nLab
Minkowski metric

Context

Riemannian geometry

Gravity

Contents

Idea

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

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

Definition

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

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

η(v 1,v 2)=η μνv 1 μv 2μ(v 0) 2+j=1p(v j) 2. \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{3,4,6,10}p+1 \in \{3,4,6,10\} the quadratic form corresponding to the Minkowski metric may be identified simply with the determinant function on 2×22 \times 2 hermitian matrices with coefficients in one of the four real normed division algebra 𝕂{,,,𝕆}\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.

Revised on November 9, 2017 09:44:06 by Urs Schreiber (88.64.51.223)