An affine space of dimension n on $\mathbb R$ is defined to be a non-empty set $E$ such that there exists a vector space $V$ of dimension n on $\mathbb R$ and a mapping

$\phi:E \times E \rightarrow V,\space\space\space (A,B) \mapsto \phi(A,B):=\vec {AB}$

that obeys the following properties:

(i) For any point $O \in E$, the function

$\phi_O: E \rightarrow V,\space\space\space M \mapsto \vec {OM}$

is bijective.

(ii) For any triplet $(A,B,C)$ of elements of $E$, the following relation holds:

$\vec {AB} + \vec {BC} = \vec {AC}.$

Some formulation of special relativity formulates that spacetime is an affine space. Others that formulates as a manifold. It is true that an affine space is flat manifold, but not all flat manifolds are affine space.

My question is why can we formulate spacetime as an affine space? What I am asking if someone could give me real experiment that satisfies the axioms of an affine space


1 Answer 1


For Galilean and Special Relativity, the "Law of Inertia" suggests that an affine space is an appropriate model of spacetime.

For more details, consult these articles by Andrzej Trautman.

From Warsaw U's Physics page for https://en.wikipedia.org/wiki/Andrzej_Trautman ,

See "Galilean spacetime" text page 20 and onward. Page 26 introduces "affine space".

(page 26)
However, this greatly complicates the geometry of Space-time, so that, for the present, we shall leave gravitation aside.

Under this assumption, we have the f‌irst law of dynamics, which says that
(1) there exists a preferred class of motions, called free motions,
(2) there exist reference frames relative to which the free motions have no acceleration.
Usually, point (1) is further clarif‌ied by saying that a body is in free motion when no external influences act upon it. Point (2) is usually formulated in a somewhat dif‌ferent manner, namely, that free motions are rectilinear and uniform relative to certain reference frames called inertial frames. The two formulations are shown to be equivalent if spacetime is furnished with a geometrical structure which admits the concepts of rectilinearity and uniformity. Such a structure is provided by the aff‌ine space.

Aff‌ine Space An af‌fine space is a pair $(E, V)$, where $E$ is a set and $V$ a vector space, with a mapping $"+": E \times V \rightarrow E$, such that $V$ acts in $E$ as an (Abelian) group of transformations...

from http://trautman.fuw.edu.pl/publications/scientific-articles.html ,
see also Trautman's 1964 lecture (text page 101)
http://trautman.fuw.edu.pl/publications/Papers-in-pdf/21.pdf#page=109 and http://trautman.fuw.edu.pl/publications/Papers-in-pdf/22.pdf


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