Simultaneous Events (Galilean)

up:: Galilean Space-time Structure

Given a Galilean Space-time Structure, we say that two points in the Affine Space $A$ ─ $P,Q\in A$─ are simultaneous if $t(\overrightarrow{PQ})=0$, where

$$t:V\to \mathbb{R}$$

is the time functional, a linear map which measures time intervals between events (separated by vectors in $V$).

Since Fixing a point in an affine space induces a vector space, and An affine space with a fixed point is isomorphic to its underlying vector space, then fixing $P\in A$ induces a time functional with respect to $P$:

$$\begin{array}{rl}{\tilde{t}}_P:V& \to \mathbb{R}\\ & \overrightarrow{PQ}\mapsto t(\overrightarrow{PQ})\end{array}$$

Since The dimension of the kernel plus the dimension of the image equals the dimension of the domain, we know that $\ker {\tilde{t}}_P$ has dimension $3$: it is the subspace of all points in $A$ which are simultaneous to $P\in A$.

Also, A Galilean Space can be partitioned into simultaneity hyperplanes, since one can create an Equivalence Relation between events:

$$\forall P,Q\in A:P\sim Q⟺t(Q-P)=0⟺\overrightarrow{PQ}\in \ker (t)$$

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References

• ARNOL’D, Vladimir Igorevich. Mathematical methods of classical mechanics. Springer Science & Business Media, 2013.
• Notes on Mathematical Physics for Mathematicians - Daniel Tausk