Talk:General covariance

message: 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)
replies: text: —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)
thread_title:  Vehement original-research template 

message: 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)
replies: text: It seems to me that this article should not be treated as a physics article but as a history of physics article.  At this point I don’t think anyone seriously considers “general covariance” a useful tool for thinking about physics, and in fact the phrase “diffeomorphism covariance” seems like it could be easily confused with the very real concept of “diffeomorphism invariance” in general relativity.  This distinction is the heart of what the MTW quote is trying to say and it should be the content of this article in my opinion, potentially noting Einstein’a original interpretation and how modern sources view it as irrelevant. INLegred (talk) 04:05, 19 October 2023 (UTC)
thread_title:  Shouldn't "general covariance" be clearly defined in this article?