Elementary questions about connections.

[Paulibus]

Elementary questions about connections.

I’ve been looking at the Wikipedia article on affine connections with a view to better
understanding the choice of Riemannian geometry for our best theory of gravity — a founding
choice made by Einstein nearly a century ago.

Sadly for me the Wikipedia article seems to be have been written for ‘streamlined folk who think in slogans (here symbols) and talk with bullets (here formulae)’ (vide George Orwell). I’m un-streamlined and I’d therefore like to ask a few primitive questions unrestricted to the simple distortion (called curvature) of, say, a two-dimensional Euclidean space exemplified by a two-dimensional, uniformly curved spherical surface.

Am I correct in supposing that for physics purposes, connections are used to extend algebraic descriptions of geometry and analyses needing calculus to beyond a ‘local’ region, inside which deviations from Euclidean geometry are imperceptible?

If so, I guess that some other kinds of connection would be needed algebraically to describe
and analyse, if deemed necessary, other distortions from Euclidean geometry. For example,
distortions like that of a helical two-dimensional geometry, say a multi-storey car park floor-surface geometry. Or of the surfaces of Klein bottles and Moibus strips. Or of scale-changes, gauged by some agreed protocol, as in a lattice where temperature varies and expansion is measured with an Invar ruler or by counting lattice steps.

In such cases, I suppose, not only could distortions classified as ‘curvature’ be measured by
orientation changes of a vector carried around a closed circuit (by parallel transport using an
affine connection) and specified by a Riemann tensor: but different distortions might produce
different changes (like vertical discontinuities in a car park) in mathematical objects carried
around circuits that would be expected to close in the ‘local’ Euclidean limit.

Could someone tell me what the connections that would be used in such cases are called and what mathematical objects would be used to specify such distortions? And are
connections other than 'affine and Levi-Civita' inappropriate for use in a theory of gravity because they would not yield a Newtonian inverse square law locally in a Euclidean limit?

Or for a more streamlined reason?

 

[atyy]

Using the Levi-Civita connection makes the metric diag(-1,1,1,1), and its first partial derivatives vanish at a point, providing one part of the equivalence principle (write the non-gravitational laws in tensor form in SR, replace partial derivatives by covariant derivatives).

http://arxiv.org/abs/1007.3937
http://arxiv.org/abs/1008.0171

Nowadays, it is usually considered that Einstein did not "derive" GR from principles, unlike what he did for SR. A better derivation of GR is that it is a massless spin 2 field.

http://arxiv.org/abs/gr-qc/0611100
http://arxiv.org/abs/gr-qc/0411023
http://arxiv.org/abs/1105.3735

 

[bcrowell]

The connection is a local thing. An example like a Klein bottle is locally the same as ordinary Euclidean space. It differs topologically from Euclidean space. You can have a two-surface with the topology of a Klein bottle but zero curvature everywhere.

In such cases, I suppose, not only could distortions classified as ‘curvature’ be measured by orientation changes of a vector carried around a closed circuit (by parallel transport using an affine connection) and specified by a Riemann tensor: but different distortions might produce different changes (like vertical discontinuities in a car park) in mathematical objects carried around circuits that would be expected to close in the ‘local’ Euclidean limit.

Your example of the car park sounds like it has boundaries at the edges of the ramp. A boundary *is* something that is locally detectable, but it can't be described by a connection.

There's the notion of a manifold. Basically a manifold is something that locally looks like Euclidean space. By itself, the manifold has no notion of measurement. It only has topological properties. A coffee cup is the same manifold as a doughnut. If a space has a boundary, it's not a manifold. For instance, the closed half-plane $y \ge 0$ is not a manifold, because there is no neighborhood around a point like (7,0) that looks Euclidean. However, the open half-plane y>0 is a manifold because it has no points that are boundary points.

Physically, we want to describe spacetime as a manifold because we expect the laws of physics to be the same, locally, no matter where we go.

By adding a connection onto a manifold you define a notion of (local) measurement.

The Einstein field equations only deal with curvature, which is a local thing. They don't directly predict anything about topology, but their predictions about curvature can indirectly constrain the possible topologies. For example, it is possible that our universe is flat and has the topology of Euclidean four-space, but it's also possible that it's flat and has some other topology: http://en.wikipedia.org/wiki/Topology_of_the_universe

 

[pervect]

If so, I guess that some other kinds of connection would be needed algebraically to describe
and analyse, if deemed necessary, other distortions from Euclidean geometry. For example,
distortions like that of a helical two-dimensional geometry, say a multi-storey car park floor-surface geometry.

