Talk:General covariance/Archive 1

Archive 1

Clarification of Critique

Hi, EMS, I am still very new to Wikipedia, so please bear with me.

I did not add the "clean-up" tag, and I guess I didn't read the article very carefully first time around, because now I see some more objections, in addition to the one mentioned above. So I'd have to agree with the other critic that the article should be rewritten essentially from scratch. Some points to bear in mind:

0. The sentence

"The wave equation (which describes the behavior of a vibrating string) is classically written as:

$U(x,t)=f(kx+ct)+g(kx-ct)$

for some functions f, g and some scalars k and c."

is seriously misleading. In fact, $U(x,t)=f(kx+ct)+g(kx-ct)$ is not the wave equation! Rather, this is D'Alembert's general solution to the one-dimnensional wave equation.

1. The article should be rewritten to discuss the notion of covariance of a differential equation under a transformation group. Thus, in modern physics/math, we can have Lorentz covariance, diffeomorphism covariance, $SU(2)$ covariance (with the group action being understood), etc. For example, the original formulation of Maxwell's equations turns out to be Lorentz covariant; this is obvious when one writes the equations in modern form as

$F_{,a}^{ab}=4\pi J^{a}$

$F_{ab,c}+F_{bc,a}+F_{ca,b}$

However, these equations are not diffeomorphism covariant, because if you apply a more general difffeomorphism than a Lorentz transformation, they assume a new form. But if we change the partial derivatives to covariant derivatives, we do get a set of diffeomorphism invariant equations,

$F_{;a}^{ab}=4\pi J^{a}$

$F_{ab;c}+F_{bc;a}+F_{ca;b}$

It turns out that this formulation can be used to define EM on curved spacetimes.

2. The term "general covariance" is in fact archaic, so the article should really be called "diffeomorphism covariance".

3. The EFE is a tensor equation, hence automatically diffeomorphism covariant, but while this is a very important property, it is not "the defining characteristic" of GTR. Indeed, competitors such as scalar-tensor theories are also diffeomorphism covariant.

4. You said "classical formulations [of EM] involve a privileged time variable". I think you might mean that Maxwell was not aware of the Lorentz covariance of his field equations, and did not know either Einstein's kinematic or Minkowski's geometric interpretation of the significance of this mathematical fact. In fact, the classical formulation of EM is mathematically equivalent to the first set above, and since this is Lorentz covariant, it does not have a privileged time variable. Rather, it has a privileged notion of non-accelerating frame. It is true that Maxwell didn't know this, however.

5. I plan to rewrite the article sometime in the next few weeks, after I have read some more math articles to get some more ideas for how to write a good math article. I can already see that it is much easier to write a new article from scratch than to try to fix a seriously flawed old one! So I am considering a "solution" which involves writing a new article on "covariance [transformation geometry]" or something like that. I am also planning to write about related topics such as the point symmetry group of a differential equation.

For the moment I have just added a citation to a good discussion of "general covariance" in a well-known and widely available gtr textbook.

--CH | Talk

Chris -

This article is not mine. My only contribution to it is the removal of a couple of sentences that I found to be ridiculous. I strongly advise that you look at the history of a page (available by clicking the "history" tab) before assigning any blame for it's contents. It is also better to refer to "this article" instead of "you".

Also be advised that I know that the clean-up tag is not yours. That is why I added that query under it's own tag insted of adding to an existing section as I am doing here. I just felt that this article needed little cleaning up in so far as it's structure went. I won't quibble with you about the contents, but that is not what I see that tag as addessing.

As for doing a rewrite, I say to go for it. You are going to bring out new and worthwhile facts in anything that you work on.

--EMS | Talk 20:22, 8 Jun 2005 (UTC)

Hi, EMS, sorry for any confusion. For what it's worth, I don't remember assuming you were the original author. I probably addressed my second comment to you because you replied to my first comment, which I probably took to indicate some interest in improving the article --CH

Hi

I dont think general covariance has anything to do with diffeomorphism. Please verify

(anon comment added from IP 203.200.95.130)

Hi, 203.200.95.130, the mainstream view nowadays is indeed that general covariance should be understood as a synonym for the more precise term diffeomorphism covariance. Earlier authors gave a variety of other interpretations. ---CH 21:51, 3 March 2006 (UTC)

