Simultaneous Events (Galilean)

up:: Galilean Space-time Structure

Given a Galilean Space-time Structure, we say that two points in the Affine Space ─ ─ are simultaneous if , where

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

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 induces a time functional with respect to :

Since The dimension of the kernel plus the dimension of the image equals the dimension of the domain, we know that has dimension : it is the subspace of all points in which are simultaneous to .

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

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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