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)
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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)
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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)
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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)
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a result due to the fact that the metric tensor has the matrix form
(5)
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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).