nLab Minkowski metric

Idea

The Minkowski metric is a pseudo-Riemannian metric which is completely flat in that its Riemann curvature vanishes.

For $p \in \mathbb{N}$, a Cartesian space $\mathbb{R}^{p+1}$ equipped with the Minkowski metric $\eta$ is called Minkowski spacetime of dimension $p + 1$, denoted $\mathbb{R}^{p,1} \coloneqq (\mathbb{R}^{p+1}, \eta)$.

Definition

In terms of the standard coordinates $(x^\mu)_{\mu = 0}^p$ on $\mathbb{R}^{p+1}$ the Minkowski metric is the constant rank 2-tensor which is given by the diagonal matrix $\eta = diag(-1,+1,+1, \cdots , +1)$ (or else $\eta = diag(+1,-1,-1, \cdots, -1)$).

This means that for $v = (v^\mu)_{\mu = 0}^p \in \mathbb{R}^{p,1}$ then

$$\eta(v_1, v_2) \;=\; \eta_{\mu \nu} v_1^\mu v_2 \mu \coloneqq - (v^0)^2 + \underoverset{j = 1}{p}{\sum} (v^j)^2 \,.$$

Properties

In terms of hermitian matrices.

In the dimensions $p+1 \in \{3,4,6,10\}$ the quadratic form corresponding to the Minkowski metric may be identified simply with the determinant function on $2 \times 2$ hermitian matrices with coefficients in one of the four real normed division algebra $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$.

For details see at geometry of physics – A first idea of quantum field theory the chapter spacetime.

Related concepts

• shell

• super-Minkowski spacetime

References

Named after Hermann Minkowski. See the references at Minkowski spacetime.