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## Main Question or Discussion Point

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

Lets be clear on what a geodesic is as defined using metric geometry. A geodesic is a curve which provides a

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

I just noticed the comment

gel's response to my object was

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

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

I'd like to point out a mistake I made in my first response to JesseM. I wroteIf 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...

*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 tendancy 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 retrospect this seems to be a bit different than "An affine geodesic is more general than a metric geodesic." would you agree gel/Hurkyl?In any case, a geodesic is defined more generally as a curve which parallelly transports its own tangent vector.

I just noticed the comment

Everything I know about math tells me that this is wrong. Can you justify this for me please?Thanks.gel said:A spacelike geodesic neither minimizes nor maximizes the length, even locally.

gel's response to my object was

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?gel said: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 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 textHurkyl said:(check that it satisfies the axioms of a connection!)

**Tensors & Manifolds[/i], Robert H. Wasserman,**

I have it and it seems nice. I'm wondering if this would be the best text for me to

Thanks

Pete*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