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 
: 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 the equivalent metric signature 
(i.e., 
for Minkowski space) is used.
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 change
of variables, it is possible to rewrite the Lorentzian inner product as above where
is in the direction of a given timelike vector 
with 
. Such a change of variables corresponds to a
change in reference frame. Collectively, the set of all reference frame changes form
the Lorentz group, also called the orthogonal
group 
(or 
when using the 
metric signature).
