In the Minkowski space of special relativity, a four-vector is a four-element vector x^mu=(x^0,x^1,x^2,x^3) that transforms under a Lorentz transformation like the position four-vector. In particular, four-vectors are the vectors in special relativity which transform as


where Lambda^mu_nu 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


where x^0 is the time coordinate and x=(x^1,x^2,x^3) is the (Euclidean) three-vector of space coordinates. Using this convention, the imaginary unit i is dropped and c=1 is assumed for the speed of light in the expression of the time coordinate x_0; moreover, writing x^mu=x_0+x implicitly makes use of the (1,3) metric signature and hence the


decomposition of Minkowski space is implicitly assumed in this convention. Given the alternative R^(3,1) decomposition, a four-vector would have the analogous form x^mu=(x^1,x^2,x^3,x^0)=x+x^0. Though subtle, this distinction is important when computing the norm of a four-vector x^mu.

Multiplication of two four-vectors with the metric tensor g_(munu) yields products of the form


a result due to the fact that the metric tensor g_(munu) 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]

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, x^0=t and any product of the form g_(munu)x^mux^nu 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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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." §1.7 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 Web Resource.

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