Talk:Metric tensor/Archive 1

Archive 1

Request

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)

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)

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

$L=\int _{0}^{1}{\sqrt {-(dx^{0})^{2}}}=\int _{0}^{1}idx^{0}=i$ (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 :

$L=\int _{1}^{0}{\sqrt {-(dx^{0})^{2}}}=\int _{1}^{0}-idx^{0}=i$ (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)

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)

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)

Accessibility

Resolved

I just read this article, but I still have a rather basic question; what the heck is a metric tensor? I understand that you have to give a precise definition for College math people and such, but could you put a working definition in layman's terms so that those of us that don't know much about college math can understand it? Thanks! Ahudson 18:50, 6 February 2007 (UTC)

Sorry, this article is in such terrible shape. It badly needs an overhaul. A metric tensor is an object defined on a manifold (like a sphere or torus or just ordinary 3-dimensional Euclidean space) that allows one to define geometric properties of that space; things like volume, lengths of curves, angles between curves and so on. Ultimately, it can tell you how lumpy, flat, or curved a space is. To be more precise, every point on a manifold has a vector space associated with it called the tangent space. A metric is something which defines a scalar product (dot product/inner product) on each tangent space. These scalar products aren't allowed to jump widely from point to point but must vary smoothly as one goes around the manifold. Moving from this basic definition to understanding geometric properties takes a lot of work and is the subject of Riemannian geometry. -- Fropuff 19:18, 6 February 2007 (UTC)

Ok, thanks! I think I get it now. Ahudson 16:47, 2 March 2007 (UTC)

intro reworking — v1

A metric tensor is an object defined on a manifold (like a sphere or torus or just 3-dimensional Euclidean space) that allows one to define geometric properties of that space, such as volume, lengths of curves, and angles between curves. It can also be used to determine how curved a space is at any given point.

Every point on a manifold has a vector space associated with it called the tangent space. A metric is something which defines a scalar product (dot product/inner product) on each tangent space. These scalar products vary smoothly as one goes around the manifold. Kevin Baas^talk 23:14, 3 March 2007 (UTC)

Relation to energy inner product

In functional analysis there's an energy inner product, written $\langle Bu|v\rangle$, which appears to be the same sort of expression as $g_{ij}\,dx^{i}/dt\,dx^{j}/dt$. That is, both are vector-matrix-vector products that yield a scalar. Is it accurate to say that the B in an energetic inner product is the metric tensor for that space? —Ben FrantzDale 13:05, 16 April 2007 (UTC)

kind of. one can be view these via a general construction of inner product spaces (see, for example, positive definite kernel). the simplest is an inner product on Rⁿ induced by a positive matrix. assigning a positive matrix smoothly to (the tangent space of) each point in a manifold gives a Riemannian metric. if your operator B, acting on some Hilbert space H, is bounded and > 0, then <x, Bx> is an inner product on H. the correponding norm "renorms" H. for example, if you do this in R², IIRC the unit circle in this new norm is some ellipse with axes parallel to the eigenvectors of B. the operator B you refer to seems to be unbounded in general, but the idea is the same, although i doubt it's called a "tensor" in functional analysis literature. Mct mht 01:32, 17 April 2007 (UTC)

Non-L2 norms?

It looks like there is an assumption that $\|dx\|={\sqrt {g_{ij}dx^{i}dx^{j}}}$. This is the case with a Hilbert space, but not with a Banach space in general. Could one have an equivalent of a metric tensor for norms involving other powers? For example, something like a max norm,

$\|dx\|:=\max _{i}g_{ij}dx^{j}$

or a one-norm,

$\|dx\|:=\sum _{i}g_{ij}dx^{j}.$

So, is this something that gets done, or is a metric tensor really only used for $L_{2}$-like norms? —Ben FrantzDale 03:17, 27 April 2007 (UTC)

looks to me like the norms you have there won't be given by a tensor; there's something funny with those expressions (maybe the tensor and the Jacobian got mixed up?). but yes, there are non-Hilbert space norms. see Finsler manifold. Mct mht 04:05, 27 April 2007 (UTC)

Cool. That's exactly what I meant. I'll cross link this page with that one.

