# Mathematical description of systems of reference - classical mechanics vs special relativity

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 $$$$g_{ij}=g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)=\eta_{ij}=\begin{cases}-1&\text{if }0=i=j\\ 1&\text{if }0 is true if we choose an orthonormal basis of the translation space of $$E^3$$?
• Alternatively, (since it is also common to model spacetime as a four-dim. affine space $$M$$ with a bilinear form $$g$$ on the translation space) what about requiring $$$$g(e-O^4,e'-O^4)=-c^2\langle t-O^1,t'-O^1\rangle_1+\langle p-O^3,p'-O^3\rangle_3$$$$ if $$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 $$A\to\mathbf{R}^n$$.

• For this, you may have a look at E. Gourgoulhon, Special Relativity in General Frames. – Massimo Ortolano Jan 10 at 14:28
• 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. – Qmechanic Jan 10 at 14:56
• @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. – Filippo Jan 10 at 15:10
• @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. – Filippo Jan 10 at 15:25