Geodesics in metric geometry and affine

[pmb_phy]

In another thread the subject came up regarding whether the affine connection was more "general" in defining geodesics than the metric tensor. Hurkyl provided an illustrative example in the post https://www.physicsforums.com/showpost.php?p=1783469&postcount=116

Hurkyl - Let me take this opportunity to complement you on what I consider to be a clever example. I'm hoping to gain some of your expertise in differential geometry by being here. Too bad it doesn't work by osmosis huh? Lol!

gel - Thank you for raising the subject and sticking to your position while still being professional in your attitude. I have great admiration for people who can do that. Well done sir!

I didn't want to discuss it in the original thread because it would have taken the topic too far off topic. However this obviously deserves more attention. As such it seems that I could very well have been wrong on my position. I want to explore this here and discuss its relevance to relativity. It started with JesseM wondering about spacelike geodesics so I'll start with his question which was - https://www.physicsforums.com/showpost.php?p=1782432&postcount=88

If we have two points with a spacelike separation, is it meaningful to talk about a spacelike "geodesic" between them? It's not obvious to me that the concept of spacelike geodesics would make sense. Thinking about flat spacetime, if you draw a straight line between two events with a spacelike separation, which would presumably be the geodesic if one exists at all, I don't think this path would minimize the value of ds integrated along it...

I'd like to point out a mistake I made in my first response to JesseM. I wrote It actually does minimize the length of the path. That is wrong.

Lets be clear on what a geodesic is as defined using metric geometry. A geodesic is a curve which provides a stationary value for the worldlines arclength. Note: I dislike the use of the term "length" because it has the tendency to make some people think in Euclidean terms. But since the use is common I will be using it in this thread.

Now let's consider a spacelike worldline which is straight. Since such a worldline satisfies the geodesic equation it follows that it gives a stationary value for the arclength of the path. If there are any concerns about that, either from the mathematical point of view or the physical point of view let us address that here and clear it up.

Question for JesseM - What was it that gave you the impression that a straight spacelike worldline is not geodesic?

Now let me address my concern about affine connection defined geodesics being more "general" than metric defined ones. Allow me to take this from the start so that it can be seen where I was comming from and so that the reader can follow my line of reasoning. The topic was raised when I gel made the following statement

In any case, a geodesic is defined more generally as a curve which parallelly transports its own tangent vector.

In retrospect this seems to be a bit different than "An affine geodesic is more general than a metric geodesic." would you agree gel/Hurkyl?

I just noticed the comment

A spacelike geodesic neither minimizes nor maximizes the length, even locally.

Everything I know about math tells me that this is wrong. Can you justify this for me please?Thanks.

gel's response to my object was

when I said "more generally" I was referring to the fact that geodesics only require the concept of parallel transport to be defined. This can be defined by a metric, but only requires the existence of a connection, which makes it the more general definition.

I disagreed with this as can be seen from the discussion which followed. However it became apparent that gel may be right in the sense that there may be situations in which a variation can't even be performed such as those situations where the manifold is one dimensional. Was that your point gel or merely an example of your point?

I guess that's enough for now, i.e. enough to get the ball rolling. Hurkyl - One of my main goals here is to undertand your notation, the example you gave and the definitions you used. I don't recall ever seeing such a definition. You stated

(check that it satisfies the axioms of a connection!)

Where can I find these axioms? I'll look in Wald. Do you have, or are familiar with, the following text

Tensors & Manifolds[/i], Robert H. Wasserman, Oxford University Press, (1992)

I have it and it seems nice. I'm wondering if this would be the best text for me to suffer through. If you know of it do you have an opinion of it? Anybody?

Thanks

Pete

 

[Mentz114]

Pete,

something that's available now and touches on affine geometries is 'Gravitation, gauge theories and differential geometry' by Eguchi, Gilkey and Hanson which is available as a arXiv reprint. See page 241, Cartan structure equations.

Also, 'On the gauge aspects of gravity' by Gronwald and Hehl is available as gr-qc/9602013.

M

 

[pmb_phy]

Pete,

something that's available now and touches on affine geometries is 'Gravitation, gauge theories and differential geometry' by Eguchi, Gilkey and Hanson which is available as a arXiv reprint. See page 241, Cartan structure equations.