As for the norms I wrote above, what do you mean that they "won't be given by a tensor"? Is my nomenclature wrong? Certainly I could find the maximum or sum of $g_{ij}dx^{j}$, right? Is $g_{ij}$ not strictly-speaking a tensor in that usage? Thanks again. —Ben FrantzDale 04:24, 27 April 2007 (UTC)

looks like they would violate the axioms of a norm, say positivity.

what i meant was that, locally you have a positive definite matrix G > 0 that induces an inner product on the tangent space V. denoted by (V <, >_G). let J be the Jacobian, so G = J^TJ, and Ran(J) be its range, then (V <,>_G) is really isomorphic, in the sense of inner product spaces, to (Ran(J), <, >) where <, > is the usual Euclidean inner product. so abandoning <,>_G on V is abandoning the usual <,> on Ran(J). and to introduce a new norm, perhaps one should be talking about J. for example, the ∞-norm on Ran(J) would be

$\|dx\|:=\max _{i}|J_{ij}dx^{j}|.$

maybe someone would be willing clarify for us these issues. Mct mht 04:56, 27 April 2007 (UTC)

That makes sense. I forgot to include absolute value in there. Thanks. —Ben FrantzDale 06:23, 27 April 2007 (UTC)

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)

Yes, this equation is correct. Here is the reasoning: (x₁(t), ..., x_n(t)) is the equation describing the the curve in the local coordinate system, a ≤ t ≤ b. Assume that functions x_i(t), 1 ≤ i ≤ n are differentiable in the open interval a < t < b. Let us now calculate the distance, ds, between two infinitesimaly close points t and t + dt on this curve. If this curve were in n-dimensional Euclidean space, E_n then (using Einstein summation convention)

$ds={\sqrt {({dx_{i} \over dt}dt)({dx_{i} \over dt}dt)}}={\sqrt {{dx_{i} \over dt}{dx_{i} \over dt}}}dt$.

(This equation is generalization of Pythagoras theorem to n-dimensions). In a general Riemann space we have to add the metric tensor:

$ds={\sqrt {g_{ij}{dx_{i} \over dt}{dx_{i} \over dt}}}dt$

And, finally, the length, L, of the curve segment between a and b is the integral of ds from a to b.

TomyDuby 07:56, 14 September 2007 (UTC)

accidental removal?

something funny happened in [this edit], especially considering it was Michael Hardy. material got removed that shouldn't have been. i am gonna leave a message on his talk page. Mct mht 00:22, 20 September 2007 (UTC)

How to calculate the matric tensor G?

I miss a simple explanation how to do this. TomyDuby 07:58, 14 September 2007 (UTC)

To compute the metric tensor from a set of equations relating the space to cartesian space (g_ij = δ_ij: see Kronecker delta for more details), compute the jacobian of the set of equations, and multiply (outer product) the transpose of that jacobian by the jacobian.

$G=J^{T}J$

Was that helpful? Kevin Baas^talk 01:59, 17 September 2007 (UTC)

Yes, it is helpful.

Thanks. TomyDuby 21:07, 21 September 2007 (UTC)

Question related to metric tensor...

How do the dot products of covariant basis vectors produce a contravariant component of the metric tensor and vice versa? I.e why do the variance types switch through taking the dot product of basis vectors? I don't see it coming out in the math. Thanks.

If I understand your question, you're misusing "dot product". A dot product takes two vectors and gives you a scalar (which has no variance type). Rather, I think you mean to talk about taking the contraction of the basis vector with the metric.

In index notation, we could write this as $g^{\mu \nu }v_{\nu }=v^{\mu }$, where $v_{\nu }$ is the covariant form of the basis vector. Then, $v^{\nu }$ is the contravariant form. Alternatively, we could get the covariant form from the contravariant form: $g_{\mu \nu }v^{\nu }=v_{\mu }$.

Now note that, because of how the metric components transform, you get two extra factors of the Jacobian or inverse Jacobian, which is why the variance type switches. That's not hard to check. (Note that some times it's confusing to have indices on basis vectors to indicate which basis vector you're dealing with, so I've left them off. Is that where your confusion was coming from?) --131.215.123.98 (talk) 17:32, 25 November 2007 (UTC)

Thanks for replying. I think you might be answering my question, but it's going a few steps over my head. Let's back up a bit, and I'll first make more clear the context with which I'm using to approach this metric tensor concept and where my confusion is arising. Specifically, I'm reading this article http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM2002211716.pdf He is saying there that the dot products of any two of the bases in one of the variances gives you a corresponding component in the metric tensor (of the opposite variance). It is at the bottom of page 20 and I can't see where this conclusion comes from based on any of the previous material. It is the sole point in the whole article I can't wrap my head around. Cheers. —Preceding unsigned comment added by 160.39.130.184 (talk) 03:24, 26 November 2007 (UTC)

