Notation: In the following, $E^n$ denotes an euclidean space of dimension $n$ (an affine space with inner product $\langle\,\cdot\,,\,\cdot\,\rangle$ on the translation space).

The answer to this question gives a very precise description/definition of systems of reference in classical physics, where the existence of absolute time - a map $T\colon M\to E^1$ defined on spacetime $M$ - is postulated: A system of reference is a map $\pi\colon M\to E^3$ (the codomain $E^3$ is called the rest space of the system of reference) such that $M\to E^1\times E^3,\ p\mapsto (T_p,\pi_p)$ is a bijection (please follow the link for more details). We conclude:

A system of reference in classical physics allows us to identify spacetime with the cartesian product of a $3$-dim. and a $1$-dim. affine space ("space and time").

My hope is that we can extend this concept to special relativity by making some modifications.

My thoughts:

One possible approach might be to define a system of reference in special relativity as a bijection $\Phi\colon M\to E^1\times E^3$ with some nice properties:

  • If we assume that $M$ is $4$-dim. manifold, the bijection $\Phi$ and the affine structure on $E^1$ and $E^3$ allow to construct a chart $x\colon M\to\mathbf{R}^4$.$^1$ What about requiring that the equation \begin{equation} g_{ij}=g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)=\eta_{ij}=\begin{cases}-1&\text{if }0=i=j\\ 1&\text{if }0<i=j\\ 0&\text{else}\end{cases} \end{equation} is true if we choose an orthonormal basis of the translation space of $E^3$?
  • Alternatively, (since it is also common to model spacetime as a four-dim. affine space $M$ with a bilinear form $g$ on the translation space) what about requiring \begin{equation} g(e-O^4,e'-O^4)=-c^2\langle t-O^1,t'-O^1\rangle_1+\langle p-O^3,p'-O^3\rangle_3 \end{equation} if $O,e,e'\in M$, $\Phi(O^4)=(O^1,O^3)$, $\Phi(e)=(t,p)$ and $\Phi(e')=(t',p')$?

I am a beginner in SR, so I hope this isn't complete nonsense. If anything, this is a starting point and needs an elaboration.

Any comments, corrections or different approaches are welcome.

$^1$ If $A$ is a $n$-dim. affine space with translation space $V$ and $O\in A$, \begin{align} F\colon A&\to V\\ P&\mapsto P-O \end{align} is a bijection and if $(v_1,\ldots,v_n)$ is a basis of $V$, \begin{align} G\colon \mathbf{R}^n&\to V\\ (\lambda_1,\ldots,\lambda_n)&\mapsto\sum_{i=1}^n\lambda_i\cdot v_i \end{align} is a bijection, too. Thus, the composition $G^{-1}\circ F$ is a bijection $A\to\mathbf{R}^n$.

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    $\begingroup$ For this, you may have a look at E. Gourgoulhon, Special Relativity in General Frames. $\endgroup$ – Massimo Ortolano Jan 10 at 14:28
  • $\begingroup$ Comment about terminology: Note that many authors call relativity for classical, since it is not a quantum theory. For this reason it is often better to call non-relativistic theory for Newtonian or Galilean rather than classical. $\endgroup$ – Qmechanic Jan 10 at 14:56
  • $\begingroup$ @Qmechanic Thank you for justifying the edit you made! I understand that my initial title - "classical physics vs special relativity" - was unclear, thus I appreciate the edit. However, "Newtonian" seemed to restrictive to me. "Classical mechanics" seems like a good solution to me in this case, I hope you agree. $\endgroup$ – Filippo Jan 10 at 15:10
  • $\begingroup$ @MassimoOrtolano Thank you very much for the EXCELLENT reference, I wish I knew this book much earlier. Section 3.4 seems to be what I was looking for. $\endgroup$ – Filippo Jan 10 at 15:25

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