Understanding General Covariance & Relativity Principle

[PAllen]

I'm just pointing out the difference between local and global diffeomorphism, you can check it on any text about differential geometry if you haven't heard about it.
General covariance as you are using it referring to coordinate transformation invariance is not to be confused with diffeomorphism invariance, a coordinate transformation is not a diffeomorphism (lacks the bijectivity).

I disagree. Show me a discussion of a coordinate transform that isn't smooth and bijective. That is part of its definition. If it maps two points to one it is not a coordinate transform.

 

[PeterDonis]

I disagree. Show me a discussion of a coordinate transform that isn't smooth and bijective. That is part of its definition. If it maps two points to one it is not a coordinate transform.

Well, the transformation between isotropic coordinates and standard Schwarzschild coordinates on Schwarzschild spacetime is usually referred to as a coordinate transformation, but it's not bijective; it maps two values of the isotropic radial coordinate to a single value of the Schwarzschild radial coordinate. Isotropic coordinates double-cover the region outside the horizon. Strictly speaking, I think that means that only the transformation from one *patch* of isotropic coordinates to Schwarzschild coordinates is a diffeomorphism; or, to put it another way, the "coordinate transformation" between isotropic and Schwarzschild coordinates defines *two* diffeomorphisms, not one.

 

[PAllen]

Well, the transformation between isotropic coordinates and standard Schwarzschild coordinates on Schwarzschild spacetime is usually referred to as a coordinate transformation, but it's not bijective; it maps two values of the isotropic radial coordinate to a single value of the Schwarzschild radial coordinate. Isotropic coordinates double-cover the region outside the horizon. Strictly speaking, I think that means that only the transformation from one *patch* of isotropic coordinates to Schwarzschild coordinates is a diffeomorphism; or, to put it another way, the "coordinate transformation" between isotropic and Schwarzschild coordinates defines *two* diffeomorphisms, not one.

Discussions of this I've seen always address the double cover problem. To treat it as true coordinate transform, you have to address by restricting your scope of analysis. Physicists may occasionally be sloppy about this, but it doesn't change the definition.

See, for example: http://en.wikipedia.org/wiki/Coordinate_transform

[Edit: The way I look at this is to say that isotropic coordinates are really two coordinate patches that each cover the exterior SC geometry: call them isotropc-large-r and isotropic-small-r. overlapping coordinate patches on a manifold are routine. It is only slightly strange that here we have two patches covering exactly the same set of points. Then, there are two coordinate transforms:

SC-exterior-patch <-> isotropic-large-r-patch
SC-exterior-patch <-> isotropic-small-r-patch
]

 

[TrickyDicky]

I disagree. Show me a discussion of a coordinate transform that isn't smooth and bijective. That is part of its definition. If it maps two points to one it is not a coordinate transform.

It is enough for a function with being injective not to map 2 points to 1.
Also this extract from "Spacetime, geometry and gravity (progress in mathematical physics)" textbook seems to confirm what I'm claiming:
"A Word of Warning
One should never, never confuse a diffeomorphism with a coordinate transformation.
A point in a manifold may be described by two charts defined in its
neighbourhood. The coordinates in these respective charts may be, say, xi and yi.
These numbers refer to the same point p. A diffeomorphism Φ maps all points of
the manifold into other points of the manifold. And barring exception a point p
is mapped to a different point q = Φ(p). The points q and p may happen to lie
in the same chart but their coordinates refer to two different points. The relationship
yi = xi + ξi above is therefore not a coordinate transformation but just a
local coordinate expression of the diffeomorphism φ when it happens to be close
to identity.
This caveat is necessary because in many texts this distinction is not emphasized
enough. Physicists define vectors or tensors as quantities which ‘transform’
in a certain way. The formula which gives a change in the components of a vector
when coordinates are changed and the formula above which gives the components
of a pushed-forward vector at q in terms components of the original vector components
at p are similar. Maybe that is why this confusion is prevalent."