Also, 'On the gauge aspects of gravity' by Gronwald and Hehl is available as gr-qc/9602013.

M

Thank you my friend! I appreciate that information! :)

Pete

 

[Fredrik]

Where can I find these axioms? I'll look in Wald.

I don't think Wald explained it very well. A connection is just a map

$$(X,Y)\mapsto\nabla_X Y$$

that takes two vector fields to one vector field and satisfies

$$\nabla_{(fX+gY)}Z=f\nabla_X Z+g\nabla_Y Z$$

$$\nabla_X(Y+Z)=\nabla_X Y+\nabla_X Z$$

$$\nabla_X(fY)=Xf\cdotY+f\nabla_X Y$$

where f and g are real-valued functions defined on the manifold (i.e. scalar fields expressed without coordinates). You can find this definition on the Wikipedia page for "affine connection".

The map $Y\mapsto\nabla_XY$ is called a covariant derivative. I think Wald chose to define "covariant derivative" instead of "connection" when he was going to explain parallel transport. Maybe he even went directly for the covariant derivative associated with a coordinate system (i.e. the choice $X=\partial_\mu$). I don't remember, it's been a while since I read that part of Wald.

 

[Hurkyl]

Hurkyl - Let me take this opportunity to complement you on what I consider to be a clever example. I'm hoping to gain some of your expertise in differential geometry by being here. Too bad it doesn't work by osmosis huh? Lol!

It turns out that circles (and more generally, spheres and their higher-dimensional brethren) are the bane of anything done 'locally'; after a while, you get used to looking at them when searching for counterexamples!

The basic point is that winding around the circle often forces an extra constraint upon whatever you're studying -- in the case I was presenting, it forced all fields to be periodic. So while it was possible for me to solve the differential equation on any 'local' neighborhood contained in the circle, I was unable to patch those solutions together to obtain a global solution.

Incidentally, that's roughly how I constructed the counterexample; I chose two points A and B on the circle and chose different ways for parallel transport to act on the two different ways to get from A to B. (Well, I used an unspecified function z rather than choosing a specific action) I ran into a little difficulty, but I had gotten far enough to allow me to guess at a sufficiently general form for the connection, giving (after some algebraic tinkering to make it work): $\nabla_X Y = X Y' + X Y z$ for some scalar field z. I simply chose z = 1 for simplicity.

Incidentally, I wouldn't consider myself an expert; I've never attempted to study this subject in-depth. I simply read a lot, so I'm familiar with a wide variety of things.

Note: I dislike the use of the term "length"

I do too. Alas, I don't know any good alternative.

I just noticed the comment

A spacelike geodesic neither minimizes nor maximizes the length, even locally.

Everything I know about math tells me that this is wrong. Can you justify this for me please?Thanks.

In some (t,x,y) coordinates on 2+1-dimensional Minkowski spacetime, consider the path (0,0,0)-->(0,4,0). (i.e. the directed line segment originating (0,0,0) and terminating at (0,4,0)) It has length 4.

Now, consider the path (0,0,0)-->(1,2,0)-->(0,4,0). It has length 2 sqrt(3) ~ 3.5

Now, consider the path (0,0,0)-->(0,2,1)-->(0,4,0). It has length 2 sqrt(5) ~ 4.5


This is a symptom of the fact the metric tensor is not positive definite. Generally, we insist upon some positivity constraint on these kinds of objects. For example:

1. The distance function in a metric space¹ is required to satisfy $d(P, Q) \geq 0$, with equality only in the case that P = Q
2. The measure in a measure space is required to satisfy $\mu(S) \geq 0$ for any measureable set S
3. Inner products, in many contexts, are required to satisfy the positivity axiom $\langle v, v\rangle \geq 0$
4. In Riemannian² gemetry, the metric tensor is required to be an inner product on the tangent spaces, so it also must satisfy $g(v, v) \geq 0$ except in the case that v = 0

1: Not to be confused with the notion of a manifold equipped with a metric tensor
2: I'm specifically excluding pseudo-Riemannian geometry.

These positivity constraints are fairly powerful, and one of the main features required to prove many of the familiar properties we expect of them. The triangle inequality, for example.


