Minkowski spacetime

In physics, Minkowski spacetime (or Minkowski space; ) is the main mathematical description of spacetime in the absence of gravitation. It combines inertialspace and timemanifolds into a four-dimensional model.

The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others, and said it "was grown on experimental physical grounds".

Minkowski spacetime is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. Minkowski spacetime differs from four-dimensional Euclidean space insofar as it treats time differently from the three spatial dimensions.

In 3-dimensional Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by rotations, reflections and translations. When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations is called the Poincaré group. Minkowski's model follows special relativity, where motion causes time dilation changing the scale applied to the frame in motion and shifts the phase of light.

Minkowski spacetime is a pseudo-Euclidean space equipped with an isotropic quadratic form called the spacetime interval or the Minkowski norm squared. In Euclidean space, the distance between any two distinct points is greater than zero, but in Minkowski spacetime, the interval between two distinct events can be zero. This occurs when one event is on the light cone of the other. Using the polarization identity the quadratic form is converted to a symmetric bilinear form called the Minkowski inner product, though it is not a geometric inner product.

The group of transformations for Minkowski spacetime that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is the Lorentz group (as opposed to the Galilean group).