3rd paragraph confusion

Currently reads:

The principle of general covariance was formulated by Einstein who wanted to extend the Lorentz covariance in Special Relativity to non-inertial frames. All known physical theories such as electrodynamics must necessarily have a generally covariant formulation.

which is very confused, IMO. It suggests (without being explicit) that general covariance implies Lorentz covariance, whereas Lorentz covariance is a separate property (i.e. we can write down generally covariant equations that are not Lorentz covariant). I also suggest deleting the words "must necessarily" from the last sentence. --Michael C. Price ^talk 22:39, 11 July 2006 (UTC)

Merger

It's done. Article is a hopeless mess right now. ---CH 22:03, 3 March 2006 (UTC)

I found this page as a redirect for the general principle of relativity. I see the connection, but I think the redirect is a bit too mathematical. Comment: I believe the general principle states that gravitational acceleration is equivalent to inertial acceleration. I believe Einstein's goal was to base his theory of gravity on this principle. In actuality, Einstein's theory ended up being based on the equivalence of gravitational mass and inertial mass. Despite being called general relativity, Einstein's theory contradicts the general principle.— Preceding unsigned comment added by Danras (talk • contribs) 03:19, 18 March 2006 (UTC)

I see that Danras (talk · contribs) is responsible for this this veritable farrago of misinformation, which rather speaks for itself.---CH 07:39, 29 August 2006 (UTC)

General Covariance is not General Relativity

I added the "covariance is not relativity" bit as this aspect was sadly lacking. It makes no sense to discuss covariance without highlighting this fundamental misunderstanding.— Preceding unsigned comment added by Kevin aylward (talk • contribs) 09:34, 14 October 2006 (UTC)

I finally decided to remove the “SR doesn’t deal with accelerating frames” bit, as this is simply not true. SR handles accelerating frames fine. However, SR does not deal with gravitation. In addition, as noted in the remarks, general covariance has zero relevance to GR, except historical interest. Kevin aylward 07:55, 4 January 2007 (UTC)

General covariance may not imply Einstein's theory of gravitation; but Einstein's theory of gravitation cannot be effectively expressed without using generally covariant notation. JRSpriggs 11:27, 4 January 2007 (UTC)

Sure, but that’s irrelevant. Its absolutely fundamental that General Covariance has absolutely nothing to do with the (physical) content of General Relativity. Einstein made a mistake, its about time this fact was universally given the weight that it is due. General Relativity is about "no prior geometry". It needs to be fully appreciated that co-ordinate free descriptions, i.e the laws of physics are independent of co-ordinate systems is a mathematical tautology. End of story. However, they are not independent of actual real physical frames. You cannot transform away an arbitrary gravitational field. “Relativity” is simply not true, according to “General Relativity”.— Preceding unsigned comment added by Kevin aylward (talk • contribs) 16:36, 04 January 2007 (UTC)

(1) How do you feel about the paragraph now that I have changed it to the following?

The general principle of relativity (i.e. general covariance) as used in GR, is that the laws of physics should be formulated in the same way in all reference frames. In special relativity, some (inertial, i.e. non-accelerating) reference frames have a preferred status which does not apply in general relativity.

(2) Electromagnetism can be expressed in equations which do not include the metric, except in the constitutive equations for a vacuum. So EM is truly "relative".

(3) Real physical observers MUST assume a coordinate condition on the metric to construct their frames of reference. It is an interesting question -- what is the best way to do that? JRSpriggs 04:14, 5 January 2007 (UTC)

[responding to point (1)]
What can I say?...have a thorough read of the JD Norton paper noted in the external links. Again, General Relativity is not equal to general covariance. In GR, all reference frames are *not* equivalent. Its that simple. Its irrelevant that all co-ordinate systems are mathematically equivalent, i.e. the equations can be written co-ordinate free. General Relativity should be renamed to something else, as it frames are not all relative. There are preferred physical frames in GR, despite their being no preferred co-ordinate systems. A physical frame is not the same as a co-ordinate reference. This causes never ending confusion. Again see JD Norton.— Preceding unsigned comment added by Kevin aylward (talk • contribs) 12:05, 23 January 2007 (UTC)

As an encyclopedia, it is not for us to overturn the almost universal usage "General Relativity". Sometimes I personally call it "Einstein's theory of gravity", but if we started doing that here people would wonder whether we meant the same thing as GR. And although some reference frames may be more convenient for describing the metric than others, we do not have the same sharp distinction that exists in SR between inertial and non-inertial frames. JRSpriggs 06:46, 24 January 2007 (UTC)

Copyright problem removed

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Tensors in Classical Electrodynamics are not general tensors

It is said that:

"A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,[1] and is usually expressed in terms of tensor fields. The classical (non-quantum) theory of electrodynamics is one theory that has such a formulation."