However, in pseudoRiemannian geometry, we omit the positivity requirement on the metric tensor. In particular, the Minkowski metric tensor is not positive definite, nor are any of the signature +--- metric tensors we use in General Relativity.

In some heuristic sense, this means perturbing a path in some directions has a positive effect, and in others yields a negative effect.

In general, lots of things become a lot more subtle when you omit positivity constraints, and instead deal with indefinite objects. This happens even in fairly elementary settings -- for example, a convergent infinite sum of positive numbers always converges absolutely, allowing you to do lots of things without chaing the value of the sum (such as rearranging its terms). However, a convergent infinite sum of mixed positive and negative numbers can converge conditionally, which forces you to be fairly careful about how you manipulate it.

I guess that's enough for now, i.e. enough to get the ball rolling. Hurkyl - One of my main goals here is to undertand your notation, the example you gave and the definitions you used. I don't recall ever seeing such a definition. You stated

Where can I find these axioms? I'll look in Wald. Do you have, or are familiar with, the following text

I learned most of what I know about differential geometry from Spivak's Differential Geometry. I really like the format of book 2, in which it presents several different formulations of the subject, and each one is used in turn to recreate the basic notions (e.g. geodesics, curvature, derivatives) and prove the basic theorems.

Wikipedia's definition seems adequate to me. (I don't have Spivak handy, having returned it to the library since I haven't looked at it in a while) Is that specifically the notation you were looking for?

 

[gel]

I think it's also worth pointing out that geodesics in Riemann monifolds are defined by more than just locally minimizing the distance.

If f:R->M is a geodesic on manifold M, then it is required that the length of the curve between any two points s,t is proportional to |t-s|. Equivalently, the magnitude of the tangent vector g(df/dt,df/dt) is constant. This is automatically true if you use the definition that it parallel transports its tangent vector.

In Hurkyls example (of a connection on the circle), the reason why the connection couldn't be a metric one is that parallel transport around the circle and back to the start rescales the vector, so g(df/dt,df/dt) can't be constant for any metric g.

However, the geodesics in his example only differ from the standard geodesics on the circle by a reparameterizing their paths.
Makes me wonder what would be the simplest example of an affine connection on a manifold whose geodesics are not given by a metric, even after reparameterizing them. Maybe geodesics on the Heisenberg group? Is there a 2d counterexample?

 

[gel]

I think it's also worth pointing out that geodesics in Riemann monifolds are defined by more than just locally minimizing the distance.

If f:R->M is a geodesic on manifold M, then it is required that the length of the curve between any two points s,t is proportional to |t-s|. Equivalently, the magnitude of the tangent vector g(df/dt,df/dt) is constant. This is automatically true if you use the definition that it parallel transports its tangent vector.

In Hurkyls example (of a connection on the circle), the reason why the connection couldn't be a metric one is that parallel transport around the circle and back to the start rescales the vector, so g(df/dt,df/dt) can't be constant for any metric g.

However, the geodesics in his example only differ from the standard geodesics on the circle by a reparameterizing their paths.
Makes me wonder what would be the simplest example of an affine connection on a manifold whose geodesics are not given by a metric, even after reparameterizing them. Maybe geodesics on the Heisenburg group? Is there a 2d counterexample?

 

[pmb_phy]

In some (t,x,y) coordinates on 2+1-dimensional Minkowski spacetime, consider the path (0,0,0)-->(0,4,0). (i.e. the directed line segment originating (0,0,0) and terminating at (0,4,0)) It has length 4.

Now, consider the path (0,0,0)-->(1,2,0)-->(0,4,0). It has length 2 sqrt(3) ~ 3.5

Now, consider the path (0,0,0)-->(0,2,1)-->(0,4,0). It has length 2 sqrt(5) ~ 4.5

Oh man! I knew that too. Oy! I blame it on the pain killers. :)

Of course! What gel was saying is that there the path is stationary. However I tend to doubt this locally. If I recall correctly the proper time between two events (as an example consider two timelike worldlines) which are infinitely close to each other is a maximum. But I think whoever said that was thinking of a particular path. E.g. for a geodesics on a sphere it would be the shortest distance between two points a distance dl apart. Lifgarbagez mentions something similar and briefly in his mechanics text, i.e. for closely spaced points the action really is minimized rather than stationary.

Pete