The squared norm of a four-vector 
is given by the dot
product
 |
(1)
|
where 
is the usual vector dot product in Euclidean space.
Here, the notation 
is merely a shorthand for the more descriptive expression
 |
(2)
|
where 
denotes the Lorentzian inner product in
so-called Minkowski space, i.e., 
with metric signature 
assumed throughout.
One result of the above formula is that the squared norm of a nonzero vector in Minkowski space may be either positive, zero, or negative. If 
, the four-vector 
is said to be timelike; if
, 
is said to be spacelike;
and if 
,
is said to be lightlike.
The subset of Minkowski space consisting of all vectors whose squared norm is zero
is known as the light cone; moreover, one often distinguishes
between positive and negative
lightlike vectors, as well as distinguishing between positive
and negative timelike vectors.
As suggested above, the four-vector norm is nothing more than a special 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)
|
whereby the definitions of timelike, spacelike, and lightlike vectors are made analogously. 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).
