# Why is the metric used in physics a squared norm? [duplicate]

The length metrics used in physics are usually a squared norm, like the following: $$ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu}. \tag{1}$$

What other kinds of continuous metrics could we define? Why not the following multi-linear candidates? \begin{align} ds &= g_{\mu} \, dx^{\mu}, \tag{2} \\[1ex] ds^3 &= g_{\mu \nu \lambda} \, dx^{\mu} \, dx^{\nu} \, dx^{\lambda}, \tag{3} \\[1ex] \vdots \\[1ex] ds^n &= g_{\mu_1 \mu_2 \mu_3 \dots \mu_n} \, dx^{\mu_1} \, dx^{\mu_2} \, dx^{\mu_3} \dots dx^{\mu_n}. \tag{4} \end{align} What are the basic rules that "forces" physics to use the "squared" metric (1)?

• Does this answer your question? Why does the metric have to be bilinear? Feb 27, 2020 at 18:02
• @JohnRennie, you're right that my question is a duplicate. However, I don't find the answers very illuminating or satisfying.
– Cham
Feb 27, 2020 at 19:00
• Feb 27, 2020 at 19:31
• @Cham the next step is to place a bounty on the previous question stating what you are looking for in an answer. This will encourage new answers to the previous question. Feb 27, 2020 at 20:23