I believe this is incorrect, however. A tensor is defined with respect to some group of transformation. In the case of GR, tensors are ``general tensors: they behave like tensors (see differential geometric definition) under any smooth coordinate transformation. In the case of Classical Electrodynamics, ``tensors are not general tensors, but ``Poincaré tensors. I suggest the following version of the text above:

"A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,[1] and is usually expressed in terms of tensor fields."

Correct me if I'm wrong.

Mcasariego (talk) 21:17, 1 August 2015 (UTC)

Actually electrodynamics is readily formulated in true tensorial form, although it is not often presented that way at the undergraduate level. Instead of E-vectors and B-vectors, the field components are better combined into a 4x4 tensor with the E components associated with the time dimension and the B components associated with the spatial dimensions. Instead of 3D "cross-product", ordinary tensor product is used, along with expressing relevant fields as tensor densities (so that they track parity). The whole "unified field theory" project originated by Einstein depended on the true-tensor formulation of the Maxwell theory.

I think what you may be referring to is the fundamental symmetry group GL(4,R) that is the basis for general covariance in such theories. Classical electrodynamics, properly expressed, is manifestly invariant under GL(4,R) transformations (that is, smooth but otherwise arbitrary automorphisms of spacetime). — DAGwyn (talk) 11:47, 5 August 2015 (UTC)

Thanks for your response. You are right. What happened is that when I read "classical electrodynamics" I translated it to the special relativistic case, in which $F_{\mu \nu }$ is defined as a Lorentz tensor. But of course, in Einstein-Maxwell theory $F_{ab}$ is a general tensor and in that sense classical electrodynamics is generally covariant, the SR case being just a local subcase obtained by using Riemann normal coordinates. — Mcasariego (talk) 12:17, 5 August 2015 (UTC)

Referance Frames and Coordinate Systems

Changed “coordinate systems” to reference frames”. The phrase is wrong and confuses virtual mathematical coordinates systems with real physical frames. Coordinate systems have no physical implications whatsoever. Real physical frames, moving do. — Preceding unsigned comment added by Kevin aylward (talk • contribs) 11:09, 4 June 2020 (UTC)

@Kevin aylward: Please put new talk page messages at the bottom of talk pages and sign your messages with four tildes (~~~~) — See Help:Using talk pages. I already asked you to do that on my talk page: User talk:DVdm#Coveriance. You were also asked twice on your user tlak page to do that.

I changed it back ([1]). It's in the cited source: "This assertion that the laws are valid independent of the choice of the coordinate system, if only such coordinate systems are considered that move rectilinearly and uniformly against each other, this we call the special relativity principle."

Comments from others are welcome. - DVdm (talk) 11:29, 4 June 2020 (UTC)

Pedantic Quibble

The author of this article says that

$U(x,t)=f(kx+ct)+g(kx-ct)$

is the wave equation. But of course this is the D'Alembert general solution to the one-dimensional wave equation, not the equation itself, is often written in coordinate-free notation as

$\Box U=0$

(where $\Box$ is the Laplace-Beltrami operator) or in conventional PDE notation (using a Cartesian coordinate chart) as

$U_{tt}=U_{xx}$

I propose to modify this page to correct this, if no-one objects.

User:Hillman

off course --MarSch 13:57, 24 June 2005 (UTC)

Clean-up tag

Would someone mind explaining how this article needs to be cleaned up? Other than an extraneous paragraph (which I have removed), I see nothing wrong with the structure of this article. I wonder if a wave equation is the right example myself, but I will leave it up to Hillman's judgement as to what to do with that.

It is my opinion that this article is of the right size and structure, being a coherent explanation of general covariance and a simple example. I think that the best way of cleaning it up is to drop the needs-cleaning-up tag.

--EMS | Talk 19:09, 2 June 2005 (UTC)