Understand the definition of frame and inertial frame in Arnold's Galilean spacetime definition

Question

In Arnold's Mathematical Methods of Classical Mechanics, we define the physical space time as a four dimensional affine space with associated Galilean structure. I understand this part.

Now what I'm not clear after reading the next section in the book is:

1. What's the definition of a frame of reference, i.e. the formal definition of the mapping from the physical spacetime to $\mathbb{R}^4$. I guess the mapping needs to be bijective and preserving the Galilean structure.

2. What's the definition of an inertial frame? Is it an additional axiom that states there's a special frame mapping? Or does the concept of inertial frame already emerge from the Galilean spacetime definition?

Answer 1

With the definition of the Galilean structure written in your comment, an inertial reference frame can be safely defined as a Cartesian coordinate system with respect to the affine structure of $A^4$ with origin $O$ and axes $e_0,e_1,e_2,e_3$ such that

(1) $\langle e_0,dt\rangle =1 $

(2) $\langle e_i, dt \rangle = 0$ for $i=1,2,3$.

With these requirements, it is not difficult to prove that, considering a pair of these Cartesian coordiante systems associated to different choices of the basis and the origin, the transformation laws are, in fact, the Galilean transformations

$$x'^0 = x^0 + c \: (= t+k)$$ $$x'^k = c^k+t v^k + \sum_{j=1}^3R^k_j x^j \:, \quad k=1,2,3$$ where $v^k, c^k \in \mathbb{R}$ and $[R^k_j] \in O(3)$.

A general reference frame $x'^b$ is connected to an inertial one $x^a$ through a generic motion of the non inertial frame with respect to the inertial one: $$x'^0 = x^0 + c\:,$$ $$x'^k = c^k(t) + \sum_{j=1}^3R^k_j(t) x^j \:, \quad k=1,2,3$$ where $c^k(t)$ and $[R^k_j](t) \in O(3)$ are arbitrary smooth functions.