# Can I fix a point in Minkowski space to give it a vector space structure?

I looked up the term Minkowski space on Wikipedia. It said

There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogenous space of the Poincaré group with the Lorentz group as the stabilizer.

In their book Metric Affine Geometry, Snapper and Troyer state on page 59:

It cannot be stressed enough that the affine space $X$ is not a vector space. Its points cannot be added and there is no way to multiply by scalars. No point in $X$ is preferred; they all play the same role. In particular, there is no point in $X$ which makes a better origin for a vector space than any other point.

The situation changes radically if we choose a point $c$ in $X$ and keep it fixed. It is now possible to make $X$ into a left vector space over $k$ by using the one-to-one mapping $f$ from $X$ onto $V$ defined by $f(x) = \overrightarrow{c,x}$ for each $x \in X$. All we do is carry the vector space structure of $V$ over to $X$ by means of the mapping $f$.

So here's my question: As I understand it, it makes sense to think of Minkowski space as an affine space since the basic principle of Special Relativity is that no point in $X$ is a preferred reference frame. But does that mean it is then impossible to "fix" a point in $X$ as Snapper and Troyer say can be done? In other words, is there any physical meaning to the idea of fixing a point in the affine space or is that impossible according to SR?

Obviously I am trying to use a mathematician's idea to interpret what can be done physically with Minkowski space.

• Is this the same idea by which one can define an "absolute" space in classical mechanics? If it is, then it probably runs into the same problem as in classical mechanics: there is no physical mechanism that can pick c! All such "fix points" are equally unphysical, even though nothing stops us from using them in all of our calculations. In the end all they are expressing by their equal meaninglessness is the very degeneracy that makes the theory "relative" to begin with. Having said all of this, there is a high likelihood that I am completely misunderstanding the idea. – CuriousOne Dec 28 '14 at 8:21
• I had similar thoughts, but I really have no idea so that's why I posted the question. – Stan Shunpike Dec 28 '14 at 8:46
• Does the text give any more context? How are the authors exploiting the vector space property of this construction? – CuriousOne Dec 28 '14 at 8:53
• Sure, let me see if I can provide some. On page 6-7, they say " the axiom system for $n$-dimensional affine space over a division ring $k$ consists of a nonempty set $X$, an $n$-dimensional left vector space $V$ over the division ring $k$, and an "action" of the additive group of $V$ on $X$. The elements of $X$ are called points and are denoted $x,y,z,...$; the elements of $V$ are called vectors and are denoted by $A,B,C,...$. Scalars are ways written on the left of vectors. The affine space defined by $X,V,k$ and the action of the additive group of $V$ on $X$ will be denoted by $(X,V,k)$. – Stan Shunpike Dec 28 '14 at 9:05
• (Continued) In case $k$ is the field of the real numbers $\Bbb{R}$, $(X,V,\Bbb{R})$ is called real affine space. – Stan Shunpike Dec 28 '14 at 9:07

## 1 Answer

Fixing a point is more or less like fixing a coordinate system on your affine space. Then you can identify $X$ with $V$ as stated in the book, where the fixed point $c\in X$ is mapped to the origin of $V$. In other words, fixing a point $c$ in $X$ is like glueing a copy of $V$ onto $X$ in such a way that $O\in V$ overlaps with $c\in X$. As far as the Lorentz group is considered then the coordinates (i.e. the component of the glued copy of $V$ onto $X$) really behave like vectors, but this is no longer the case under more general transformations (consider for instance translations, or the action of the ray inversion from the conformal group).