How are spatial coordinate systems in physics defined?

Grothendieck once asked "What is a meter?" (https://golem.ph.utexas.edu/category/2006/08/letter_from_grothendieck.html). This innocent sounding question, made me to think about how coordinate systems are defined in physics.

How are coordinate systems in physics defined, for example in special relativity where the coordinate system is assumed to be given.

I have tried a possibility, and was asking myself, if there are other mathematical ways to define a coordinate system. (Thanks for your help):

Here is a possible

For the definition of a physical affine basis $$B = (v_1 = p_1-p_0, v_2 = p_2-p_0, v_3=p_3-p_0)$$ of the spatial space $$\mathbb{R}^3$$, there has to be an a priori defined basis $$\hat{B}$$, so that $$B$$ can refer to this basis $$\hat{B}$$.

Observer $$A$$ at point $$p=p_0$$ can do the following:

• Choose nearby points $$p_1,p_2,p_3$$ in spatial space, without actually to give those points any a priori coordinates and define "affine vectors": $$v_1 = p_1 p_0, v_2 = p_2 p_0, v_3 = p_3 p_0$$
• It is possible for the observer $$A$$ to measure the distance between two (nearby) points: $$d_{ij} = d(p_i,p_j), 0\le i,j \le 3$$
• Oberser $$A$$ then can compute the gram matrix: $$G = (g_{ij}), g_{ij} = \frac{1}{2}(d_{0i}^2+d_{0j}^2-d_{ij}^2), 1\le i,j \le 3$$ we assume by the Schönberg criterion, that this matrix is a positive definite matrix, and hence especially a Gram matrix.
• Observer $$A$$ then can do Cholesky decomposition of the Gram matrix : $$G = C C^T, C = (x_1,x_2,x_3)$$
• Observer $$A$$ can do the Gram-Schmidt procedure to get an orthonormal basis: $$e_1,e_2,e_3$$ given $$x_1,x_2,x_3$$.
• Hence a posteriori observer $$A$$ can choose the points $$p_i$$ in such a way that $$G=\mathbf{1}$$, since $$\left< e_i,e_j \right> = \delta_{ij},$$ where $$\delta$$ is the Kronecker $$\delta$$.

Observer $$B$$ at point $$q$$ can do the same procedure, to get the orthonormal Basis $$\mathbf{1} = G_q = C_q C_q^T$$.

By the unitary freedom of square roots, there exists an orthogonal matrix $$O_{pq}$$ such that $$C_p O_{pq} = C_q$$ hence:

$$O_{pq}=C_q C_p^{-1} = C_q C_p^T$$

From this last equation we get:

• $$O_{pq}^T = O_{pq}^{-1} = C_p C_q^{-1}= O_{qp}$$
• $$O_{pp} = \mathbf{1}$$
• $$O_{pr} O_{rq} = O_{pq}$$

Setting $$w_{pq} := \log(O_{pq})$$ we get

• $$w_{pq} = -w_{qp}$$
• $$w_{pp} = \mathbf{0}$$
• $$w_{pr} + w_{rq} = w_{pq}$$

Hence one might define the affine vectors between the points $$p,q$$ as $$w_{pq}$$. The distance between $$p,q$$ might be given as:

$$d(p,q) = |w_{pq}|_F = |\log(O_{pq})|_F = |\log(C_q C_p^T)|_F$$

where $$|.|_F$$ denotes the Frobenius norm.

Also asked here in case it is not appropriate for this forum: https://mathoverflow.net/questions/409500/how-are-spatial-coordinate-systems-in-physics-defined

In physics, coordinates are defined as a diffeomorphism between an open subset of spacetime and an open subset of $$\mathbb{R}^N$$.