Wow

This article is breathtakingly bad. It lacks the one thing it should without question have: a clear mathematical definition of what a metric tensor is. Namely, a section of the tensor product of the cotangent bundle of M with itself, which at each point gives us a positive-definite symmetric bilinear form (i.e. an inner product). See e.g. http://planetmath.org/encyclopedia/MetricTensor.html. The introductory paragraph should not attempt a precise mathematical definition, rather giving the intuition of the idea. But later on, at least a paragraph needs to be devoted to exactly what a metric tensor is, why it's of type (0, 2), how we get from g to a matrix using coordinates, etc. And this should probably be the first body paragraph, before talking about applications -- how to calculate things with metrics and so on. I would also note that the article as it stands does things in coordinates wherever it can, which is a remarkably bad idea. Sometimes coordinates are necessary, but it's better, more canonical, less cluttered, easier to understand, to do things coordinate-free when possible. For example, instead of saying the cosine of an angle is g_iju^iv^j/sqrt(g_iju^iu^j*gijv^iv^j) (by the way, the absolute values are not needed), why not say it's g(u, v)/sqrt(g(u, u)*g(v, v))? I would note that this article used to have a "definition" section, some months ago, which Michael Hardy for some reason removed. Can't exactly fathom why that would be. Somebody want to fix this article? If there are no takers, I'll try and have a go at it, but I don't really have much time to devote to it and it wouldn't get done any time soon. Kier07 (talk) 16:56, 9 May 2008 (UTC)

Go for it. —Ben FrantzDale (talk) 02:13, 10 May 2008 (UTC)

Why not go back to the pre-"Michael Hardy" point and use that as a new starting point?RandomTool2 (talk) 16:45, 10 May 2008 (UTC)

(0,2) tensor

what is a (0,2) tensor? This should be explained or have a wikilink. Thanks. 131.111.28.28 (talk) 12:50, 20 May 2008 (UTC)

While I have never encountered this (0,2) terminology, I would assume that it refers simply to the fact that the (covariant) tensor has 2 lower, and 0 upper indicesRandomTool2 (talk) 19:08, 1 June 2008 (UTC)

Connectedness

If this article is not going to assume that the manifold is connected, then it should at least address the definition of the value n that is used without mention. Are we assuming the manifolds are equidimensional? RobHar (talk) 19:19, 3 August 2008 (UTC)

Yes, each manifold is an n-dimensional manifold for some natural number n. JRSpriggs (talk) 04:28, 4 August 2008 (UTC)

The article Manifold allows for non-connected manifolds with connected components with different dimensions. Anyway, Silly rabbit has taken care of my comment as it pertains to this article. Thanks. RobHar (talk) 05:14, 4 August 2008 (UTC)

clarity

In 4 years,

"To compute the metric tensor from a set of equations relating the space to cartesian space (gij = δij: see Kronecker delta for more details), compute the jacobian of the set of equations, and multiply (outer product) the transpose of that jacobian by the jacobian."

G = J^T J

has become:

The induced metric tensor for a smooth embedding of a manifold into Euclidean space can be computed by the formula

G = J^T J

where J denotes the Jacobian of the embedding and J^T its transpose.

The following was inserted into the intro, as if to clarify:

In other terms, given a smooth manifold, we make a choice of positive-definite quadratic form on the manifold's tangent spaces which varies smoothly from point to point. The manifold, equipped with the metric tensor (the varying choice of quadratic form), is called a Riemannian manifold and in this context the metric tensor is often called a Riemannian metric.

And everything from "Arbitrary Coordinates" on down was added to the article.

Now I have nothing to say about anything below "arbitrary coordinates", but i take issue with the other two things: I don't see them as improvements.

The first part: most people reading the article have no idea what an "induced" metric tensor is, what a "smooth embedding" is, why it is relevant, nor what "euclidean space" is. Or how to interpret the formula given. Though i'm sure many a reader comes to the article with a set of things they easily recognize as "equations" that relate the space to "cartesian space", a concept they've very familiar with, and want to know how to compute the "metric tensor" from them. To those people, the new version of the paragraph isn't much help.