The math I'm familiar with assumes that the underlying geometry is a manifold, which is the sort of geometry that's physically interesting.

The detailed definition of a manifold is rather complex, a quick description would be that a manifold looks like open subsets of R^n that can be "sewn together" smoothly at the seams.

Smooth sewing implies you don't sew folds together (no darts, for instance, if you think of the clothing analogy).

You'd have to exclude the origin to make a helical multi-sheet geometry a manifold (like the geometry of the log of a complex number) but I think it would be a manifold if you did that and also used only real coordinates.

I'm not sure about the "car park" geometry, I can't quite envision what you mean.

 

[Paulibus]

I've taken too long in replying to the three illuminating comments on my post. I appreciate the trouble you took, atyy, Pervect and bcrowell. Thank you all.

Your comments on locality still have me a bit buffaloed, bcrowell. What exactly is meant by local? I floated the idea that a ‘local’ region is one "inside which deviations from Euclidean geometry are imperceptible". This definition, garnered from my reading of the distinction between local and global in relativity, seemed to me at first to conform with your describing a manifold as "something that locally looks like Euclidean space" , a something which, when augmented by a connection, enables the "notion of (local) measurement" and hence the rule over physical measurements of Einstein's field equations that "only deal with curvature, which is a local thing". But now I'm not so sure I have the accepted notion of "local".

Currently, I think of a connection as an arcane mathematical device whose bottom-line purpose (in the case of general relativity) is to provide a ratcheting kind of tool for tracing out (by prescribing how to parallel-transport tangent vectors) the paths followed by photons and freely gravitating particles through a spacetime that is specifically curved by mass/energy.

Curved meaning in a mathematical sense a differentiable mapping of an open set of real numbers into a manifold? Differentiable meaning no discontinuities or local jerks?

Although spacetime geodesics are not the only game in town when it comes to, say, understanding astronomical phenomena, I know that for astrophysicists it's useful in practice to treat the planet Mercury as a single particle falling along a spacetime geodesic when calculating the advance of it's perihelion. And the concept of a geodesic is essential for relating the gravitation lensing of remote galaxies to intervening masses. Or for understanding why rings form around planets, and accretion discs around stars and black holes. That's why I, as an unstreamlined non-mathematician, would like to understand connections.

Two other remarks. I'm still intrigued by the thought that since curvature is not the only way that spacetime can be imagined to be distorted, mass/energy may not be the only imaginable distorting agent --- just the rather feeble one, perhaps. And what, I ask again, about distortions involving changes of scale?

That bad multi-storey car park example of a two-space distortion I gave led to confusion. Sorry. Such a distortion was better described (by Pervect) as a "helical multi-sheet geometry", which of course needn't have confusing edges or boundaries. It's also the lattice geometry associated with a helical kind of crystal defect (one aptly named a "screw dislocation").

 

[dextercioby]

[...]Nowadays, it is usually considered that Einstein did not "derive" GR from principles, unlike what he did for SR. A better derivation of GR is that it is a massless spin 2 field.

http://arxiv.org/abs/gr-qc/0611100
http://arxiv.org/abs/gr-qc/0411023
http://arxiv.org/abs/1105.3735

Many thanks for the Deser reference. Didn't know he put it on arxiv. I was in possession of the original Feynman calculation from his GR lectures and the 2000 BRST derivation by Henneaux & collaborators.

 

[Ben Niehoff]

Here is how I think of connections; this method can encompass all of the issues you've brought up:

First of all, at every point of a manifold you can construct the tangent space. The tangent space is the vector space spanned by all the tangent vectors. The "tangent" terminology comes about because if you were to embed your manifold in a higher-dimensional flat space, the "tangent space" would be precisely the hyperplane that lies tangent to your manifold at a given point.

A connection is a mathematical object that describes how the tangent space at nearby points are related. It essentially gives a gluing procedure, saying "vector X at point A corresponds to vector Y at point B". This is why it's called a "connection", because it describes how nearby tangent spaces are connected to each other.

