I Affine Space: Understanding the Difference from Ordinary Space

[Arman777]

I am reading this book and in there the spacetime defined as a manifold such that an affine space of dimension 4. I am having trouble to understand the affine space. I made some reasearch but I couldn't grasp the idea of it. In the books its also stated that " We are familiar with the structure of affine space of dimension 3. It involves the notion of points that can be joined two by two by vectors."

So is it just ordinary space ? Or it has some additional differences/properities.

 

[Dale]

An affine space is basically a vector space without an origin.

 

[Arman777]

An affine space is basically a vector space without an origin.

Hmm I see thanks than

 

[robphy]

From an old post of mine https://www.physicsforums.com/threads/position-vector.99312/post-826856

Here's some physical motivations on this issue concerning an affine space vs. a vector space.

A space of positions in a plane (or the space of times on a line) is an affine space. With no point being physically distinguished from any other, an affine space is more natural than a vector space. In an affine space, there is no sense of addition of elements of this space. (If you attempt to add two elements, the sum depends on the choice of an "origin" [which does not exist in an affine space]. As was mentioned, one can introduce an origin by introducing a coordinate system. Then, the sum now depends on a choice of coordinate system.) There is, however, a sense of subtraction... the "difference of two positions [in an affine space]" is a vector... the displacement vector. (The difference does not depend on a choice of [or existence of] an origin.)

later in that thread https://www.physicsforums.com/threads/position-vector.99312/post-827363

Read the discussion here http://www.acs.ucalgary.ca/~alphy/MAP/workinprogr/Pos/ItemDescr.htm
It raises an additional point that "positions" can't be scaled (i.e. scalar multiplied), as well as can't be added. Let P denote the position ("location") of an object. It makes no sense (that is, there is no physical, coordinate-independent interpretation) to multiply P by 2, or to add another position Q to form P+Q. Thus, the space of such P's is not a vector space. P is not a vector. [Note, I am not referring to assigning coordinates to P. If you wish, you can assign a triple of numbers to P, effectively introducing a coordinate system. Then, then triple (0,0,0), the origin of that imposed coordinate system, is a position, call it "oh" O. One can draw an arrow from O to P... this is a displacement vector, OP, from the origin of the imposed coordinate system. Of course, another assignment of coordinates locates the new origin at another point O', which yields a different displacement vector O'P. Certainly, 2(OP) is generally different from 2(O'P), and (OP+OQ) is generally different from (O'P+O'Q). However, (OQ-OP)=(O'Q-O'P)=PQ... the choice of coordinate origin and coordinate system is irrelevant when forming displacements between two positions P and Q.]

http://mathworld.wolfram.com/AffineCoordinates.html is probably relevant to this discussion.

If I recall correctly, there is a section in Bamberg and Sternberg's

Let me make a comment, which may or may not help:
not all "configuration spaces" are vector spaces... however, the "velocity space" (the tangent space) is a vector space.

Finally, https://www.physicsforums.com/threads/position-vector.99312/post-827372

As a postscript, look at this paper by Wald on teaching General Relativity: https://arxiv.org/abs/gr-qc/0511073
In particular, the first paragraph says...

goes against what students have been taught since high school (or earlier): namely, that
“space” has the natural structure of a vector space.

..and on page 4:

However, very few students have any inkling that, in nature, the points of space and/or
the events in spacetime fail to have any natural vector space structure. Indeed, the con-
cept of a “vector” is normally introduced to students early in their physics education
through the concept of “position vectors” representing the points of space! Students are
taught that, given the choice of a point to serve as an “origin”, it makes sense to add
and scalar multiply points of space. The only significant change introduced by special
relativity is the generalization of this vector space structure from space to spacetime: In
special relativity, the position vector ~x representing a point of space is replaced by the
“4-vector” xμ representing an event in spacetime. One can add and/or scalar multiply
4-vectors in special relativity in exactly the same way as one adds and/or scalar multiplies
ordinary position vectors in pre-relativity physics.
This situation changes dramatically in general relativity, since the vector space char-
acter of space and/or spacetime depends crucially on having a flat geometry. In general
relativity, it does not make any more sense to “add” two events in spacetime than it would
make sense to try to define a notion of addition of points on the surface of a potato.

 

[PAllen]

While defining 'position vectors' for an affine space requires adding an origin, displacement vectors can be added and scaled, and normalized to direction vectors. The vector space of all displacement vectors from a given point can be seen as generalizing to the tangent space at a point in GR, as long as one is clear that the displacements are in the tangent space, not in the manifold.

 

[Orodruin]

Put a bit differently, an affine space is in essence a space such that any two points have a well-defined difference vector that represents the displacement between the points.