As to the second part: I can tell right away that english isn't the author's first language. And again, people shouldn't be expected to know what a "form" is, what a "tangent space" is, etc. And if I were to take a poll of how many people those two sentences makes any sense to, I know that I would get a very, very, very small set of hands. I know I wouldn't be raising mine.

I'm quite compelled to revert the first part back four years, and remove the second part. Kevin Baas^talk 01:58, 17 September 2007 (UTC)

i am going to revert the last 4 edits (adding back the passage and removing one wrong edit). i didn't write that passage but it seems fine and is an accurate desccription of what the metric tensor is, as opposed to merely how it's calculated. saying the author's first language doesn't seem to be english is an inexplicable cheapshot and just bs. not understanding something ain't a valid reason to chuck it in this case. Mct mht 23:59, 19 September 2007 (UTC)

No, I wasn't making a cheapshot, I being serious. It honestly looks to me like the author's first language isn't english. The mistakes in the construction of the sentences were those that are commonly made when someone is speaking/writting in a language that is not their native language:

• Miswording or not knowing idioms: "In other terms" should be "In other words".
• Redundancy that adds confusion showing a poor understanding of proper sentence syntax: "smooth manifold...which varies smoothly from point to point" - using proper english rules for the interpretation of syntax, this tells the reader that there is something in the sentence that varies smoothly from point to point, but it is not the "smooth manifold". However, the sentence does not make clear what that noun is.

Additionally:

• the "In other terms (sic)..." sentence is NOT a restatement of the sentence that comes before it; i.e. it is NOT "in other words". again, confusing the reader.
• Each sentence has more than 7 ideas in it, thus breaking the rules of good writing style. I'd guess that the author's native language is German, as I've heard their sentence syntax is pretty loose and their sentences often relatively complex.

Kevin Baas^talk 22:31, 20 September 2007 (UTC)

Just a quick remark on the "other terms" locution. This is indeed a well-known idiomatic expression in the English language. See, for instance, Google scholar which turns up over 40,000 hits. It seems from the Scholar that it tends to turn up mainly in scientific contexts, and I'm guessing that you don't have a science background from the overall timbre of your objections to the article. siℓℓy rabbit (talk) 16:44, 6 August 2008 (UTC)

Oh, and I put my minor corrections back. I assume your reverting them was a mistake, as their pretty non-controversial edits. But I left the disputed text at the status quo so that it can be discussed peacefully. Kevin Baas^talk 22:34, 20 September 2007 (UTC)

Todo list

Here is a list of things I would like to add to the article. I may continue to add to this list, or strike through them as I go as appropriate. siℓℓy rabbit (talk) 03:19, 11 August 2008 (UTC)

1. Add a discussion of applications in modern mathematics and physics
2. Discuss the induced metric on an embedded submanifold in Euclidean space at first, and then in a more general manifold equipped with a metric.
3. Something on Clifford algebras seems appropriate.
4. A variety of generalizations should be discussed, including Finsler manifolds, Hermitian metrics, and infinite-dimensional generalizations of the metric tensor concept.
5. The volume form (in coordinates at least, with an indication of how to get this intrinsically)

Vector transformation

I'm not sure if I should post this here or in Talk:Inner product space. When I see $g_{ij}x^{i}y^{j}$ I want to think about it as transforming x and y and then taking their good-old-fashioned inner product. But it seems like that transformation, assuming g is self-adjoint, is going to be ${\sqrt {g}}$. That is, consider

${\begin{bmatrix}1&0\end{bmatrix}}{\begin{bmatrix}2&0\\0&2\end{bmatrix}}{\begin{bmatrix}1\\0\end{bmatrix}}={\begin{bmatrix}1&0\end{bmatrix}}{\begin{bmatrix}2\\0\end{bmatrix}}=2$.

If I wanted to transform the vectors separately first, I would have to do

${\begin{bmatrix}1&0\end{bmatrix}}{\begin{bmatrix}{\sqrt {2}}&0\\0&{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}{\sqrt {2}}&0\\0&{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}1\\0\end{bmatrix}}={\begin{bmatrix}{\sqrt {2}}&0\end{bmatrix}}{\begin{bmatrix}{\sqrt {2}}\\0\end{bmatrix}}=2$.

