Initial insights into the concept of the metric

A few days ago, I read Professor Gao Xian’s new book “Classical Mechanics”, and had my first basic understanding of the concept of “metric”.

The key point is that Professor Gao combines some of our existing concepts with more advanced mathematical tools, introducing new concepts in a more concise and understandable way.

Generalization of the Pythagorean Theorem

We know that in flat Euclidean space, the distance between two points can be expressed using the Pythagorean theorem: \(\begin{equation} \label{eq1} \mathrm{d}s^2=\mathrm{d}x^2+\mathrm{d}y^2, \end{equation}\) or in polar coordinates: \(\begin{equation} \label{eq2} \mathrm{d}s^2=\mathrm{d}\rho^2+\rho^2\mathrm{d}\varphi^2. \end{equation}\)

In other spaces, as long as we take infinitesimal distances, making the space approximately flat and linear, the Pythagorean theorem still holds. For example, on the surface of a sphere: \(\begin{equation} \label{eq3} \mathrm{d}s^2=R^2\mathrm{d}\theta^2+R^2\sin^2\theta\,\mathrm{d}\varphi^2. \end{equation}\)

Express the Pythagorean theorem in quadratic form

Although the Pythagorean theorem holds in different spaces and coordinate systems, their forms seem to be not very unified (such as $\eqref{eq1}$, $\eqref{eq2}$, and $\eqref{eq3}$). However, each term is a square term. This prompts us to think of the quadratic forms we learned in linear algebra. So we try to write the above three Pythagorean theorems in the form of quadratic matrices: \(\begin{align} \mathrm{d}s^2&=\begin{pmatrix}\mathrm{d}x&\mathrm{d}y\end{pmatrix}\begin{pmatrix}1&0\\0&1\end{pmatrix}\begin{pmatrix}\mathrm{d}x\\\mathrm{d}y\end{pmatrix},\\[12pt] \mathrm{d}s^2&=\begin{pmatrix}\mathrm{d}\rho&\mathrm{d}\varphi\end{pmatrix}\begin{pmatrix}1&0\\0&\rho^2\end{pmatrix}\begin{pmatrix}\mathrm{d}\rho\\\mathrm{d}\varphi\end{pmatrix},\\[12pt] \mathrm{d}s^2&=\begin{pmatrix}\mathrm{d}\theta&\mathrm{d}\varphi\end{pmatrix}\begin{pmatrix}R^2&0\\0&R^2\sin^2\theta\end{pmatrix}\begin{pmatrix}\mathrm{d}\theta\\\mathrm{d}\varphi\end{pmatrix}. \end{align}\) A glimpse of Dingzhen reveals: \(\begin{equation} \mathrm{d}s^2\equiv\mathrm{d}\boldsymbol{\rho}^\mathrm{T}\,G\,\mathrm{d}\boldsymbol{\rho}. \end{equation}\) With a unified form, we can write a general form: \(\begin{equation} \mathrm{d}s^2=\begin{pmatrix}\mathrm{d}\rho^1&\cdots&\mathrm{d}\rho^s\end{pmatrix}\begin{pmatrix}g_{11}&\cdots&g_{1s}\\\vdots&\ddots&\vdots\\g_{s1}&\cdots&g_{ss}\end{pmatrix}\begin{pmatrix}\mathrm{d}\rho^1\\\vdots\\\mathrm{d}q^s\end{pmatrix}. \end{equation}\) So we define the matrix $g_{ab}$ as the metric. Then the above three examples and other metrics for spaces/coordinate systems become clear :P.

So in summary, the metric is the generalization of the Pythagorean theorem to infinitesimal distances.