GR Metric Meaning: Unpacking Confusion

[Rob Woodside]

Sorry for the confusion, you are right. It is contraction and the indices must be upstairs and downstairs. I've seen things like Ricci = Tr Riemann meaning to contract the first and third indices or what ever convention one is using. I guess one should write Ricci = Con Riemann, but I've never seen it. At least I didn't use "Spur"

Pete, yes I 'd like a copy of Ohanian's paper, if it is not too much trouble. I mentioned this thread to a friend and he said there is nothing local about the curvature ! I exploded with: "The whole point of curvature was that the rotation of a vector parallel transported around a small arbitrary loop was given LOCALLY by the curvature tensor" He responded with: "It depends on second partials of the metric and that's non local" In reply to "Wadayamean?", he pointed out that you can't MEASURE curvature at a point. It requires the separation of neighbouring geodesics and that occurs over a small region and not at a point. The mathematical notion of local is "at a point". The physicist's idea of local is "in a small neighborhood" To sum up: the Curvature tensor is well-defined at each point and changes little over a small neighbourhood and is measured by geodesic separation (or vector rotation around a loop, etc) over this neighbourhood. Does this help or merely add more confusion?

 

[pmb_phy]

Pete, yes I 'd like a copy of Ohanian's paper, if it is not too much trouble.

It's on the way. Check your e-mail. I sent 3 articles. One is Ohanian's article. The other is a criticism of Ohanian's article by Walstad. The last is Ohanian's response to Walstad.

I mentioned this thread to a friend and he said there is nothing local about the curvature !

Yep. Been there, done that. I had the same argument with other people too.

I exploded with: "The whole point of curvature was that the rotation of a vector parallel transported around a small arbitrary loop was given LOCALLY by the curvature tensor" He responded with: "It depends on second partials of the metric and that's non local"

That is entirely an incorrect statement.

In reply to "Wadayamean?", he pointed out that you can't MEASURE curvature at a point.

He's showing that he thinks that local means at a single point where in fact the term local, as the term is used in differential geometry, refers to something in a small neighborhood. In fact recall the definition as given in Differential Geometry, by Kreyszig

(page 2) local – A geometric property is called local, if it does not pertain to the geometric figure as a whole but depends only on the form of the configuration in an (arbitrarily small) neighborhood under consideration

The author actually gives an example stating "For instance, the curvature of a curve is a local property." which readily extends to Riemann curvature as being a local property referring to a small neighborhood. Therefore since derivatives are defined in a limit in small region and not with regard to a single point only it follows that "local" applies to a point and "nearby" surrounding points.

It requires the separation of neighbouring geodesics and that occurs over a small region and not at a point.

Exactly. Bravo.

The mathematical notion of local is "at a point".

Sorry but I have to disagree with that assertion.

Does this help or merely add more confusion?

I wasn't confused in the first place. I was merely pointing out that this point is the subject of debate in GR.

Pete

 

[Rob Woodside]

Thanks Pete, I'll check for the papers on Monday at the office.

I'm surprised at Kreyszig. A derivative senses changes at nearby points and can't be found without knowing what goes on at the nearby points, but it lives at a specific point. His notion of local seems to be in contrast with global. Live and learn.

 

[pmb_phy]

Thanks Pete, I'll check for the papers on Monday at the office.

I'm surprised at Kreyszig. A derivative senses changes at nearby points and can't be found without knowing what goes on at the nearby points, but it lives at a specific point. His notion of local seems to be in contrast with global. Live and learn.

Kreyszig is doing the typical thing - He's speaking about what is normally considered "local" in mathematics, i.e. an $\epsilon$ neighborhood. An $\epsilon$ neighborhood is not a global region. How else could anything be a local property? A propertys is not something a point on a curve has.

This is not unique to Kreyszig. See -- http://mathworld.wolfram.com/Local.html

In the end what really matters is what an author means by "local" when they use it. But its nice to stick to well defined terms rather than to redefine terms to fit our ideas of them.

How would you define local property?

Pete

 

[jcsd]

Now I'm confused I always though that local was defined in the limit.

 

[pmb_phy]

Now I'm confused I always though that local was defined in the limit.

Same here.

If anyone finds a mathmatical sourc/text which states otherwise please let me know.

Pete

 

[Rob Woodside]

How would you define local property?

Pete

The property lives at the point but depends on the neighbourhood. The derivative from first year calculus is the paradigm.