 

[TrickyDicky]

IMO part of the confusion comes also from the fact that in a diffeomorphism one can associate two coordinate transformations, one the inverse of the other, or that given two parametrizations ψ and \phi, the composition of the inverse of one with the other (wich is a diffeomorphism) is sometimes called a change of coordinates.

 

[PAllen]

It is enough for a function with being injective not to map 2 points to 1.

But then the reverse transform will not be injective. A coordinate transform must be invertible.

Also this extract from "Spacetime, geometry and gravity (progress in mathematical physics)" textbook seems to confirm what I'm claiming:
"A Word of Warning
One should never, never confuse a diffeomorphism with a coordinate transformation.
A point in a manifold may be described by two charts defined in its
neighbourhood. The coordinates in these respective charts may be, say, xi and yi.
These numbers refer to the same point p. A diffeomorphism Φ maps all points of
the manifold into other points of the manifold. And barring exception a point p
is mapped to a different point q = Φ(p). The points q and p may happen to lie
in the same chart but their coordinates refer to two different points. The relationship
yi = xi + ξi above is therefore not a coordinate transformation but just a
local coordinate expression of the diffeomorphism φ when it happens to be close
to identity.
This caveat is necessary because in many texts this distinction is not emphasized
enough. Physicists define vectors or tensors as quantities which ‘transform’
in a certain way. The formula which gives a change in the components of a vector
when coordinates are changed and the formula above which gives the components
of a pushed-forward vector at q in terms components of the original vector components
at p are similar. Maybe that is why this confusion is prevalent."

There is nothing in this quote that I interpret as suggesting that a coordinate transform is not bijective.

 

[TrickyDicky]

But then the reverse transform will not be injective. A coordinate transform must be invertible.


There is nothing in this quote that I interpret as suggesting that a coordinate transform is not bijective.

Because neither that quote nor me are arguing against that.
We are talking about passive coordinate transformations in the context of GR and general covariance.

 

[TrickyDicky]

To be clear, the quote was focusing on the GR coordinate transformations only.
Certainly, generally speaking, coordinate transformations in differentiable manifolds are bijective and thus diffeomorphisms.

 

[PeterDonis]

I'm not sure it's worth getting hung up on whether or not anything called a "coordinate transformation" is a diffeomorphism. The key point appears to me to be that, if we are talking about general covariance, we are talking about what kinds of transformations leave physical laws invariant. That question is independent of whatever exact definition we adopt for "coordinate transformation".

Take isotropic vs. Schwarzschild coordinates as an example. However we want to label the transformation between the two, clearly it leaves the EFE, which is the relevant physical law, invariant. More precisely, once we have decided which range of the isotropic radial coordinate (0 -> m/2 or m/2 -> infinity) we are going to map to the range 2m -> infinity of the Schwarzschild radial coordinate, whichever choice we make, the mapping leaves the EFE invariant. If we go the other way, we are obviously going to have to choose *which* range of the isotropic radial coordinate we map to, but whichever choice we make, again the inverse mapping will leave the EFE invariant.

 

[lugita15]

1) Einstein originally hoped that general covariance, or the principle of general relativity, would be analogous to the principle of relativity in SR. The most fundamental feature of the relativity principle of SR is that 'inside a box' you truly cannot distinguish one state of inertial motion from another. Obviously, within a box, you can tell if you are accelerating. The hope was that at least you could say that you can't distinguish acceleration from gravity: thus, even if you feel an inertial force inside a box, you still can't tell your actual state of motion. However, you certainly can tell you are not 'inertial'.

Well, I think what Einstein says, in Relatvity: the Special and General Theory, is relevant here. He is discussing a passenger in a railway carriage after the brake has been applied:

It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: “My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the Earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced.”

Is there anything wrong with this point of view? Also, is there anything trivial or tautological about the physical principle being illustrated here? To my mind, it does seem to have some substantive physical content, and this physical content seems to be a legitimate extension of Galileo's principle of relativity.

 

[zonde]

Is there anything wrong with this point of view? Also, is there anything trivial or tautological about the physical principle being illustrated here? To my mind, it does seem to have some substantive physical content, and this physical content seems to be a legitimate extension of Galileo's principle of relativity.

This is equivalence principle. It is quite clear and important in GR.

But have you any quote from Einstein where he is talking about "general covariance"? Just "covariance" maybe?

 

[lugita15]

This is equivalence principle. It is quite clear and important in GR.

But have you any quote from Einstein where he is talking about "general covariance"? Just "covariance" maybe?

zonde, in this passage Einstein is defending the "general principle of relativity", which is the original term for general covariance. The section is about how the equivalence principle can be used as an argument for the general principle of relativity.

See section 20:
http://www.bartleby.com/173/

There are other sections of the book that you may also find interesting.

 

[zonde]

Hmm, so I have mixed up "general principle of relativity" with "equivalence principle".

But I rather like this "general principle of relativity". So I can pretend like this "general principle of relativity" and "general covariance" are two different things and speak only about "general principle of relativity" as basic postulate of GR.

And if someone wants to say that "general covariance" is the same thing as "general principle of relativity" then please define it first (about what kind of "covariance" we are talking and how it is generalized) and then prove this equivalence. Something like that I think.

 

[atyy]

zonde, in this passage Einstein is defending the "general principle of relativity", which is the original term for general covariance. The section is about how the equivalence principle can be used as an argument for the general principle of relativity.

See section 20:
http://www.bartleby.com/173/

There are other sections of the book that you may also find interesting.

Einstein was wrong, because all theories can be formulated in generally covariant form, even special relativity.

The Equivalence Principle is the same as the so-called "gauge principle": they are minimal coupling prescriptions.

Classically, GR is usually considered to consist of the EP and "no prior geometry". An example of a relativistic theory of gravitation which has the EP and prior geometry is Nordstrom's second theory (which came before GR).

From the quantum point of view, GR is a relativistic spin-2 field (probably with some caveats).

 

[PAllen]

zonde, in this passage Einstein is defending the "general principle of relativity", which is the original term for general covariance. The section is about how the equivalence principle can be used as an argument for the general principle of relativity.

See section 20:
http://www.bartleby.com/173/

There are other sections of the book that you may also find interesting.

I don't see how this can be distinguished as a separate principle from the principle of equivalence.

In fact, in Einstein's "The Meaning of Relativity" which is his book for a technical audience, this exact discussion is referred to only as the principle of equivalence. (page 57-58 of my edition).

 

[TrickyDicky]

So should we all agree that in GR the general principle of relativity, general covariance and the principle of equivalence are three names for the essentially same concept?

 

[Haelfix]

So should we all agree that in GR the general principle of relativity, general covariance and the principle of equivalence are three names for the essentially same concept?

It depends how you define them. Again, not all textbooks define these things in an equivalent manner, which is why most of these discussions go in circles (especially if we pull things from several sources). The equivalence principle has several different (inequivalent) forms and so you see sometimes there are additional assumptions that lie within a given definition. This gets confusing rapidly.

The first two names as far as I know, are almost always taken to be the same thing and I don't think its possible to ever make GC and the principle of equivalence entirely independant.

Anyway...

 

[TrickyDicky]

I'm not sure it's worth getting hung up on whether or not anything called a "coordinate transformation" is a diffeomorphism. The key point appears to me to be that, if we are talking about general covariance, we are talking about what kinds of transformations leave physical laws invariant. That question is independent of whatever exact definition we adopt for "coordinate transformation".

Take isotropic vs. Schwarzschild coordinates as an example. However we want to label the transformation between the two, clearly it leaves the EFE, which is the relevant physical law, invariant. More precisely, once we have decided which range of the isotropic radial coordinate (0 -> m/2 or m/2 -> infinity) we are going to map to the range 2m -> infinity of the Schwarzschild radial coordinate, whichever choice we make, the mapping leaves the EFE invariant. If we go the other way, we are obviously going to have to choose *which* range of the isotropic radial coordinate we map to, but whichever choice we make, again the inverse mapping will leave the EFE invariant.

The wikipedia entry "Active and passive transformation" clears up any confusion, clearly in the case of GR's general covariance transformations and your examples it is the passive ones we are dealing with(change of basis), and they certainly leave the EFE unchanged. These transformations are not diffeomorphisms, they are local diffeomorphisms though.

 

[atyy]

Einstein was wrong, because all theories can be formulated in generally covariant form, even special relativity.

The first two names as far as I know, are almost always taken to be the same thing and I don't think its possible to ever make GC and the principle of equivalence entirely independant.

OK, let me take my statement back about Einstein being wrong. It does seem that while GC implies neither the EP nor "no prior geometry", the EP requires GC to be stated (eg. "comma goes to semicolon" rule for derivatives).

 

[PeterDonis]

The wikipedia entry "Active and passive transformation" clears up any confusion, clearly in the case of GR's general covariance transformations and your examples it is the passive ones we are dealing with(change of basis), and they certainly leave the EFE unchanged. These transformations are not diffeomorphisms, they are local diffeomorphisms though.

We had a thread about this a while back; as I recall there was general agreement that "passive diffeomorphisms" (changing coordinate charts on the same manifold) leave the EFE invariant, but we didn't reach any real consensus about "active diffeomorphisms", partly because we couldn't reach consensus on exactly what they were.

I put "diffeomorphisms" in quotes just now because of what you said about the passive transformations not being diffeomorphisms; however, I'm not sure that's true. Since a diffeomorphism is just a differentiable, invertible map between two manifolds, M and N, a differentiable, invertible map between two charts on the *same* manifold would meet the definition, with M = N.

 

[TrickyDicky]

We had a thread about this a while back; as I recall there was general agreement that "passive diffeomorphisms" (changing coordinate charts on the same manifold) leave the EFE invariant, but we didn't reach any real consensus about "active diffeomorphisms", partly because we couldn't reach consensus on exactly what they were.

Active diffeomorphisms are what the text quote I found calls simply diffeomorphisms, they imply mapping from points to different points. In the passive diffeomorphism that I'm calling local diffeomorphism, the point p is the same.

I put "diffeomorphisms" in quotes just now because of what you said about the passive transformations not being diffeomorphisms; however, I'm not sure that's true. Since a diffeomorphism is just a differentiable, invertible map between two manifolds, M and N, a differentiable, invertible map between two charts on the *same* manifold would meet the definition, with M = N.

Here is where the tricky part comes, coordinate changes of the passive diffeomorphism (changes of basis) kind are not necessarily invertible maps between charts(they must be injective though, to qualify as local isometries), they are only required to be invertible mappings at the tangent space.

 

[Muphrid]

Let me wander in here with some insights coming from Gauge Theory Gravity. What is GTG? It's a classical theory of gravity solely on a Minkowski background, phrased in the language of geometric algebra and calculus. The fixed background means it lacks background independence, but it shares many other features with general relativity, including "general covariance" and the "(weak) equivalance principle".

GTG's version of general covariance is called the displacement gauge principle, saying that the laws of physics are invariant under a differentiable smooth remapping of events on spacetime. This largely covers coordinate transformations and yields all the usual tensor transformation laws. This is linked to the idea that spacetime is homogeneous, for in GTG at least, it involves this notion of remapping events on the background spacetime, something that we wouldn't be free to do if spacetime weren't homogeneous.

GTG's version of the weak equivalence principle is called the rotation gauge principle, saying that the laws of physics are covariant under local rotations ("rotations" generalized to include boosts) of fields. This is a part that requires more thinking, for the metric of GR is invariant under such local, differentiable rotations of fields, but GTG keeps track of what's essentially a tetrad field, which is not invariant under such rotations. At any particular point, a local rotation can reduce the gravitational field to zero, making the effects of gravity indistinguishable from fictitious forces of a curved coordinate system, and since all physical predictions must be independent of gauge, we say that gravity is a fictitous force everywhere, exactly as in GR. The rotation gauge principle basically gives us this freedom of rotation of fields that can only make sense if we take as granted that spacetime is isotropic.

So really, as far as a new perspective to look at things in GR, I think the rotation gauge principle gives some insight into how the weak equivalence principle might be arrived at, and the relations between these ideas and the notions of spacetime being homogeneous and isotropic really shed light on what they mean (at least to me).

 

[lugita15]

Einstein was wrong, because all theories can be formulated in generally covariant form, even special relativity.

It seems to me that what is nowadays called "general covariance" is indeed a statement with no physical content. But I think Einstein was describing a physically meaningful principle.

Let me put it this way. General covariance, let's call it statement 1, states "the laws of physics are the same in all coordinate systems". A special case of this is statement 2, "the laws of physics are the same in all coordinate systems that are at rest or in uniform motion". That sounds like the Principle of Relativity, doesn't it? So how can a physically meaningless statement imply a physically meaningful statement? I think that the answer to that is that statement 2 is NOT really the principle of relativity, although it sounds similar to it. Statement 2 is a statement that any theory can be made to satisfy, whereas the principle of relativity holds for some theories but not all, e.g. Aristotelian physics. So statement 2 is to the POR as statement 1 is to ... what?

 

[PeterDonis]

Active diffeomorphisms are what the text quote I found calls simply diffeomorphisms, they imply mapping from points to different points.

And possibly also mapping between different manifolds; at least, that's what some of the papers that were linked to in the previous thread seemed to indicate.

Here is where the tricky part comes, coordinate changes of the passive diffeomorphism (changes of basis) kind are not necessarily invertible maps between charts (they must be injective though, to qualify as local isometries), they are only required to be invertible mappings at the tangent space.

Can you give a specific example of a passive diffeomorphism that is not an invertible map between charts, but *is* an invertible map between tangent spaces at each point?

 

[Muphrid]

Let me put it this way. General covariance, let's call it statement 1, states "the laws of physics are the same in all coordinate systems". A special case of this is statement 2, "the laws of physics are the same in all coordinate systems that are at rest or in uniform motion". That sounds like the Principle of Relativity, doesn't it? So how can a physically meaningless statement imply a physically meaningful statement? I think that the answer to that is that statement 2 is NOT really the principle of relativity, although it sounds similar to it. Statement 2 is a statement that any theory can be made to satisfy, whereas the principle of relativity holds for some theories but not all, e.g. Aristotelian physics. So statement 2 is to the POR as statement 1 is to ... what?

I think statement 2 is to isotropy of spacetime as statement 1 is to homogeneity of spacetime. These, combined with the posited existence of null vectors (lightlike vectors) gets you all the way to SR. The coupling of the stress energy to the curvature of spacetime gets you GR on top of that.

I admit, though, I am not entirely sure spacetime in GR is isotropic and homogeneous. At least, I haven't found any independent sources speaking to this. Nevertheless, the statement sounds logical to me. Isotropy gives us the freedom to choose a basis, which is intimately connected to Lorentz transformations, for observers in different reference frames merely make measurements with respect to different sets of basis vectors. Homogeneity gives us the freedom to choose a coordinate system, the freedom to remap spacetime with a new set of coordinate tuples. Homogeneity tells us there is no absolute place to set the origin, no absolute way to draw the coordinate axes. All choices for these are valid, and so quantities are only meaningful, only physical, if they respect that property.

 

[atyy]

It seems to me that what is nowadays called "general covariance" is indeed a statement with no physical content. But I think Einstein was describing a physically meaningful principle.

Let me put it this way. General covariance, let's call it statement 1, states "the laws of physics are the same in all coordinate systems". A special case of this is statement 2, "the laws of physics are the same in all coordinate systems that are at rest or in uniform motion". That sounds like the Principle of Relativity, doesn't it? So how can a physically meaningless statement imply a physically meaningful statement? I think that the answer to that is that statement 2 is NOT really the principle of relativity, although it sounds similar to it. Statement 2 is a statement that any theory can be made to satisfy, whereas the principle of relativity holds for some theories but not all, e.g. Aristotelian physics. So statement 2 is to the POR as statement 1 is to ... what?