In other words, the inner product space you get by using a metric tensor, g, is the same as what you'd get by transforming all vectors by ${\sqrt {g}}$. Is that a reasonable way of thinking about it? —Ben FrantzDale 06:17, 30 April 2007 (UTC)

essentially, yes. G is > 0, so G = B*B for some invertible B. there's a natural isomorphism between (V, <,>_G) and (Ran(B), <,>). take v in (V, <,>_G) to Bv in (Ran(B), <,>). this map preserves the inner product and so is an isomorphism. replace B by the positive square root G^1/2 and we get something that seems pretty close to what you're saying.

same thing goes through when G is ≥ 0. but now <,>_G may be degenerate and Ran(B) may be a proper subspace. the map is then [v] -> Bv, where [v] is an equivalence class modulo the degenerate subspace. Mct mht 06:41, 30 April 2007 (UTC)

Thanks. I'm not sure what you mean by Ran(B). What operator is that? —Ben FrantzDale (talk) 14:14, 25 August 2008 (UTC)

Duah, it's Range (mathematics). Now that all makes sense. Thanks again. —Ben FrantzDale (talk) 16:51, 25 August 2008 (UTC)

Redundant whitespace in equation number

To JRSpriggs:

The original author, siℓℓy rabbit, of the equations does not oppose to remove redundant whitespace follow each {{EquationRef}} as you can see in User_talk:Silly_rabbit#Metric_tensor and User_talk:Justin545#Go_ahead. Beside, I think the article consistency and appearance are more important than the source clarity. Typical editor would not add redundant whitespace within equation numbers in general as you can see in this example. - Justin545 (talk) 02:02, 6 February 2009 (UTC)

To Justin545: OK. I am sorry that I did not understand the purpose of your change earlier. JRSpriggs (talk) 08:11, 7 February 2009 (UTC)

jacobian of cartesian to polar

in the portion where the metric tensor in euclidean space in polar coordinates is derived, shouldn't the jacobian be multiplied times the identity matrix and then times its inverse? rather than times its transpose —Preceding unsigned comment added by 131.156.85.112 (talk) 15:38, 4 September 2009 (UTC)

No. Your suggestion implicitly assumes that the metric tensor is a mixed tensor with one covariant index and one contravariant index. The metric tensor actually has two covariant indices. JRSpriggs (talk) 17:23, 5 September 2009 (UTC)

Relation to metric (mathematics)

Shouldn't this article say something about the relation between the metric tensor and the metric of a metric space? In fact, it may be helpful to lay readers to mention in the lead that the metric tensor is (in some way) the infinitesimal form of the the metric function. Anybody know a good source that discusses this in a coherent way?TimothyRias (talk) 11:23, 18 January 2011 (UTC)

Hi Tim. I've incorporated your idea into the lead. The section on arc length should be expanded to include a paragraph or two on Riemannian manifolds and metrics as well. Regarding your question, if I recall, Cartan's Geometry of Riemannian spaces takes this point of view, although it is not easy reading (so probably not good as a source for expository ideas). Sławomir Biały (talk) 12:46, 18 January 2011 (UTC)

Statement in lead: "smallest length"

The (newly modified) statement in the lead, "The curve connecting two points that has the smallest length is called a geodesic", is not in general correct, and in particular is incorrect with an indefinite metric tensor such as in general relativity. A possible correction would be to replace "smallest" with "extremal". — Quondum^☏ 22:19, 24 August 2012 (UTC)

The paragraph in question is talking about the metric space structure on Riemannian manifolds. Bringing indefinite metrics up here is just wrong and confusing. And regardless of signature, it's obviously nonsense to talk about curves of "least or greatest" length. Sławomir Biały (talk) 23:38, 24 August 2012 (UTC)

My perspective is clearly in disagreement with the lead with respect to the metric tensor being defined specifically in relation to a Riemannian manifold (A manifold equipped with a metric tensor is known as a Riemannian manifold.), and by implication to being positive-definite (i.e. a Riemannian metric). Is a metric tensor positive-definite by definition, or isn't it? You'll see that the article geodesic duly qualifies the applicability of the case of a "shortest path" to a Riemannian metric, and the lead of this article should do the same. — Quondum^☏ 04:54, 25 August 2012 (UTC)

definition of r_u and r_v

I find the definition of r_u and r_v a bit short ("...and the subscripts denote partial derivatives...") and completely missed it when I tried to understand the article. I (thought I) am an advanced beginner, and an a bit more explicit explanation would have saved me looking up the topic in my old lecture notes... 178.10.184.164 (talk) 18:14, 26 September 2012 (UTC)

Yes, defining them without using the symbols is rather terse and unobvious. I've tweaked it. — Quondum 18:33, 26 September 2012 (UTC)

Unobvious inverse-square law

Therefore, a far away observer (at infinity) would feel a gravitational pull proportional to

${\frac {GM}{r^{2}}},$.

