Zoran Skoda
Galilean structure

A Galilean structure is a 5-tuple G=( 4, 4,μ,t,(,))G = (\mathcal{B}^4, \mathbb{R}^4, \mu, t, (,)) where ( 4, 4,μ)(\mathcal{B}^4, \mathbb{R}^4, \mu) is a 4-dimensional affine space, t: 4t : \mathbb{R}^4 \rightarrow \mathbb{R} is a nonzero \mathbb{R}-linear map and (,):kert×kert 0(,) : ker\, t \times ker\,t \rightarrow \mathbb{R}_{\geq 0} is a real inner product on 3-dimensional \mathbb{R}-vector space kertker\,t.

The elements of 4\mathcal{B}^4 will be called events. For any two events aa and bb we define the time interval between them to be t(ba)t(b - a). If the time interval between events aa and aa' is 00 then we say that aa and aa' are simultaneous events. Fix an event aa. The set a 3\mathcal{B}^3_a of all events which are simultaneous with aa is simply a+kerta + ker\,t. Action of 4\mathbb{R}^4 on 4\mathcal{B}^4 thus restricts to an action of kertker\,t on a 3\mathcal{B}^3_a which therefore becomes a 3-dimensional affine space.

If a,aa,a' are simultaneous events we can speak of the space distance between them. It is simply the length of the parallel displacement aaa' - a with respect to the norm induced by inner product on kertker\,t:

(1)d 3(a,a)=aa=(aa,aa). d_3(a,a') = \| a' - a \| = \sqrt{(a' - a, a' - a)}.

However there is no natural choice of space distance between the events which are not simultaneous. We can not even say for two such events if they took place at the same point. Indeed, imagine that we define such a notion. Then the naturality would mean that that notion is determined solely in terms of Galilean structure. In other words it should be preserved by the maps between Galilean structure which preserve it in the following sense.

A map A: 4 4A : \mathcal{B}^4 \rightarrow \mathcal{B}^4 is called a Galilean transformation if it preserves the Galilean structure. More precisely, AA is an affine map such that

(2)t(ba)=t(A(b)A(a)) t(b - a) = t(A(b) - A(a))

and (ba,ca)=(A(b)A(a),A(c)A(a))(b - a, c - a) = (A(b) - A(a), A(c) - A(a)) for any three events a,b,c 4a,b,c \in \mathcal{B}^4.

Definition. Let vkertv \in ker\,t. The uniform motion A v,aA_{v,a} with velocity vv cooccuring with event aa is the Galilean transformation given by

(3)calA v,a(a+w)=a+w+t(w)v. {\cal A}_{v,a}(a + w)= a + w + t(w)v\,.

The problem with non-simultaneous events above can now be restated as assertion that for any two non-simultaneous events a,b 4a,b \in \mathcal{B}^4 and any event bb' simultaneous with bb, the uniform motion A v,aA_{v,a} where v=bbt(ba)kertv = \frac{b' - b}{t(b-a)} \in ker\,t leaves aa intact and changes bb into bb'. Thus if aa and bb ‘took place’ at the same point by some magic definition, that notion will be ruined by the Galilean transformation mentioned.

Lemma. Let pqp \neq q be two points in some affine space \mathcal{B} over a division ring kk. The set

(4) 1(p,q)={p+α(qp)|αk} \mathcal{B}^1(p,q) = \{ p + \alpha (q-p) \,|\, \alpha \in k \}

has a natural structure of a left 1-dimensional kk-vector space and therefore also the natural structure of a 1-dimensional kk-affine space. The inclusion of sets ι: 1(p,q)\iota : \mathcal{B}^1(p,q) \rightarrow \mathcal{B} is an affine map.

Lemma. Let pp and qq be two non-simultaneous events in a Galilean structure G=( 4, 4,μ,t,(,))G = (\mathcal{B}^4, \mathbb{R}^4, \mu, t, (,)). Then for every point r 4r \in \mathcal{B}^4 there is exactly one point r 4r' \in \mathcal{B}^4 simultaneous with rr and belonging to 1(p,q)\mathcal{B}^1(p,q).

Created on July 13, 2009 at 20:40:32. See the history of this page for a list of all contributions to it.