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
(1)
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Rewriting (where by definition), the norm in (0) can be written as
(2)
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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
(3)
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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 .