I don`t understand this `Therefore`...

I've removed this as it is an unobvious result. The previous matrix may also fall into the same category, though it might serve as an example of a metric. — Quondum 00:22, 21 June 2013 (UTC)

Necessarily on a manifold?

The concept of a metric tensor makes perfect sense as a tensor (as opposed to a tensor field). While the most notable uses may require it to be continuous on a manifold, it seems that its definition should be as a tensor on a vector space, rather than only as the section of a bundle on a manifold (I don't even properly understand the latter phrase). One could define it fairly simply as a nondegenerate symmetric bilinear form on a vector space (with no reference to tangent spaces or manifolds). If desired, one could then impose the condition of continuity over a manifold if extended to the context of differentiable manifolds as a metric tensor field, although even then, the continuity constraint seems like an unnecessary part of the definition, just as a function is not defined to be continuous. The a priori restrictions appear to achieve nothing (other than convenience of not having to do it later in contexts such as general relativity), so one must ask whether this the only notable definition of a metric tensor. I understand that its motivation originally comes from the need to define a quantity that integrates to a path length, but since its use extends very notably to finding the magnitude of vectors at points on the manifold with no reference to the differential structure, or equivalently giving a natural mapping of vectors to covectors, the less restrictive definition seems to be well-motivated. — Quondum 15:14, 29 June 2013 (UTC)

Let's not go there. The manifold case is by far the most common and easiest to motivate. Mathematicians already study quadratic forms and symmetric bilinear forms as such. Sławomir
Biały 15:23, 30 November 2015 (UTC)

Equation 3 derivation

Could someone please add more steps to the derivation of the matrix transformation equation (3). This seems to be a critical result but the derivation via the chain rule isn't immediately clear. Perhaps have the steps laid out in a hidden/expandable section. Thank you. 203.110.235.6 (talk) 22:40, 28 August 2016 (UTC)

Opening sentence

If one is going to call the metric tensor a function (which feels odd, but it certainly technically correct), hadn't you better say it takes a point AND two tangent vectors at that point? It is simply not correct to say that it is a function that takes two tangent vectors. (For example, you can't plug in two tangent vectors from different points).76.121.187.193 (talk) 22:52, 5 July 2012 (UTC)

The lead specifies a "function defined on a manifold", which is to say that it is defined at (or as a function of) every point of the manifold. Given that it is a point-specific function, it already conforms to your requirement. Where the vectors come from is immaterial to the definition, but implicitly they would be from a tensor field over the manifold. I think the wording is fine as is. — Quondum^☏ 07:43, 6 July 2012 (UTC)

No, I don't think it is fine. It lacks clarity and depends on the reader to be aware of what a metric tensor on a manifold is already in the opening sentence.

The whole lead also relies on the perpetuated confusion of tensors and tensor fields. I'd recommend beginning with metric tensor on a vector space, then generalizing to metric tensor fields on manifolds, and finally to introduce the perpetuated confusion of terminology (sometimes tensor = tensor field) making the novice aware of it. Also, the Euclidean metric on ℝⁿ could well be introduced in the lead. YohanN7 (talk) 12:26, 30 November 2015 (UTC)

I don't think this is a good idea. A metric tensor does take as input a pair of tangent vectors to a manifold (at the same point), and outputs a number in a way that generalizes the dot product. This is what the first sentence conveys. The second sentence says that the metric tensor is used to measure distances. Starting with the metric tensor on Euclidean space would be very misleading: readers would get the idea that this is the main example of a metric tensor, when in actuality we are interested in metrics on manifolds (e.g., surfaces). Technical niceties like the distinction between tensors at a point and tensor fields are not likely to be so important to the average reader, and already are dealt with in the more technical parts of the article. Sławomir
Biały 12:59, 30 November 2015 (UTC)

Functions on manifolds (like in the first sentence) take points in the manifold for input, not vectors from nowhere. The function may then output a new function (a tensor) accepting vectors from the appropriate tangent space as its input. This is why I think it is potentially confusing. The confusion between tensors and tensor fields is everywhere if you insist on that first sentence being correct.

