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.