The squared norm of a four-vector 
is given in the standard basis using
the 
signature as
 |
(1)
|
and using the 
signature as
 |
(2)
|
where 
is the usual vector dot product in Euclidean space
and 
denotes the Lorentzian inner product in
so-called Minkowski space, i.e., 
with metric signature 
assumed throughout. Note 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).
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)
|
for signatures 
and 
respectively, and where the definitions of timelike,
spacelike, and lightlike
vectors are made analogously.
