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# Minkowski Metric

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

 (1)

where the convention is used, and the indices run over 0, 1, 2, and 3, with the time coordinate and the space coordinates.

 (2)

gives the line element

 (3) (4)

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

 (5) (6)

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

 (7)

where is a Lorentz tensor. It also satisfies

 (8)
 (9)
 (10)

The metric of Minkowski space is diagonal with

 (11)

and so satisfies

 (12)

The necessary and sufficient conditions for a metric to be equivalent to the Minkowski metric are that the Riemann tensor vanishes everywhere () and that at some point has three positive and one negative eigenvalues.

Euclidean Metric, Line Element, Lorentz Tensor, Lorentz Transformation, Minkowski Space

## References

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 38, 1972.

Minkowski Metric

## Cite this as:

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