The standard Lorentzian inner product on 
is given by
 |
(1)
|
i.e., for vectors 
and 
,
 |
(2)
|
endowed with the metric tensor induced by the above
Lorentzian inner product is known as Minkowski space
and is denoted 
.
The Lorentzian inner product on 
is nothing more than a specific case of the more general
Lorentzian inner product 
on 
-dimensional Lorentzian
space with metric signature 
when negative squares are
listed first. In this more general environment, the inner product of two vectors 
and 
has the form
 |
(3)
|
The Lorentzian inner product of two such vectors is sometimes denoted 
to avoid the possible confusion of the angled brackets with the standard Euclidean
inner product (Ratcliffe 2006). Analogous presentations
can be made if its metric signature is reported
with positive squares first as 
(i.e., 
for Minkowski space).
The four-dimensional Lorentzian inner product is used as a tool in special relativity, namely as a measurement which is independent of
reference frame and which replaces the typical Euclidean notion of distance.
For a four-vector 
in Minkowski
space, the variables 
, 
, and 
can be thought of as space variables
with 
as the time variable. In various literature, the time
variable is sometimes labeled 
; moreover, when used in general relativity, either of 
or 
may be used where 
denotes the speed of light and where 
denotes the imaginary
unit (Misner et al. 1973). For simplicity, the formula (2)
uses the conventions of real time coordinates and
appropriately-chosen units so that the speed of light has the value 
.
For a vector 
, the sign of 
determines the type of 
. In particular, if 
, then 
is spacelike; if 
, then 
is lightlike; and if 
, then 
is called timelike. After a coordinate transformation, it is possible
to rewrite the Lorentzian inner product as above where 
is in the direction of a given timelike vector 
with 
. Such a coordinate
transformation corresponds to a change in reference frame. Collectively, the
set of all reference frame changes forms the Lorentz
group, also called the orthogonal group 
(or 
when using the 
metric signature).
