Minkowski Metric

The Minkowski metric, also called the Minkowski tensor, is the flat pseudo-Riemannian metric on four-dimensional Minkowski space whose components in standard coordinates are given 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.

The Euclidean metric

(2)

gives the line element

			(3)
			(4)

For a timelike curve, the Minkowski metric gives the proper time

			(5)
			(6)

with the chosen sign convention (Carroll 1997, pp. 68-71).

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 tensor of Minkowski space is diagonal with

(11)

and so satisfies

(12)

On a connected four-dimensional smooth manifold without boundary, a smooth metric tensor is locally equivalent to the Minkowski metric iff the Riemann tensor vanishes everywhere () and, at some point, has three positive and one negative eigenvalues (Carroll 1997, pp. 77-78).