Minkowski Metric

The Minkowski metric, also called the Minkowski tensor or pseudo-Riemannian metric, is a tensor eta_(alphabeta) whose elements are defined by the matrix

where the convention c=1 is used, and the indices alpha,beta run over 0, 1, 2, and 3, with x^0=t the time coordinate and (x^1,x^2,x^3) the space coordinates.

The Euclidean metric

gives the line element

while the Minkowski metric gives its relativistic generalization, the proper time

The Minkowski metric is fundamental in relativity theory, and arises in the definition of the Lorentz transformation as

where Lambda^alpha_beta is a Lorentz tensor. It also satisfies

The metric of Minkowski space is diagonal with

and so satisfies

The necessary and sufficient conditions for a metric g_(munu) to be equivalent to the Minkowski metric eta_(alphabeta) are that the Riemann tensor vanishes everywhere (R^lambda_(munukappa)=0) and that at some point g^(munu) has three positive and one negative eigenvalues.

See also

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References

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Cite this as:

Weisstein, Eric W. "Minkowski Metric." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MinkowskiMetric.html

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