Talk:Metric tensor

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please do not delete this page. although there is an alternative approach to differential geometry, the component-based approach is fundamental to understanding the 'modern' approach, and the metric tensor is the fundamental definition of Riemannian geometry. What are the goals of this encyclopedia? what should they be? to be esoteric and create what some very few people might find to be 'elegant' and 'precise', or to make information accessible? I believe that it is the latter. Furthermore, I do not believe that the two goals are mutually exclusive. I believe, rather, that writting in a clear language that can readily be understood is a form of eloquence and 'perfection', and should be a priority. I am reminded of early medicine, when the professors turned the science of medicine into an esoteric and pedantic rite in pursuit of the luster of exclusive power. I would hate to see mathematics go the same way.

I like your attitude. I don't suppose you know what a tensor product is? See my comment on Talk:Tensor product. By the way, Kevin, you should sign your entries on talk with ~~~~, which is automatically replaced with a signature like the following. -- Tim Starling 04:23 Mar 14, 2003 (UTC)

perhaps we should explain the implicit summation and products of differentials more? - Gauge 05:41, 31 Jul 2004 (UTC)


Contents

[edit] Support for keeping this page

This page is simple, clear, and essentially self-contained. Browsing from the General Relativity entry, I was much happier with this page than with most other tensor-related explanations, which were so extravagantly reference-dependent as to be useless. I have a pretty solid general math and physics background, and doubt that a much more demanding presentation would serve a significant number of readers. Peter 19:50, 4 Feb 2005 (UTC)

I wonder if anybody has thought about the divide and conquer approach for writing mathematical objects like equations. It requires only the existence of parallel computers normally used in business (say stores).

Benjamin Cuong P. Nghiem bcnghiem@hotmail.com


Question:

Shouldn't be there a Sigma symbol in the first equation on this page? Under the square root. Something doesn't seem right without it in comparision to equation just under "The length of a curve reduces to the familiar calculus formula:" paragraph.

The summation convention is in force; so the Σ is implied. Charles Matthews 15:57, 4 August 2005 (UTC)

[edit] Is the first equation correct?

While I understand that the Einstein convention of summation over repeated indices is being employed in the first equation (so there is no need for the sigma); are those dt denominators within the square root right? 203.52.176.26 03:06, 6 December 2005 (UTC)


[edit] Trouble with non-metrics

For the Minkiwsi case as well as for the Schwarzschild as well as for any "metric" from relativity theory, the metric tensor is not positive definite and especially the formula for the length of a curve does not apply in the form given in the article for a proper, positive definite metric simply because the root function is not well-defined for non-negative values and using the principal square root convention takes us just into some different kind of trouble.

So I've cut the calculation that leads to imaginary lengthes from the article and put it here:

" Indeed, the distance between A = (0,0,0,0) and B = (1,0,0,0) is

L = \int_0^1 \sqrt{-(dx^0)^2} = \int_0^1 i dx^0 = i (with the principal square root convention, where \sqrt{-1} is equal to i and not -i).

We can check that in this case, the distance from A to B is equal to the distance from B to A :

L = \int_1^0 \sqrt{-(dx^0)^2} = \int_1^0 -i dx^0 = i (in this case, dx0 is negative).

"

The Infidel 19:37, 18 February 2006 (UTC)

The calculation as such (despite not done by me ;-) looks correct. Only the formula is wrong, but I havn't found a relayable reference with a correct one. The Infidel 17:24, 19 February 2006 (UTC)

[edit] Pseudo-Riemannian metrics

They are not "non-metrics"; they are pseudo-Riemannian. In the case of Minkowski and Schwarzschild, we have a particular type of pseudo-Riemannian: the "Lorentzian" metrics. These are all valid metrics. It is important that the general page on metric tensors includes all types of metric. I've changed the introduction to reflect these facts. I think I've also made it clearer.

I corrected a few flat-out errors. Also, because we can deal with metrics without coordinates, I took the coordinates out of the introduction. However, since coordinates are very useful in dealing with metrics, I just moved the coordinate expressions down a section. I think the page looks much better now, besides being more accurate. I hope you folks approve. MOBle 13:04, 22 February 2006 (UTC)

Not quite so. The bilinear forms that are not positive-semidefinite don't give rise to a mertric. On the other hand, physicist call the bilinear form a metric tensor without any regards of its signiture and often leave out the word "tensor" for convenience. I think this can be called separation by common language. The Infidel 18:40, 22 February 2006 (UTC)

[edit] Riemannian metrics

The term "Riemannian metric" now redirects to "Riemannian manifold". I think this will make curious math students happier. This way we can all be happy.

Tajmahall 07:08, 12 September 2006 (UTC)

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