Specifically, the connection describes how to relate the tangent space at some point x to the tangent space an infinitesimal distance away at x + dx. Since these tangent spaces are infinitesimally close together, this relation must be a linear map from $T_xM$ (the tangent space at x) to $T_{x+dx}M$. The tangent spaces have the same dimension, so the map must be a member of $GL(n, \mathbb{R})$, the group of nxn invertible matrices (where n is the dimension of the manifold).

So fundamentally the connection must involve some kind of "machine" that accepts a given direction V (i.e., a vector) as input, and then spits out the linear map that links $T_xM$ to $T_{x + dx}M$, where dx is an infinitesimal step taken in the direction indicated by V. Such a machine that eats a vector and spits out some information is called a "one-form"; specifically in this case, we need a "matrix-valued one-form", since our one-form must tell us what linear transformation relates the two tangent spaces. We can call this one-form $\omega^a{}_b$, where the indices {a,b} indicate the components of the matrix. Then given some vector $V \in T_xM$, we can write

$$\omega^a{}_b (V)$$
to indicate sticking the vector V into $\omega^a{}_b$ to obtain the specific matrix that links us from $T_xM$ to the tangent space infinitesimally nearby in the V direction. Then this one-form $\omega^a{}_b$ essentially contains all the information to tell us how our manifold is put together.

The one-form $\omega^a{}_b$ might have some additional properties, as well. For example, it might have torsion, which is a measure of how much the tangent space twists around the direction of motion, and can be related to screw dislocations in crystals. It might also have curvature, which is related to what happens to a vector if you move it in a small loop, which is related to the motion of Foucault's pendulum as it travels in a loop around the globe. And if the manifold has a "metric" (which is a way to measure the lengths of vectors), then $\omega^a{}_b$ can have the property of non-metricity, which means that the lengths of vectors change as they are moved along from one tangent space to the next, or the property of metricity, which is where the vectors do not change length as they are moved around.

Classical Riemannian geometry considers only the case where $\omega^a{}_b$ is metric and torsion-free. This corresponds most nearly to our intuition of what "geometry" is. We can imagine building a manifold out of a bunch of tiny rods of fixed length (much like a geodesic dome). This means that we specify a metric on the manifold. Then we might imagine that all notions of curvature must come solely from our information about distances; a geodesic dome is curved only because its rods are assembled in a specific way, and not for any other reason. This is the same as demanding that a connection be metric and torsion-free.

But this need not be the case, and later mathematicians generalized the idea of geometry to include torsion. In principle, one could do away with the idea of metricity, also, but this is less useful, generally speaking. However, in the area of differential topology, we wish to construct quantities that hold for any connection, including non-metric ones, because then such quantities depend only on the topology of the manifold, and not on how we have chosen to represent that manifold mathematically.

 

[Paulibus]

Thanks for your very lucid outline of what a connection is, and what it does, Ben Niehoff. Your paragraph:

The one-form might have some additional properties, as well. For example, it might have torsion, which is a measure of how much the tangent space twists around the direction of motion, and can be related to screw dislocations in crystals. It might also have curvature, which is related to what happens to a vector if you move it in a small loop, which is related to the motion of Foucault's pendulum as it travels in a loop around the globe. And if the manifold has a "metric" (which is a way to measure the lengths of vectors), then can have the property of non-metricity, which means that the lengths of vectors change as they are moved along from one tangent space to the next, or the property of metricity, which is where the vectors do not change length as they are moved around.

covers most of the ground that I've been puzzling about, as you intended. I understood torsion, but non-metricity was something new for me, and describes (I think) the changes of scale I was enquiring about, for example changes that happen when some parts of a solid become hotter and expand more than others. Thanks again for this clarification.

The whole idea of a manifold as a description of geometry seems to me to rely on the continuous and smooth nature of the set of real numbers, carrying these qualities over from algebra to geometry. However such a set, per se, looks to me like a very convenient but nevertheless invented concept that specifically excludes discrete breaks in continuity, which are certainly not physically evident on the mesoscopic scale of the familiar dimensions we inhabit, measure and model with general relativity. But nature may be trickier than we know.

I sometimes wonder if continuity features inevitably on smaller scales inaccessible to inspection. If so, would the connection machine still work, say in approaches like the Causal Dynamic Triangulation models (Renate Loll) of spacetime that I've seen discussed elsewhere?

 

[Goldbeetle]

Ben,
that was a great summary on connections!
Thanks.
Goldbeetle