Starting with things defined on vector spaces, then generalizing to tensor fields is the way e.g. John M. Lee mostly consistently does it (not only for this concept). Understanding what the thing is on a single tangent space (inner product or dot product) is necessary for understanding what the thing is in general i m o. You probably argue that the first sentence does the job. I don't agree.

The Euclidean metric is also the first example in Lee's book. I can't see why this would be inappropriate. It is the simplest example. It is also a full-blown tensor field on the manifold ℝⁿ. YohanN7 (talk) 13:33, 30 November 2015 (UTC)

Certainly, if we were doing a formal definition, then we would start by defining an inner product in a vector space, and then allow that to vary from point to point. But the first sentence of a lead is not supposed to be structured like a formal definition. It is supposed to be read by all readers of an article. As such, the first few sentences should convey the idea that a metric tensor allows vectors to be measured and, by integration, to allow lengths on manifolds to be computed. This conveys the essence of what a metric tensor is. If we were instead to say something like "A metric tensor is like an inner product on $\mathbb {R} ^{n}$", I feel that this leaves out the most fundamental aspect of things. Regarding the use of the word "function" here, the metric tensor is a function: it takes two vectors, and returns a number. That's a function. It's defined on a manifold. I see nothing wrong with this, unless we try to over-analyze it and make the sentence say something it's not. Sławomir
Biały 15:21, 30 November 2015 (UTC)

I maintain that the opening sentence is bad. Functions on X expect arguments x ∈ X, nothing will change that, but I have seen worse. As you say, a metric tensor is a function. It takes two vectors as argument. Yes, yes, yes. Is it a function on the manifold? No, no, no. Taking this in combination with, by the mathematician, the understood terminological confusion of tensors and tensor fields, the opening sentence has at least the potential of not being optimal, no? I am not going to push my point further. It is, after all, just a sentence. YohanN7 (talk) 10:15, 1 December 2015 (UTC)

Yes, functions have inputs and outputs. The first sentence says that the inputs of this function are tangent vectors and the output is a scalar. That's about as clear as anyone can expect. Sławomir
Biały 10:52, 1 December 2015 (UTC)

(ec) The first sentence also says that the function is a function on the manifold. That's about as contradictory as anyone can expect. But I wouldn't normally expect this from you. Why so stubborn? It is just a sentence YohanN7 (talk) 11:01, 1 December 2015 (UTC)

I'm astonished that you think I am not usually this stubborn.  ;-P Sławomir
Biały 23:24, 1 December 2015 (UTC)

Hopefully this resolves both of your issues. Sławomir
Biały 10:57, 1 December 2015 (UTC)

Much better. YohanN7 (talk) 11:04, 1 December 2015 (UTC)

I am trying really, really hard not to get involved in this, but I'll say something anyway. We can never make the experts and the beginners totally happy with the first sentence. So it must be decided whether the first sentence is for the experts or for the beginners. In my opinion, the first sentence is for the beginners who have heard of a metric tensor and just want to get a first clue. The experts can look further down the page. Here's an idea for a first sentence for beginners.

A metric tensor is an inner product which can be defined differently at each point of a manifold.

Then for a second sentence, I might try this kind of thing.

It is called a metric tensor because it can be used to measure the length of a vector at any point of the manifold, and this length can be integrated along a curve to compute the length of a curve in a curved space.

Then I would distinguish the physics interpretation.

In special and general relativity, a metric tensor is a kind of inner product which can make the lengths of vectors either positive or negative, whereas in Riemannian geometry, the lengths are always positive.

