Talk:General covariance - Misplaced Pages

This is an old revision of this page, as edited by Anita5192 (talk | contribs) at 02:13, 20 April 2023 (→Vehement original-research template: Reply.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Revision as of 02:13, 20 April 2023 by Anita5192 (talk | contribs) (→Vehement original-research template: Reply.)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)

Physics: Relativity Start‑class High‑importance

	Physics portal This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics
Start	This article has been rated as Start-class on Misplaced Pages's content assessment scale.
High	This article has been rated as High-importance on the project's importance scale.
	This article is supported by the relativity task force.

To-do list for General covariance: edit · history · watch · refresh · Updated 2006-03-03

Tasks for expert:

completely rewrite article, using J. D. Norton review as a guide
expand and link to Hole argument

Archives

This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 5 sections are present.

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)

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)

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 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:

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

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 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

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) 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)

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)

Copyright problem removed

One or more portions of this article duplicated other source(s). The material was copied from: http://philsci-archive.pitt.edu/archive/00002532/01/equivalencecopyedited.pdf. Infringing material has been rewritten or removed and must not be restored, unless it is duly released under a compatible license. (For more information, please see "using copyrighted works from others" if you are not the copyright holder of this material, or "donating copyrighted materials" if you are.) For legal reasons, we cannot accept copyrighted text or images borrowed from other web sites or published material; such additions will be deleted. Contributors may use copyrighted publications as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Misplaced Pages takes copyright violations very seriously, and persistent violators will be blocked from editing. While we appreciate contributions, we must require all contributors to understand and comply with these policies. Thank you. Ozob (talk) 21:38, 19 December 2009 (UTC)

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, 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, 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)

No such thing as Principle of GR

The article is misleading and wrong, because there is no such thing as a Principle of General Relativity because the laws of physics are simply not the same for accelerating frames where fictitious forces must be introduced. The article must be rewritten to explain this. — Preceding unsigned comment added by 47.201.179.7 (talk) 05:41, 15 January 2017 (UTC)

I will add this to the Remarks section:

General Relativity is actually only a theory of gravitation, and is not in any sense a general theory of relativity(G. H. Keswani, Br. J. Philos. Sci. 16, 276 (1966)). There exists nowhere in nature a principle of general relativity; the laws of physics are simply not the same for observers in acceleration where fictitious forces must be introduced. The term Covariance in general relativity refers only to a mathematical formalism, and is not used in the same sense as the term in special relativity. — Preceding unsigned comment added by 47.201.179.7 (talk) 14:52, 15 January 2017 (UTC)

Actually, according to the general principle of relativity, if a physical observer is forcibly accelerated or rotates, and experiences apparent gravitational field effects ("fictitious forces"), by then applying the general principle, background inertial observers are supposed to see corresponding field distortion effects due to the noninertial behaviour of the first observer's mass relative to background (Einstein 1921).

In other words, we start a chain of argument by introducing fictitious forces, but then we iterate that argument using the general principle, and use the GPoR to modify the initially-defined geometry, and by the end, the fictitious forces are no longer fictitious - they are associated with real intrinsic distortions of the spacetime metric, that exist for all observers.

So ... If we rotate, we experience an apparent field. In the first stage of the argument, this field is not yet a "true" field, because it can be removed by switching back to an inertial coordinate system. But this is not yet correct general relativity! Applying the GPoR then requires our colleague who sees us rotating, to also see the relative rotation of our mass to be associated with a twist in spacetime, with the twist existing in all frames. As a sanity-check, think: suppose that the field experienced by a rotating body really was fictitious: if we moved back to an inertial frame, the field would disappear, and there would be no physical rotating dragging effects around rotating bodies. But Gravity Probe B showed that the rotational dragging effect of the Earth was physically real, so ... NOT fictitious. GP-B is evidence that the general principle of relativity really does seem to be supported by Nature.

Unfortunately, Einstein's general theory is a logically incoherent mess, and doesn't work consistently as geometry. It has some great ingredients, but Einstein's attempted geometrical implementation of the GPoR was junk. This is why, when textbook authors try to "explain" the theory and try to make sense of the mess, different authors will seize on different aspects of the theory and end up producing "provably correct" interpretations that somehow manage to conflict. It's because Einstein accidentally constructed a pathological system. ErkDemon (talk) 04:27, 3 July 2020 (UTC)

... This makes writing wiki pages about Einstein's general theory a bit challenging: if the page genuinely reflects the theory, it will be contradictory and incoherent, and people will blame the authors ... but if the page's arguments are thorough and consistent and make sense, they will not be a correct representation of the theory. :) ErkDemon (talk) 04:27, 3 July 2020 (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 (). 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)

Vehement original-research template

There is currently a template at the top of the page with somewhat harsh language. Is this acceptable? I am tempted to remove it, but its message may be valid even though the language used to express it is probably not, and its issues have not yet been resolved. What, if anything, should we do?—Anita5192 (talk) 01:34, 11 September 2021 (UTC)

Fixed—This was removed, 10:39 12 September 2021, but the editor who removed it did not post anything about it here.—Anita5192 (talk) 02:13, 20 April 2023 (UTC)

Shouldn't "general covariance" be clearly defined in this article?

I feel this article is currently rather vague and not that useful for anyone who wants to know what terms such as "general covariance", "general invariance" and "diffeomorphism invariance" actually mean. There is a good reason for this since different sources (e.g. popular text books) define these terms differently. However, the current article does not address this. I feel unsure that one meaning of "general covariance" is being discussed or whether several different notions are being referred too in this article without disentangling them.

The problems of the article start with the first sentence that states that "In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations." The first issue is that this implies that "general covariance", "diffeomorphism covariance" and "general invariance" all mean refer to the same notion. The second issue is that "the invariance of the form of physical laws under..." does not have a clear meaning (at least for me). I'm not sure when a physical law is supposed to have the same form as supposed to being the same. In general relativity the physical laws are the same in all coordinate systems. By this I mean that the Einstein equations as a set of differential equations for the metric are not changed in anyway if I go from one coordinate system to another. I'm happy if this is what is meant by the same form, however, I'm aware that "general covariance" can refer to merely writing an equation in terms of covariant derivatives such that as expressed with covariant derivatives it has the same form in all coordinate systems (Sean Carrol defines in this way in his book on GR). For example if I write Maxwell's equations down in special relativity (i.e. in the absence of gravity) I could express them using covariant derivatives so that they have the same form in all coordinate systems expressed. These means the same in terms differential equations that involve both the electromagnetic field tensor and the metric tensor. However, Maxwell's equations are a set of physical laws that govern the electric and magnetic fields but not physical laws for the metric tensor. To find the components of the electric and magnetic fields we have to specify the coordinates to determine which form the metric takes.

To put it differently suppose I want to know whether some electric fields E_i(x) and magnetic fields B_i(x) are solutions to the Maxwell's theory (let's say in vacuum). To answer that I have to put them into the Maxwell equations to see if they are a solution but if I have written out the equations in a form that is independent of the coordinates this means the equations depend also on the metric. So to answer the question I have to specify the metric too which is the same as saying which coordinate system I'm using. In general relativity I can ask if a metric obeys Einstein's law of gravity but there I just put the metric in the Einstein equations without referring to any coordinate system.

To make Maxwell's theory in the absence of gravity also have the property that we make no reference to coordinates one can state a "law of physics" that the Riemann tensor vanishes everywhere. Then one has the Maxwell equations plus equations that that determine the metric which are then laws which are invariant under changes of coordinates. This could also be what is meant by the "invariance of the form of physical laws". But aren't such laws actually independent of coordinate systems rather than simply of the same form when expressed in a coordinate language?

I would propose to give clear definitions of general covariance and related concepts contrasting different definitions to give the reader a clear picture. I would recommend that the paper "Coordinates, observables and symmetry in relativity" https://arxiv.org/abs/0711.2651 by Hans Westman and Sebastiano Sonego as the clearest paper on this subject. The paper by Norton that is cited is also good and covers many of the same points. However, the paper Westman and Sonego is much clearer in my opinion. Finbar1984 (talk) 03:20, 17 April 2023 (UTC)

Categories: