equipped with a nondegenerate, symmetric bilinear form with signature (+,-,-,-). Without loss of generality, the signature may alternatively be chosen as (-,+,+,+) (see sign conventions). The metric is indefinite along the light cone of zero distance. Sometimes, this vector space is denoted  to emphasize the signature of the inner product. In physics, Minkowski space is sometimes denoted as . 

Table of contents

1 History 2 Structure 3 Locally flat spacetime 4 Related Topics 5 References

History

Around 1907 Hermann Minkowski realized that the special theory of relativity, introduced by Albert Einstein in 1905, could be mathematically described using a four-dimensional spacetime, which combines the dimension of time with the three space dimensions. The Lorentz transformations of special relativity can be represented as generalized rotations in Minkowski space.

Structure

Relative to the standard basis the inner product is given by

Vectors in Minkowski space are classified according to the sign of their norm squared. Vectors are said to be timelike or spacelike if their norms squared are positive or negative respectively. Vectors with zero norm are called null or lightlike. This terminology comes from the use of Minkowski space in the theory of relativity. The set of all lightlike vectors constitutes what is called the light cone. Note that all of these notions are independent of a choice of basis.

The standard basis for Minkowski space consists of one timelike and three spacelike unit vectors, although it is possible to construct bases with one or more null vectors. A basis consisting entirely of lightlike vectors is called a null basis.

Locally flat spacetime

For physical systems, the use of the Minkowski metric over finite distances applies only in the Newtonian limit of systems without significant gravitation. In the case of significant gravitation, spacetime is in general curved. Nevertheless, even in such cases, Minkowski space applies in a infinitesimal region surrounding any point in curved spacetime, except in the infinitesimal vicinity of gravitational singularities.

Related Topics

• Euclidean space
• Metric tensor
• World line
• Lorentz group, Poincaré group
• General relativity, Pseudo-Riemannian manifold
• Erlanger program