Lorentzian Space

Lorentzian n-space is the inner product space consisting of the vector space R^n together with the n-dimensional Lorentzian inner product.

In the event that the (1,n-1) metric signature is used, Lorentzian n-space is denoted R^(1,n-1); the notation R^(n-1,1) is used analogously with the metric signature (n-1,1).

The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector x=(x_0,x_1,...,x_(n-1)) has the form


Rewriting x=x_0+x^_ (where x^_=(x_1,x_2,...,x_(n-1)) by definition), the norm in (0) can be written as


In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in n-dimensional Lorentzian space into types based on the sign of their squared norm, e.g., as spacelike, timelike, and lightlike. The collection of all lightlike vectors in Lorentzian n-space is known as the light cone, which is further separated into lightlike vectors which are positive and negative lightlike. A similar distinction is made for positive and negative timelike vectors as well.

Sometimes, the n-dimensional Lorentzian norm is written |·|=|·|_L to avoid confusion with the standard Euclidean norm; one may also write u degreesv for the Lorentzian inner product of two vectors u and v.

Lorentzian space comes up in a number of contexts throughout pure and applied mathematics. In particular, four-dimensional Lorentzian space R^4=R^(1,3) is known as Minkowski space and forms the basis of the study of spacetime within special relativity. What's more, the collection

 F^n={x=(x_0,x_1,...,x_n) in R^(n+1):|x|_L=-1}

consisting of all vectors in R^(n+1) having imaginary Lorentzian length forms a two-sheeted hyperboloid of vectors x=x_0+x^_ satisfying the identity x_1^2-|x^_|^2=1; upon identifying antipodal vectors of F^n (or, equivalently, upon discarding the negative sheet of vectors which satisfy x_0<0), one arrives at the so-called hyperboloid model for hyperbolic n-space H^n.

See also

Inner Product Space, Light Cone, Lightlike, Lorentzian Inner Product, Metric Signature, Negative Lightlike, Negative Timelike, p-Signature, Positive Definite Quadratic Form, Positive Definite Tensor, Positive Lightlike, Positive Timelike, Quadratic, Quadratic Form Rank, Spacelike, Sylvester's Inertia Law, Sylvester's Signature, Timelike

This entry contributed by Christopher Stover

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Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1105, 2000.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, 1973.Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer, 2006.

Cite this as:

Stover, Christopher. "Lorentzian Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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