Rigorous definition of frame of reference I'm looking for a mathematical definition of frame of reference. Most of the textbooks I have seen take it for granted and they just refer to some set of spacetime coordinates. A more mathematical definition I found defines spacetime $E$ as an affine space over $\mathbb{R}^4$ and then defines a frame of reference as a 5-tuple $(P,v_1,v_2,v_3,v_4)$ where $P\in E$ and $(v_i)$ is a basis of $\mathbb{R}^4$. But with this definition I cannot make sense of the phrase "a frame moving with respect to another".
 A: Note that a central idea of special relativity is that you can't define a frame of reference with respect to anything but another frame of reference. Just keep in mind that this doesn't make the space any less general.
Provided that $(v_i)$ is an acceptable basis, this is precisely the correct definition. Lets take $P=(0,0,0,0)$, where the coordinates are like $(ct,x,y,z)$. Then take $v_1=(\gamma,\beta \gamma,0,0)$, $v_2=(\beta \gamma,\gamma,0,0)$, $v_3=(0,0,1,0)$, $v_4=(0,0,0,1)$. Then, since $P$ is zero, we can let $(v_i)$ be a nonaffine linear transformation. As usual with a change of basis ("the new coordinates for a column vector v are given by the matrix product $M^{-1}v$"), the matrix that will change our basis is the inverse of $M=(v_1,v_2,v_3,v_4)$ where each basis vector is a column vector, or:
$M^{-1}=\begin{pmatrix}\gamma & -\beta \gamma & 0 & 0\\-\beta \gamma & \gamma & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1 \end{pmatrix}$
the usual Lorentz transform. (In general an acceptable basis $v_i$ would be the composition of a rotation and a lorentz transformation)
So what gives us the right to say this represents a frame moving with respect to another? If a particle in coordinate system $(v_i)$ is at position $s=(ct',0,0,0)$, then its position in the standard basis would be $Ms=(ct'\gamma,ct'\beta\gamma,0,0)=(ct,x,y,z)$. So since the position is linear, to find its velocity we find: $\frac{x}{t}=\frac{ct'\beta\gamma}{t'\gamma}=c\beta$. (I calculated $\frac{x}{t}$ because its meaning is physically clearer than $\frac{x}{c t}$.) Therefore the basis $(v_i)$ given represents a coordinate system moving at velocity $\beta c$ with respect to the current coordinate system.
A: Any rigorous definition in the context of the theory of relativity must of course be stated in terms of its primitive notions ( http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity ; see also http://en.wikisource.org/wiki/Space_and_Time ): namely in terms of 


*

*distinguishable "perceptible substantial points" (in the following for short: "participants"), and    

*"coincidences" (or "events") which are distinguishable by identifying which participants took part, and which did not.
A frame of reference for a given set $S$ of events is thereby, first of all, any set $K$ of participants
(1) where for each event of set $S$ there is one participant in set $K$ who took part in it, and
(2) where no two participants of set $K$ took part in the same event of set $S$
(Thus for each event of set $S$ there is precisely one participant in set $K$ who took part in it.)
Additional geometric relations between members of such a set $K$, further allowing to characterize this frame of reference as a system, may be obtained by considering participants who did not belong to set $K$ but who also took part in (several of) the events $S$.
Addendum -- More recently I learned:
if the given set $S$ of events constitutes a Lorentzian manifold then any corresponding frame of reference described as a set $K$ above is also referred to as a timelike congruence.
