The four-vector norm is a conventional name for the Lorentzian length of a four-vector, rather than a norm in the positive-definite sense. Its square is , the Lorentzian inner product of with itself. For , where , the squared Lorentzian length in the standard vector basis using the convention is

and using the convention it is

where is the usual vector dot product in Euclidean space and denotes the Lorentzian inner product in so-called Minkowski space, i.e., with either overall sign convention. The convention has one negative and three positive directions, while the convention has one positive and three negative directions. The Lorentzian inner product of two such vectors is sometimes denoted to avoid the possible confusion of the angled brackets with the standard Euclidean inner product (Ratcliffe 2006).

The squared Lorentzian length of a nonzero vector in Minkowski space may be positive, zero, or negative. With the convention, if , the four-vector is said to be timelike; if , is said to be spacelike; and if , is said to be lightlike. With the convention, the timelike and spacelike sign conditions reverse, while lightlike remains zero. The subset of Minkowski space consisting of all vectors whose squared Lorentzian length is zero is known as the light cone; moreover, one often distinguishes between positive lightlike and negative lightlike vectors, as well as distinguishing between positive timelike and negative timelike vectors.

The same construction gives the squared Lorentzian length in -dimensional Lorentzian space. In this more general environment, the Lorentzian inner product of two vectors and is of the form

for the corresponding time-first sign conventions. The definitions of timelike, spacelike, and lightlike vectors are made analogously.

See also

Dot Product, Four-Vector, Light Cone, Lightlike, Lorentzian Inner Product, Lorentzian Space, Metric Signature, Minkowski Space, Negative Lightlike, Negative Timelike, Norm, Positive Lightlike, Positive Timelike, Spacelike, Timelike

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.Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer-Verlag, 2006.

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Four-Vector Norm

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Four-Vector Norm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Four-VectorNorm.html

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