Four-Vector Norm

The squared norm of a four-vector a=(a^0,a^1,a^2,a^3)=a^0+a is given in the standard basis using the +-- signature as


and using the -+++ signature as


where a·a is the usual vector dot product in Euclidean space and <·,·> denotes the Lorentzian inner product in so-called Minkowski space, i.e., R^4=R^(1,3) with metric signature (1,3) assumed throughout. Note the Lorentzian inner product of two such vectors is sometimes denoted x degreesy to avoid the possible confusion of the angled brackets with the standard Euclidean inner product (Ratcliffe 2006).

The squared norm of a nonzero vector in Minkowski space may be either positive, zero, or negative. If a^2<0, the four-vector a^mu is said to be timelike; if a^2>0, a^mu is said to be spacelike; and if a^2=0, a^mu is said to be lightlike. The subset of Minkowski space consisting of all vectors whose squared norm is zero is known as the light cone; moreover, one often distinguishes between positive and negative lightlike vectors, as well as distinguishing between positive and negative timelike vectors.

As suggested above, the four-vector norm is nothing more than a special case of the more general Lorentzian inner product <·,·> on n-dimensional Lorentzian space with metric signature (1,n-1). In this more general environment, the inner product of two vectors x=(x_0,x_1,...,x_(n-1)) and y=(y_0,y_1,...,y_(n-1)) has the form


for signatures +-- and -+++ respectively, and where 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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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:

Stover, Christopher and Weisstein, Eric W. "Four-Vector Norm." From MathWorld--A Wolfram Web Resource.

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