FredericGos said:
Hmm...
Ok, but then i Look up the definition of 'connection' here:
http://en.wikipedia.org/wiki/Connection_(mathematics )
The first line says 'In geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner'
So the notion of a connection is described in terms of the word 'paralell'. This short circuits my brain since you just said the opposite?
Parse what was said more carefully... The connection makes precise what we mean by transporting...in a parallel and consistent manner. The active noun here is connection and it acts on the notion of "parallel" making it precise, i.e. defining it in a precise way.
and if I look at the notion of 'paralell transport' :
http://en.wikipedia.org/wiki/Parallel_transport
first line is: 'In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection'
again "parallel with respect to connection" means the connection is what decides what is "parallel".
Again, parallel is just used here. But then i look at:
http://en.wikipedia.org/wiki/Parallel_(geometry )
Under 'construction' it says 'Definition 2: Take a random line through a that intersects l in x. Move point x to infinity.'
That makes a bit more sense, but then again, further down it says:
'The angle the parallel lines make with the perpendicular from that point to the given line is called the angle of parallelism. The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space'
So, this uses the curvature... I think it makes more sense now somehow, but I can't find the
right way to explain it to myself consistently.
But anyway, is there another approach to these things that don't use the word paralell? Could we formulate the whole thing with something like 'angle preserving' etc?
Sorry if I am babbling a bit, I suspect that the main problem is really that to understand this stuff better, one should study the subject much more rigorously and from within the mathematics and not try to use 'natural language' which only confuse things.
/Frederic
No need to apologize. It is this kind of quest for comprehension that you must undergo to develop understanding of the deep physics/mathematics rather than turning the crank on the calculations.
The problem with your last reference is it is a construction in Euclidean geometry...though I think they're trying to extend it to other geometries. Firstly it is talking about global parallelism rather than a local property which the connection defines. Secondly you have to know how "line" generalizes to other geometries (usually geodesic curves) and if the construction is still valid.
But it helps because we use the fact that all smooth manifolds with a geometry (metric) defined are locally flat. Understand that when we speak of vectors on a curved manifold the vectors don't reside in the manifold but in a tangent space touching the manifold at one point. Keep in mind that we define a metric on the tangent space not on the manifold itself but as we see below we can relate metric information on the tangent space to the manifold.
Directions on the tangent space at a point can be identified with curves through that point using the coordinate derivatives and coordinate differentials. The flatness of the tangent space reflects the local flatness of the manifold. We can thus take all the curves through a point and map them to lines or vectors on the tangent space at that point. We can thus speak of angles (or pseudo-angles if the metric is indefinite) between curves at that point.
OK so far? Now naturally we can have many curves passing through a point in a given direction. So without some way to connect the tangent spaces of infinitesimally close points to our original point we have trouble figuring which curves bend more or less than others. The easiest way to approach this is to imagine a random connection and ask if it is "good" or "bad" in terms of how it relates to the local metric at each point. We first must ask what the connection does.
Given two points we can "connect" their tangent spaces by defining a linear operator T(p1,p2) mapping vectors in one to vectors in the other. We presume this operator is invertible so we can map both ways. We do this for all point pairs on the manifold and we get a "Global connection" Now we assume certain "natural" conditions on this global connection. First that it is smoothly differentiable with respect to either of the points. Second that it remains invertible at least for all points in a certain open region. Thirdly we would hope that as we move continuously along a curve we can express the connection as an integral of infinitesimal transformations as we move from point to point along that curve.
This last may be a problem as we see that for an arbitrary global connection different curves may compose to yield different integrated connections between two points. If we impose the condition that this doesn't happen then this restricts our choice too much and yields the equivalent of a flat manifold (though it may have screwy topology).
But in considering this idea we find a more general place to start. We just define what happens during infinitesimal transformations and don't worrying about a global connection. We then allow for this path dependence. What we need is a generator \Gamma_\mu(p) of a linear transformation corresponding to each basis vector \mathbf{e}_\mu(p) in the tangent space at the point p. Then by moving in a direction a certain infinitesimal distance \mathbf{dx} we transform tangent vectors by the amount dx^\mu\Gamma_\mu. Integrating along a curve then gives us a (path dependent) connection between the tangent spaces of the curves endpoints. This is the parallel transport defined by our local connection \Gamma_\mu(p).
Now there is one more condition we need on this local connection and that is that it be compatible with the local metric. In particular we would hope that it preserve the lengths of vectors and (pseudo)angles between them as we move from tangent space to tangent space. This comes down to the covariant derivative of the metric being zero. We also get the condition that curves for which the local connection preserves tangent vectors are also curves with minimal arclengths as defined by integrating with the metric. "Geodesic" in terms of connection = "Geodesic" in terms of metric.
So in summary. We define (arbitrary) parallel transport (how we map tangent space vectors to tangent space vectors when we move along a curve) by choosing an (arbitrary) local connection. We then impose conditions on this connection so that it is compatible with the metric, i.e. preserving lengths and angles. This (I think) defines the parallel transport uniquely. We won't be able to define a global connection because in general transport may be path dependent which equates to the fact that transport around a loop may yield net linear transformations on the tangent space at a point on that loop. This is reflected in the Riemann curvature. The only spaces with global connections are those with zero curvature, i.e. flat Euclidean or pseud-Euclidean manifolds.
Does this make it any clearer? I glossed over some issues such as assuming we define point dependent tangent bases \{\mathbf{e}_\mu(p)\} which are smoothly differentiable. This defines another implicit (arbitrary) connection and really the \Gamma_\mu are the difference between this one and the connection we actually want. Hence the covariant derivative D_\mu=\partial_\mu + \Gamma_\mu is what really defines the connection. This is why neither the \Gammas nor the partials are "tensorial" while the covariant derivative is.
