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