Recommendaton for Clarifying Special Relativity

[TrickyDicky]

Euclidean geometry, Minkowski spacetime, and Galilean spacetime are really "affine geometries" (often described as "a vector space that has forgotten its origin"). Indeed, there is no distinguished element in any of these spaces. Thus, one cannot add elements or scalar-multiply... although one can subtract two elements (and get a vector).

As mentioned earlier, the tangent space at each point-event is a vector space.

Very true. Taken as abstract spaces they are affine. Note however that both Euclidean and Minkowskian geometries being homogeneous allow one to freely choose a point as origin and that is what one does in physics when measuring, and in that sense it is accepted to consider them vector spaces and indeed anybody can check Minkowski space is defined as such for instance in Wikipedia and other sources.

 

[WannabeNewton]

Minkowski space-time is just $\mathbb{R}^{4}$ equipped with the bilinear form $\eta_{ab}$. $\eta_{ab}$ doesn't alter the natural vector space structure of $\mathbb{R}^{4}$; it simply adds a pseudo-inner product structure on top of the vector space structure. As a side note, every finite dimensional real vector space is a smooth manifold.

 

[Dale]

both Euclidean and Minkowskian geometries being homogeneous allow one to freely choose a point as origin

"Allow", yes. "Require", no.

If flat spacetime by itself were a vector space then two events would have a well defined vector sum. They don't. The elements of spacetime are events, and events aren't vectors, therefore spacetime is not a vector space.

Furthermore, coordinate charts do not generally form vector spaces either. Consider spherical coordinates. The r coordinate is strictly positive, so multiplying a valid coordinate by a negative number gives a point in R4 which is outside the open subset covered by the chart.

In contrast, the fields we would want to measure in physics are vectors (and tensors and scalars). As such, their components are also vectors, scalar multiples of some basis vectors. So not only can they be described without coordinates, they can only be described with coordinates if you use the coordinates to generate a basis (and obviously that isn't the only way to generate a basis).

 

[robphy]

Very true. Taken as abstract spaces they are affine. Note however that both Euclidean and Minkowskian geometries being homogeneous allow one to freely choose a point as origin and that is what one does in physics when measuring, and in that sense it is accepted to consider them vector spaces and indeed anybody can check Minkowski space is defined as such for instance in Wikipedia and other sources.

So, I found

http://en.wikipedia.org/wiki/Minkowski_space][/PLAIN] [/PLAIN]
Structure

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (−,+,+,+) ...

and

http://en.wikipedia.org/wiki/Minkowski_space][/PLAIN] [/PLAIN]
Alternative definition

The section above defines Minkowski space as a vector space. There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer. See Erlangen program.

Note also that the term "Minkowski space" is also used for analogues in any dimension: if n≥2, n-dimensional Minkowski space is a vector space or affine space of real dimension n on which there is an inner product or pseudo-Riemannian metric of signature (n−1,1), i.e., in the above terminology, n−1 "pluses" and one "minus".

Yes, once you pick a point-event to be the origin, you now have a vector space (the set of displacement vectors from your choice of origin)... without singling out a point-event, you [still] have the affine structure.

Since the universe doesn't distinguish any particular event in Minkowski spacetime [e.g. the game-winning goal at the recent Stanley Cup finals], I prefer not to impose that structure in the mathematical model... until necessary.

(One lesson I learned when studying physics is that it's good to know the MINIMAL structure needed to obtain something. Once you toss everything in [e.g. symmetries, choice of dimensionality, signature]... it's hard to see WHERE a particular feature comes from. This is important (e.g.) in quantum gravity where one tries to deconstruct what we observe and try to find the right generalizations to extend the classical theory to a quantum one.tangentially-related anecdote... My favorite math professor sternly corrected a student who said "a square is a parallelogram with four right-angles" by saying "a square is a parallelogram with ONE right-angle". This was an aha-moment for me.)

 

[Vanadium 50]

Pretty much every possible point has been made (often more than once) and we're just covering the same ground again and again. Also this thread has become a magnet for sockpuppets of banned members and other crackpots, so this thread is closed.