A four-vector in the Minkowski space of special relativity is a four-element vector 
that transforms under a Lorentz
transformation like the position four-vector.
In particular, four-vectors are the vectors in special
relativity which transform as
 |
(1)
|
where 
is the Lorentz tensor.
In the context of general relativity, four-vectors satisfy a more general transformation rule (Morse and Feshbach 1953).
Throughout the literature, four-vectors are often of the form
 |
(2)
|
where 
is the time coordinate and 
is the (Euclidean)
vector of three space coordinates.
The time coordinate is written in units with 
. The ordering of coordinates
is independent of the metric signature: time
may be first or last with either overall sign convention.
With metric signature 
and time first, the metric
tensor is 
.
Moving time to the last coordinate gives 
and preserves the same metric
signature. Reversing every sign, rather than permuting
coordinates, gives the alternative 
convention.
Contracting a four-vector with itself using the metric tensor 
gives
 |
(3)
|
a result due to the fact that the metric tensor 
has the matrix
form
![diag(-1,1,1,1)=[-1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]](/images/equations/Four-Vector/NumberedEquation4.svg) |
(4)
|
in any Lorentz frame (Misner et al. 1973). One of the immediate consequences of this Lorentzian inner product is that the squared Lorentzian length of a nonzero four-vector may be either positive, zero, or negative, corresponding to vectors which are spacelike, lightlike, and timelike, respectively.
In the case of the position four-vector, 
and any product of the form 
is an invariant known as the spacetime interval
(Misner et al. 1973).
