Lorentzian 
-space
is 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 determines a quadratic form, or Lorentzian squared length,
on Lorentzian space. For a vector 
in the convention with one negative
square, it is
 |
(1)
|
Writing 
,
where 
,
gives
 |
(2)
|
where 
is the Euclidean norm. This quadratic
form is not a norm since it can be negative
and vanishes on nonzero vectors.
Its sign classifies nonzero vectors
as spacelike, timelike,
or lightlike. The collection of lightlike vectors forms the light cone,
whose two halves consist of positive lightlike
and negative lightlike vectors.
A similar distinction is made between positive timelike
and negative timelike vectors.
The formal square root of 
is sometimes called the Lorentzian length. One may also
write 
for the Lorentzian inner product 
of two vectors.
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)
|
consisting of all vectors in 
having Lorentzian squared 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 
.
