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Following Arnold's [1] definition of the universe as an affine space $ A ^4$ with the group $\mathbb R ^4$ acting on it, we may define a galilean transformation as an affine map $g:A^4 \to A^4$ which preserves the galilean structure, i.e. which preserves time intervals and spatial distance beetween simultaneous events. “Time” is an application $t:\mathbb R ^4 \to \mathbb R $ and $P,Q\in A^4$ are simultaneous if $t(P-Q)=0$?

Furthermore, to say that a coordinate system $\varphi _1$ (read: a bijection $\varphi _1 :A^4 \to \mathbb R ^4$) is in uniform motion with respect to another $\varphi _2$ means that $\varphi_2 ^{-1}\circ \varphi _1 $ is a galilean transformation of $\mathbb R \times \mathbb R ^3 $ (to give a galilean structure to this space we take $t$ to be the projection on the first coordinate).

On the other hand, a Lorentz transformation beetween two coordinate systems in relative motion is often defined as a linear transformation of the space of coordinates: $\Lambda :\mathbb R \times \mathbb R ^3 \to \mathbb R \times \mathbb R ^3 $.

Question is: is it possible in special relativity to characterize

a) the universe as an affine space $A^4$ (with the group $\mathbb R \times \mathbb R ^3$ acting on it) with some additional structure in a way similar to the galilean universe? I think the answer is yes, the structure is given by the pseudo metric on $\mathbb R \times \mathbb R ^3$, $|x|^2 =c^2t^2 - x_1 ^2 -x _2 ^2 -x_3 ^2$. Is this sufficient to fully characterize the universe $A^4$.

b) Lorentz transformation as transformations of $A^4$ that preserve its structure. Again, is it sufficient to say that Lorentz transformations are the linear ones which preserve the space-time distance given by $|.|^2$? And why linear and not affine?

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  • $\begingroup$ I may have thrown too much questions inside, so that it isn't possible to give a direct and concise answer. If it is so, let me know that and I'll try to restate the question in a way that can be answered. $\endgroup$
    – pppqqq
    Dec 12, 2013 at 10:14

1 Answer 1

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The affine Galilean structure is assigned by the first principle of Newtonian dynamics, i.e. by giving the class of inertial reference frames in the spacetime $G^4$. On the one hand it assigns the structure of an affine space to the spacetime, on the other hand it selects a subclass of permitted transformations between reference frames. A reference frame is a bijective map $G^4 \ni P \mapsto (t(P), {\bf x} (P) )\in \mathbb R \times \mathbb R^3$

The class of allowed coordinate transformations is that including all of the transformations:

$$t' = t + c \quad(1)$$

$${\bf x}' = R{\bf x} + t {\bf v} + {\bf c}\:, \quad (2) $$

where $c\in \mathbb R$ is constant, ${\bf v}$ and ${\bf c}$ are constants in $\mathbb R^3$ and $R \in O(3)$. These are the transformation of coordinate between inertial reference frames.

In this way a foliation $\{\Sigma_t\}_{t \in \mathbb R}$ of $G^4$ turns out to be defined, made of 3-dimensional Euclidean spaces. Also a surjective map $T : G^4 \to \mathbb R$ is consequently defined, that labels the class of those manifods. The function $T$ (absolute time) coincides, up to an additive constant, with the coordinate $t$ of every inertial reference frame, so that intervals of time are absolute. The coordinates ${\bf x}$ of every inertial frame range in each absolute 3-dimensional space $\Sigma_t$ that is in common with each of these observers, including the metrical structures that are invariant changing reference frame, in view of (2).

Concerning special relativity. Yes, you can completely characterize Minkowski spacetime $M^4$ as a real affine 4D space equipped with a pseudo distance with signature $(1, -1, -1,-1)$ and a time orientation. The inertial reference frames are a subclass of the affine coordinate systems on $M^4$ where the pseudo distance assumes the canonical form you have written in your question. The transformations between these coordinate frames are the so called Poincaré orthochronous transformations.

ADDENDUM To answer the last question: The relativistic corresponding of Galileo group is Poincaré orthochronous group and not the Lorentz group. Orthochronous Poincaré transformations are the most general transformations preserving the structure of Minkowski spacetime including time orientation. These are, obviously, affine transformations as Galilean transformations are.

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  • $\begingroup$ Thank you for this great answer! A note on the second paragraph: I have no idea of what a foliation is. I guess I can think about it as the quotient $G^4/\sim$ where $\sim$ is the equivalence relation “being contemporary“? $\endgroup$
    – pppqqq
    Dec 12, 2013 at 22:19
  • $\begingroup$ Yes, in this case it is exactly what you wrote. But it is a more general notion: en.wikipedia.org/wiki/Foliation $\endgroup$ Dec 13, 2013 at 6:26

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