at rest in S, then its endpoints are located upon the ct axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x) positions of the endpoints are O and D, thus the proper length is given by OD. But in S' the simultaneous (parallel to the axis of x') positions are O and C, thus the contracted length is given by OC.

Additional geometrical considerations show, that length contraction can be regarded as a trigonometric phenomenon, with analogy to parallel slices through a cuboid before and after a rotation in E3 (see left half figure at the right). This is the Euclidean analog of boosting a cuboid in E1,2. In the latter case, however, we can interpret the boosted cuboid as the world slab of a moving plate.

In special relativity, Poincaré transformations are a class of affine transformations which can be characterized as the transformations between alternative Cartesian coordinate charts on Minkowski spacetime corresponding to alternative states of inertial motion (and different choices of an origin). Lorentz transformations are Poincaré transformations which are linear transformations (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the Lorentz group forms the isotropy group of the self-isometries of the spacetime) which are played by rotations in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean trigonometry in Minkowski spacetime, as suggested by the following table: