A four-vector a_mu is said to be lightlike if its four-vector norm satisfies a_mua^mu=0.

One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product <·,·> on Lorentzian n-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


whereby one defines a vector a to be lightlike precisely when <a,a>=0.

Lightlike vectors are sometimes called null vectors. The collection of all lightlike vectors in a Lorentzian space (e.g., in the Minkowski space of special relativity) is known as the light cone. One often draws distinction between lightlike vectors which are positive and those which are negative.

See also

Light Cone, Lorentzian Inner Product, Lorentzian Space, Metric Signature, Negative Lightlike, Negative Timelike, 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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Cite this as:

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

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