That's a rough first draft.
--Alan U. Kennington (talk) 14:14, 30 November 2015 (UTC)

I don't think this is an improvement over what's there now, for beginners or experts. The proposed revision relies on the reader knowing what an inner product is, whereas the current revision only assumes familiarity with the Euclidean dot product. Also the current revision conveys that the metric is applied to tangent vectors; it's not just an abstract inner product. I don't see why the article should dwell on the meaning specific in general relativity. There is already a separate article for that. This article concerns the main case, which is much easier to motivate and goes back at least to Gauss and Riemann, when the metric tensor gives an actual metric. Sławomir
Biały 15:27, 30 November 2015 (UTC)

The first sentence on the Wolfram site is so superior: "Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space." Andrewthomas10 (talk) 19:37, 23 February 2016 (UTC)

No superior, plainly wrong. What is true is that the metric tensor can be used to define a metric on the underlying manifold. First define length of path between points. Then take infimum over all paths connecting points. That gives a distance. Metric tensor ≠ metric. YohanN7 (talk) 08:56, 29 August 2016 (UTC)

From second paragraph: Thus the metric tensor gives the infinitesimal distance on the manifold. This is the correct view. YohanN7 (talk) 08:58, 29 August 2016 (UTC)

Clarification possible?

The definition says

More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U ...

Then, as far as I understand the next two lines, elements of X and Y end up as arguments of g_p, though g_p is only defined on the tangent space at p. Should this not go into the definition of the vector fields X and Y. And why are they defined "on U", which is exactly not the tangent space of a p\in U? It would be great if this could be clarified a bit.

As you can guess, I am not an expert, just someone trying to understand this :-) Haraldkir (talk) 13:24, 17 August 2019 (UTC)

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

This article is true to the intuition of Gauss and the other pioneers of the subject - it deals with the topic as they dealt with it at the time. This makes it more readable and understandable, more suitable for learning. Such intuition is lost almost entirely from the more formal, abstract style that characterises most WP articles in this area. Such entries are less suitable for learning, and I suspect make sense only to people who already understand the subject. Whilst I see the intellectual appeal of recasting historical fundamental works in a modern abstract form, reducing it to its elements, something really important is lost in that process which may not even occur to the mechanical abstract mind, namely the human intuition and genius of the original pioneers (here Gauss). I wish all the articles on differential geometry, general relativity and perhaps wider could somehow retain the intuition, genius and creativity of the pioneers, to help learning and to inspire others. Therefore I hope this article stays! Phrichuk (talk) 12:33, 24 January 2022

To editor Phrichuk: You are replying to a 17-years post. As far as this page is concerned, nobody has ever proposed to delete this page. So, the title of this thread refers to a non-existent discussion, and your "hope this article stays" is satisfied (if somebody would propose to delete this article, there is no chance of success of such a the proposition). D.Lazard (talk) 14:01, 24 January 2022 (UTC)

possible simplafications for the opening pharagraph

I got super confused while reading the first paragraph, the wording doesn't make sense, "as input a pair of tangent vectors" shouldn't it be "an input of a pair of tangent vectors." and if not, than why "as" and how does it apply to metric tensors. Changing As to An makes the page flow more correspondingly. Also this line should be removed, "(or higher dimensional differentiable manifold)" it doesn't really help make the paragraph make sense. While yes Metric Tensors do apply do higher dimensional differential manifolds it shouldn't really be added to an introduction paragraph in brackets. It makes the page look blocky and messy. It should either be added to the other elaborating paragraphs after the introduction or have its own sentence explaining it instead of it being in brackets. My solution is a changing of the opening, something like this, "In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes an input of a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. Metric tensors also apply to 3rd Dimension differentiable manifold as well as higher dimension differential manifolds that exist in 4D spaces. In the same way as a dot product, metric tensors are used to define the length of tangent vectors and angle between them. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold." Though my sentence uses a lot of words starting with D, making saying it or even reading it a bit of a tongue twister it makes sense, and could be simplified more with some commas.

In conclusion a simplification of the opening paragraph is very much possible, and my possible changes could make it a more friendlier page to read, as it really relies on people already knowing what a metric tensor is. JustAnEagerLearner (talk) 21:27, 24 April 2022 (UTC)

I agree that the first paragraph was pedantic. However, it was correct, as "taking as input" is a common jargon when talking of functions. Your edit is not an improvement, as "one definition of a metric tensor is a type of function" is non-sensical (a function is not a definition).

I have rewritten the first paragraph, for making it easier to understand for non-specialists. D.Lazard (talk) 10:38, 25 April 2022 (UTC)

Simplifying is good, but only if it preserves accuracy. Also, wikilinks such as [[Vectors (mathematics and physics)]] should generally be piped to avoid clutter, e.g., [[Vectors (mathematics and physics)|]]. 4D is just a special case of higher dimension, albeit an important one, and probably does not belong in the lead. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:05, 25 April 2022 (UTC)