Talk:General covariance

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)

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)

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)

Einstein's general theory already doesn't work, before we even get to covariance

The difficulty in comparing the general principle of covariance with the geometry of Einstein's general theory and its modern textbook updates is that one has to presuppose that there actually is "a geometry" and a set of geometrical rules corresponding to the theory, that the covariance principle can be compared against.

And there isn't.

Einstein's system is geometrically incoherent.

• On the one hand, the requirement for supporting gravitomagnetism and a dynamic spacetime geometry means that the case of two stars moving in straightish lines at constantish relative velocities must be a curved-spacetime problem. The region must contain gravitational and gravitomagnetic curvatures that tell us the location and mass of the stars and their relative states of motion. The metric must be dynamic, and interactive, with no prior geometry.
• On the other hand, the requirement that the system supports the SR equations of motion lets us prove geometrically that the region is flat as a function of the stars' relative velocity, and contains no gravitomagnetic fields whatsoever. SR-compliance requires a fixed prior Minkowski geometry that dictates rules to the stars without any geometrical back-reaction.

Einstein's 1916 theory is invalid as a geometrical theory of physics because it tries to support two incompatible geometries, at the same time, for the same situation. It allows us to make two incompatible predictions for the same outcome.

• If we follow Einstein and modern textbooks, and give special relativity priority over the GPoR, then there is no such thing as gravitomagnetism, and no such thing as a general principle of relativity, or a general theory of relativity.
• On the other hand, if we decide that we want a general theory of relativity, and agree with the Gravity Probe B frame-dragging result, we have to reject special relativity's Minkowski metric as being too naive to be physics.

Anyone who has failed to notice that the 1916 theory is essentially a fake -- it presents itself as a geometrical solution, and it isn't -- shouldn't even attempt to tackle the more subtle problem of covariance. ErkDemon (talk) 14:07, 7 July 2024 (UTC)

Unless I am misinterpreting your argument, claiming that GR is incoherent because the adherence to Minkowski geometry necessitates a specific frame of reference seems... irrelevant? GR is consistent with the flat Minkowski space in SR for sufficiently local reference frames, so claiming that GR is "fake" because they disagree at large scales seems akin to contemporaries calling the Pythagorean Theorem fake because it results in irrational numbers: of course it differs to the previous theories, that is why it was introduced in the first place.

Does GR as presented in 1916 presuppose some aspects of the geometry? Yes, of course. Einstein's field equations, as presented, inherently assume a 4-D spacetime metric tensor. But claiming it is geometrically incoherent seems a harsher than necessary way to call it "incomplete". If spacetime turns out to be more consistent with a higher order geometry at some scales, then GR will likely be inconsistent with those reference frames unless updated, but this would be an inconsistency with observation. At current time, the framework of GR is largely internally coherent.

Is your claim coming from the fact that Einstein himself was not presenting the general solution to the field equations in 1916? Or is there some other issue I'm missing here? TWorkman (talk) 15:14, 3 December 2024 (UTC)