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
![diag(-1,1,1,1)=[-1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FFour-Vector%2FNumberedEquation5.svg) |
(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).
See also
Four-Vector Norm, Gradient Four-Vector, Lightlike, Lorentz Transformation, Metric Tensor, Minkowski Space, Position Four-Vector, Quaternion, Spacelike, Tensor, Timelike, Vector
Portions of this entry contributed by Christopher Stover
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References
Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, p. 53, 1973.Morse, P. M. and Feshbach, H. "The Lorentz Transformation, Four-Vectors, Spinors." ยง1.7 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 93-107, 1953.Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer-Verlag, 2006.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, pp. 31 and 35, 1972.
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Cite this as:
Stover, Christopher and Weisstein, Eric W. "Four-Vector." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Four-Vector.html