Four-Vector

In the Minkowski space of special relativity, a four-vector 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 1973).

Throughout the literature, four-vectors are often expressed in the form

(2)

where is the time coordinate and is the (Euclidean) three-vector of space coordinates. Using this convention, the imaginary unit is dropped and is assumed for the speed of light in the expression of the time coordinate ; moreover, writing implicitly makes use of the metric signature and hence the

(3)

decomposition of Minkowski space is implicitly assumed in this convention. Given the alternative decomposition, a four-vector would have the analogous form . Though subtle, this distinction is important when computing the norm of a four-vector .

Multiplication of two four-vectors with the metric tensor yields products of the form

(4)

a result due to the fact that the metric tensor has the matrix form

(5)

in any Lorentz frame (Misner et al. 1973). One of the immediate consequences of this product rule is that the squared norm of a nonzero four-vector may be either positive, zero, or negative, corresponding 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).