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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

 |x|=-x_0^2+x_1^2+...+x_(n-1)^2.
(1)

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

 |x|=-x_0^2+|x^_|^2.
(2)

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}
(3)

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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References

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. https://mathworld.wolfram.com/LorentzianSpace.html

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