The four-vector norm is a conventional name for the Lorentzian length of a four-vector, rather than a norm in the positive-definite sense. Its square is 
, the Lorentzian
inner product of 
with itself. For 
, where
, the squared
Lorentzian length in the standard vector basis using
the 
convention is
 |
(1)
|
and using the 
convention it is
 |
(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 either overall sign convention. The 
convention has one negative
and three positive directions, while the 
convention has one positive
and three negative directions. 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 Lorentzian length of a nonzero vector in Minkowski space
may be positive, zero, or negative. With the 
convention, if 
, the four-vector 
is said to be timelike;
if 
, 
is said to be spacelike; and
if 
, 
is said to be lightlike. With
the 
convention, the timelike and spacelike sign conditions reverse, while lightlike
remains zero. The subset of Minkowski space consisting of all vectors
whose squared Lorentzian length is zero
is known as the light cone; moreover, one often distinguishes
between positive lightlike and negative
lightlike vectors, as well as distinguishing between
positive timelike and negative
timelike vectors.
The same construction gives the squared Lorentzian length 
in 
-dimensional Lorentzian
space. In this more general environment, the Lorentzian
inner product of two vectors 
and 
is of the
form
 |
(3)
|
for the corresponding time-first sign conventions. The definitions of timelike, spacelike, and lightlike vectors are made analogously.