Statement 2 does sound like the Principle of Relativity to me. I think the difference between statement 1 and 2 is that in statement 2, the "laws of physics" are not allowed to be written in generally covariant form. They can only be written in Lorentz covariant form - one is not allowed to use Christoffel symbols to write the "laws of physics" in the "same form". This is why statement 2 is not a special case of statement 1 where one is allowed to use Christoffel symbols to write the "laws of physics" in the "same form". Because of this difference in the definition of "laws of physics in the same form" in statements 1 and 2, statement 1 places no restriction on what theories are allowed, while statement 2 restricts one to Lorentz covariant theories.

(BTW, I did retract in post #49 my statement on Einstein being wrong .)

 

[TrickyDicky]

I admit, though, I am not entirely sure spacetime in GR is isotropic and homogeneous. At least, I haven't found any independent sources speaking to this. Nevertheless, the statement sounds logical to me. Isotropy gives us the freedom to choose a basis, which is intimately connected to Lorentz transformations, for observers in different reference frames merely make measurements with respect to different sets of basis vectors. Homogeneity gives us the freedom to choose a coordinate system, the freedom to remap spacetime with a new set of coordinate tuples. Homogeneity tells us there is no absolute place to set the origin, no absolute way to draw the coordinate axes. All choices for these are valid, and so quantities are only meaningful, only physical, if they respect that property.

There are GR solutions of the EFE that are neither isotropic not homogeneous, so none of these properties are intrinsic to GR. However in cosmology solutions are seeked that fulfill those two requirements (this is called the cosmological principle) and its main example are FRW cosmologies.

 

[lugita15]

Statement 2 does sound like the Principle of Relativity to me. I think the difference between statement 1 and 2 is that in statement 2, the "laws of physics" are not allowed to be written in generally covariant form. They can only be written in Lorentz covariant form - one is not allowed to use Christoffel symbols to write the "laws of physics" in the "same form". This is why statement 2 is not a special case of statement 1 where one is allowed to use Christoffel symbols to write the "laws of physics" in the "same form". Because of this difference in the definition of "laws of physics in the same form" in statements 1 and 2, statement 1 places no restriction on what theories are allowed, while statement 2 restricts one to Lorentz covariant theories.

Perhaps you're misinterpreting statement 2. Surely "such-and-such is true for all coordinate systems satisfying condition X" is a logical consequence of "such-and-such is true for all coordinate systems". Do you disagree with that? So statement 2 is a logical consequence of statement 1, and thus statement 2 can't be the PoR.

 

[lugita15]

I think statement 2 is to isotropy of spacetime as statement 1 is to homogeneity of spacetime.

I'm not really sure what makes you say this. But in any case, what do you think is the relation between statement 2 and the principle of relativity? Are they the same, or is one a trivial statement and the other a meaningful physical principle?

I admit, though, I am not entirely sure spacetime in GR is isotropic and homogeneous.

GR doesn't admit global spacetime symmetries, only local ones.

 

[Muphrid]

There are GR solutions of the EFE that are neither isotropic not homogeneous, so none of these properties are intrinsic to GR. However in cosmology solutions are seeked that fulfill those two requirements (this is called the cosmological principle) and its main example are FRW cosmologies.

Yeah, I was aware of that. I just began to wonder if they were really using the terms the same way as I am. Perhaps it is my usage that is unconventional, but I think isotropic and homogeneous are more often used to describe the curvature and associated distribution of stress-energy, where I'm referring to an intrinsic property of spacetime itself. For example:

So I guess I'm trying to distinguish between the isotropy and homogeneity of spacetime's curvature vs. those of spacetime itself. Perhaps there is a better (or more established) way of thinking about these concepts. Perhaps that is why "general covariance" is the prevailing terminology still.

It's interesting to me also that isotropy is almost always considered only with respect to the spatial dimensions--even in the wiki article they make clear that this is the isotropy they're speaking of when talking about the FRW metric.

It may be I have to abandon the word isotropy for this ability to perform local (i.e. position-dependent) rotations of fields (which in turn allows one to choose a local basis). I think that concept at least stands, given its connection to the weak principle of equivalence, and the same for what I've been calling "homogeneity" and its connection to general covariance.