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I read Freeman Dyson's article Missed Opportunities, in which he talked about the mathematical attractiveness of the Lorenz group compared to the Galilean group. I am reading Florian Scheck's book on special relativity, and I feel very confused because he emphasized that though $P_{G}$ is an `affine bundle', it is nevertheless not trivial.

To illustrate this he argued that since the general form of Galilean transformation is $$x'=Rx+vt+a, t'=t+s, R\in SO(3)$$

He claimed that if $v\not=0$, then the projection of the spacial coordinate is no longer invariant for the rest frame and moving frame. Does this just mean $x'\not=x?$ I am confused with his language.

A related question is what is the structure of $P_{G}$ afterall? The wikipedia article is sadly incomplete; I do not want to know what the central extension of $P_{G}$; I want to understand how to describe it mathematically (like a matrix group or as a fibre bundle). What is its topological structure? What is its group structure? What makes it so complicated as Dyson claimed? The representations of $P_{G}$ are not clear to me, either as the indices mixed up everywhere. Overall I feel very confused because I believe Galiean transformations should be easy to understand.

A strategy of unwind this ambiguity is to work with lower dimensions. The $P_{G}$ for one dimensional space should only have the form $$x'=x+vt+a,t'=t+s$$ and is only 3 dimensional as a Lie group. Group structure operates by $$(v_{2},a_{2},s_{2})(v_{1},a_{2},s_{1})=(v_{1}+v_{2},a_{1}+a_{2}+v_{2}s_{1},s_{1}+s_{2})$$ But I am not sure what kind of group this is. Is this $\mathbb{R}^{2}\rtimes \mathbb{R}^{1}$ with $$\phi_{s{1}}(v_{2},a_{2})=(v_{2},a_{2}+v_{2}s_{1})?$$ and how this generalizes to higher dimensions?

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Yes, the Galilean group and algebra is a semidirect product. For each rotation $R\in SO(3)$, there is a 6-dimensional flat manifold, fiber, for which you have to specify the velocity $v$ for the boost as well as the extra absolute translation $a$. The translations and Galilean boosts transform as vectors under the $SO(3)$ rotations which determines the commutators with the rotations; the rotations don't commute with each other. Otherwise the translations and Galilean boosts commute with each other. A typical semidirect algebra which is less constraining than a simple group e.g. Lorentz group. –  Luboš Motl Jun 8 '12 at 7:43
    
Thank you, this is very helpful. –  Bombyx mori Jun 8 '12 at 7:54

1 Answer 1

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The (connected) Galilei group in d space dimensions is the group of all block upper triangular matrices with three blocks of size $d,1,1$, whose diagonal blocks are rotation matrices of the respective dimensions (which implies diagonal entries 1 in the blocks of size 1).

The corresponding Lie algebra is therefore the Lie agebra of all block upper triangular matrices with three blocks of size $d,1,1$, whose diagonal blocks are antisymmetric matrices of the respective dimensions (which implies diagonal entries 0 in the blocks of size 1).

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