Being a bit more technical, an affine space $S$ is a space on which there exists a translation map $T: S\times V \to S$, where $V$ is a vector space. The translation map needs to satisfy:
- For any fixed $p\in S$, $T(p,\vec v)$ is a bijection from $V$ to $S$.
- Translation is associative, i.e., $T(T(p,\vec v_1),\vec v_2) = T(p,\vec v_1+\vec v_2)$.
- Translating by zero returns the same point, $T(p,0) = p$.

The difference vector between two points $p$ and $q$ is the unique vector $\vec v$ such that $T(p,\vec v) = q$. (It is unique because translation from $p$ is a bijection from $V$ to $S$.)

 

[Ibix]

Put a bit differently, an affine space is in essence a space such that any two points have a well-defined difference vector that represents the displacement between the points.

So this only applies to flat spaces, since there isn't in general a well-defined difference vector between points in a curved space? Is there a name for whatever a curved manifold is? Other than "curved manifold", I mean.

 

[Orodruin]

So this only applies to flat spaces, since there isn't in general a well-defined difference vector between points in a curved space? Is there a name for whatever a curved manifold is? Other than "curved manifold", I mean.

Yes and no. There is a priori no requirement to choose the natural connection on the affine space (i.e., saying that your geodesics are of the form $T(p,t\vec v)$ with fixed $p$ and $\vec v$ -- I cannot think of an application where you would not want to do that, but that does not mean it does not exist). Flatness generally refers to the connection you introduce. However, if you do choose the natural connection on the affine space, then it is flat.

What is wrong with "curved manifold"?

 

[Arman777]

Can we say that affine space is important since in SR $ds^2$ is invarient ? Because $ds^2$ represents the spacetime interval between two events. And any reference frame agrees on the interval , so there's no notion of an origin.

Other than "curved manifold", I mean.

In the same book, at chapter 22 it says "The base space is not necessarily an affine space but a more general structure: a differentiable manifold, as defined in Sect.7.2.1. "

 

[Orodruin]

Can we say that affine space is important since in SR $ds^2$ is invarient ? Because $ds^2$ represents the spacetime interval between two events. And any reference frame agrees on the interval , so there's no notion of an origin.

No, this is true regardless of whether the space is affine or not. Affine space is important as already the Galilean spacetime of classical mechanics is an affine space (it does not have a $ds^2$, it has a distance form and a time metric). The Minkowski spacetime of special relativity is also an affine space (there is no preferred origin, we can pick the origin in the most convenient way). A general spacetime will not be an affine space.

 

[Arman777]

Okay I underatand it now. As Dale and others pointed out its just a vector space without an origin. I thought that its due to invarience of the speed of light whlch determines the metric and the property of the space, which is different then Classical Mechanics

 

[robphy]

Okay I underatand it now. As Dale and others pointed out its just a vector space without an origin. I thought that its due to invarience of the speed of light whlch determines the metric and the property of the space, which is different then Classical Mechanics

It's more about parallel lines for geometry,
and the "Law of Inertia" for non-gravitational relativity.

For more about the foundational ideas of spacetime structure along these lines...

From Warsaw U's Physics page for https://en.wikipedia.org/wiki/Andrzej_Trautman ,
http://trautman.fuw.edu.pl/publicat...d_gravitation_Kopczynski_Trautman.pdf#page=30
See "Galilean spacetime" text page 20 and onward. Page 26 introduces "affine space".

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

and this editorial note by Trautman for the reprint of the "EPS"-paper
J. Ehlers, F. A. E. Pirani and A. Schild, The geometry of free fall and light propagation
https://link.springer.com/content/pdf/10.1007/s10714-012-1352-5.pdf

 

[Arman777]

Okay I was looking at the book and I read this,

From this paragraph I understand that there's no certain true origin, since it says origin of the considered affine system. So every observer can assign themself as the origin.

If you have an affine space the origin of a reference frame is an arbitrary point O∈MO∈MO \in M.

So @vanhees71 is right I think.

It's more about parallel lines for geometry,
and the "Law of Inertia" for non-gravitational relativity.

For more about the foundational ideas of spacetime structure along these lines...

From Warsaw U's Physics page for https://en.wikipedia.org/wiki/Andrzej_Trautman ,
http://trautman.fuw.edu.pl/publicat...d_gravitation_Kopczynski_Trautman.pdf#page=30
See "Galilean spacetime" text page 20 and onward. Page 26 introduces "affine space".

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

and this editorial note by Trautman for the reprint of the "EPS"-paper
J. Ehlers, F. A. E. Pirani and A. Schild, The geometry of free fall and light propagation
https://link.springer.com/content/pdf/10.1007/s10714-012-1352-5.pdf

Thanks I ll look at them

 

[PeterDonis]

Note to thread participants: A long sidebar discussion on affine spaces vs. vector spaces has been moved to a new "A" level thread:

https://www.physicsforums.com/threads/affine-spaces-and-vector-spaces.987545/