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Lorentzian Manifold

A semi-Riemannian manifold M=(M,g) is said to be Lorentzian if dim(M)>=2 and if the index I=I_g associated with the metric tensor g satisfies I=1.

Alternatively, a smooth manifold M^n of dimension n>=2 is Lorentzian if it comes equipped with a tensor g of metric signature (1,n-1) (or, equivalently, (n-1,1)).

See also

Metric Signature, Metric Tensor, Metric Tensor Index, Positive Definite Tensor, Pseudo-Euclidean Space, Pseudo-Riemannian Manifold, Semi-Riemannian Manifold, Semi-Riemannian Metric, Smooth Manifold, Strong Pseudo-Riemannian Metric, Strong Riemannian Metric, Weak Pseudo-Riemannian Metric, Weak Riemannian Metric

This entry contributed by Christopher Stover

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References

Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer-Verlag, 2006.Sachs, R. K. and Wu, H. General Relativity for Mathematicians. New York: Springer-Verlag, 1977.Snygg, J. A New Approach to Differential Geometry using Clifford's Geometric Algebra. New York: Springer Science+Business Media, 2012.

Cite this as:

Stover, Christopher. "Lorentzian Manifold." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LorentzianManifold.html

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