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This is an old revision of this page, as edited by Hillman (talk | contribs) at 07:39, 29 August 2006 (→Merger). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

I have removed tha article on wave equation because it has NOTHING to do with general covariance.

Pedantic Quibble

The author of this article says that

${\displaystyle U(x,t)=f(kx+ct)+g(kx-ct)}$

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

${\displaystyle \Box U=0}$

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

${\displaystyle U_{tt}=U_{xx}}$

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

User:Hillman

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

Clarification of Critique

Hi, EMS, I am still very new to Misplaced Pages, 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:

${\displaystyle U(x,t)=f(kx+ct)+g(kx-ct)}$

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

is seriously misleading. In fact, ${\displaystyle U(x,t)=f(kx+ct)+g(kx-ct)}$ 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, ${\displaystyle SU(2)}$ 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

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

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

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,

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

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

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 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 " 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)

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)

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)

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 22:39, 11 July 2006 (UTC)

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