Minkowski Metric

Prerequisites: Special Relativity. Full path (6 pages)

Intuitive
Concrete
Abstract

Intuitive

The Minkowski metric encodes the experimental observation that the speed of light has the same value in all inertial frames of reference.

A metric is a tool that allows us to measure the distance between two points. This means, a metric eats two points and spits out a number. Depending on the geometry of the space the points live in the distance can be different. In this sense, a metric tells us almost everything we need to know about the space where our points live.

The stage of special relativity is Minkowski space and the Minkowski metric is the correct tool that tells us how far two points in Minkowski space are away from each other.

Concrete

$\begin{equation} \begin{gathered} g_{\mu\nu}=\left( \begin{matrix} 1 & 0 & 0 & 0\cr 0 & -1 & 0 & 0\cr 0 & 0 & -1 & 0\cr 0 & 0 & 0 & -1 \end{matrix} \right) \end{gathered} \end{equation}$ Thus $x_0=x^0$ , $x_j=-x^j$ for $1\le j\le 3$ . For real vectors and tensors lowering and raising indices is made by $\begin{equation} A_{\mu}^a=g_{\mu\nu}A^{\nu a}\\ \quad F_{\mu\nu}^a=g_{\mu\alpha}g_{\nu\beta}F^{\alpha\beta a}\\ \quad \partial_{\mu}=g_{\mu\nu}\partial^{\nu} \end{equation}$

No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "The Minkowski Metric and Four-Vectors"

Abstract

Minkowski space is a four-dimensional real affine space $\mathbb{M}$ whose space of translations $V$ carries a nondegenerate symmetric bilinear form $\eta:V\times V\to\mathbb{R}$ of signature $(1,3)$ ; with the sign convention of the Concrete layer, whose matrix $g_{\mu\nu}$ we write as $\eta_{\mu\nu}$ here, $\eta(v,v)=(v^0)^2-\vec v^{\,2}$ in an orthonormal basis. The form is not positive definite, so it does not define a distance in the sense of metric spaces. It splits nonzero vectors into timelike ( $\eta(v,v)>0$ ), spacelike ( $\eta(v,v)<0$ ) and lightlike ( $\eta(v,v)=0$ ) vectors. The lightlike vectors form the light cone, and a choice of one of its two halves (a time orientation) separates the future from the past. For a timelike curve, $\int\sqrt{\eta(\dot x,\dot x)}\,d\lambda$ is the proper time measured by a clock that moves along it (in units with $c=1$ ).

The linear maps of $V$ that preserve $\eta$ form the Lorentz group $O(1,3)$ , defined by $\Lambda^T\eta\Lambda=\eta$ in matrix form. It has four connected components; the component of the identity, $SO^+(1,3)$ , preserves both orientation and time orientation. Adding the translations gives the isometry group of $\mathbb{M}$ , the Poincaré group $\mathbb{R}^{1,3}\rtimes O(1,3)$ . The causal structure alone determines almost all of the metric: by a theorem of Zeeman, the bijections of $\mathbb{M}$ that preserve the causal order are compositions of orthochronous Lorentz transformations, translations and dilations. So the light cone fixes the metric up to an overall constant factor.

As a pseudo-Riemannian manifold, Minkowski space is $\mathbb{R}^4$ with the constant metric tensor $\eta=\eta_{\mu\nu}\,dx^\mu\otimes dx^\nu$ in inertial coordinates. Its Riemann tensor vanishes, and it is the unique simply connected, geodesically complete, flat Lorentzian manifold of dimension four up to isometry. Nondegeneracy means that $\eta$ defines an isomorphism $V\to V^*$ , $v\mapsto\eta(v,\cdot)$ , whose components are the index lowering $v_\mu=\eta_{\mu\nu}v^\nu$ of the Concrete layer; its inverse, with components $\eta^{\mu\nu}$ , raises indices. In general relativity $\eta$ is replaced by a general Lorentzian metric $g$ , and at every point there are coordinates in which $g_{\mu\nu}=\eta_{\mu\nu}$ and the first derivatives of $g_{\mu\nu}$ vanish, so that Minkowski space is the local model of every spacetime.

The Minkowski metric is the correct mathematical tool that tells us how far two events are away from each other in special relativity.

Minkowski Metric

Intuitive

Concrete

Abstract

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