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

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?

• What is the "Galileian structure"? Commented Nov 16, 2023 at 13:58
• @ValterMoretti As defined in Arnold's book. A Galilean structure is an affine space A^4, plus a time interval linear form t: R^4 -> R, plus a distance metric p: A^3 x A^3 -> R defined only on the kernals of t (simultaneous events) Commented Nov 16, 2023 at 14:01
• The kernel is of $dt$ not $t$... Commented Nov 16, 2023 at 14:22
• Commented Nov 17, 2023 at 22:10

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.

• This is a definition of the inertial reference frame, what's the definition for reference frame in general? Commented Nov 17, 2023 at 7:55
• For example, how would a rotating frame of reference be defined? Commented Nov 17, 2023 at 10:50
• It is defined, in coordinates, as the change of coordinates $x'^0 = x^0 + c \: (= t+k)$ and $x'^k = c^k(t) + \sum_{j=1}^3R(t)^k_j x^j\:, \quad k=1,2,3$ for arbitrary smooth functions $c^k(t)$ and $R(t) \in O(3)$. Commented Nov 17, 2023 at 14:01