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.

Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation.
San Francisco, CA: W. H. Freeman, p. 53, 1973.Ratcliffe,
J. G. Foundations
of Hyperbolic Manifolds. New York: Springer-Verlag, 2006.