Lorentzian -space is the inner product space consisting of the vector space together with the -dimensional Lorentzian inner product.
In the event that the metric signature is used, Lorentzian -space is denoted ; the notation is used analogously with the metric signature .
The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector has the form
Rewriting (where by definition), the norm in (0) can be written as
In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in -dimensional Lorentzian space into types based on the sign of their squared norm, e.g., as spacelike, timelike, and lightlike. The collection of all lightlike vectors in Lorentzian -space is known as the light cone, which is further separated into lightlike vectors which are positive and negative lightlike. A similar distinction is made for positive and negative timelike vectors as well.
Sometimes, the -dimensional Lorentzian norm is written to avoid confusion with the standard Euclidean norm; one may also write for the Lorentzian inner product of two vectors and .
Lorentzian space comes up in a number of contexts throughout pure and applied mathematics. In particular, four-dimensional Lorentzian space is known as Minkowski space and forms the basis of the study of spacetime within special relativity. What's more, the collection
consisting of all vectors in having imaginary Lorentzian length forms a two-sheeted hyperboloid of vectors satisfying the identity ; upon identifying antipodal vectors of (or, equivalently, upon discarding the negative sheet of vectors which satisfy ), one arrives at the so-called hyperboloid model for hyperbolic -space .