Understanding General Covariance & Relativity Principle

[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.

 

[TrickyDicky]

And possibly also mapping between different manifolds;

Sure.

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?

No, what is invertible is the function that maps open sets containing a point of the manifold to open sets at the tangent space, the inverse map (from the tangent space at a point in M to M itself) would be the exponential map and is given by the affine connection in GR.
The map between charts fails to be invertible but it is injective, it turns out that that is all you need for vacuum solutions and for conformally flat ones (like FRW metrics).
The examples are any coordinate transformation in GR of the above mentioned kind of solutions, that is what I mentioned as the solution to the Einstein's "hole argument". He realized that: "All our spacetime verifications invariably amount to a determination of spacetime coincidences." (Einstein, 1916, p.117)
The drawback is that there are no solutions in general relativity with point particle stress-energy.

 

[TrickyDicky]

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

This is a key point. Often not well explained in GR textbooks. This leads to Killing vector fields in GR not being global(except those geometrically imposed to solutions like spherical symmetry for instance), this took me a long time to grasp.

 

[PAllen]

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?

I would say the physical special relativity principle is that rest cannot be distinguished from inertial motion (in GR, add : locally).

The simplest generalization is obviously false: accelerated motion cannot be distinguished from rest (or inertial motion). The best you have is a particular variant of the equivalence principle: you cannot locally distinguish accelerated motion from rest in a uniform gravitational field.

I think the modern view is the separate out some variant of equivalence principle, some variant of background independents or 'no prior geometry', and the principle relativity (SR sense, local). Then say there is no such thing as the 'general principle of relativity'. This is my view.

Then, general covariance is simply the result of formulating a theory in a coordinate independent manner.

 

[atyy]

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.

Yes, if statement 2 taken to be a special form of statement 1 (general covariance), then it places no restriction on what theories are possible, ie. it is not the PoR.

In order for something with the same words as statement 2 to be the PoR, it cannot be a special form of statement 1. In particular, it must use a definition of "laws of physics have the same form" in which Christoffel symbols are not allowed.

Basically statement 1 is less restrictive than the PoR because it allows use "fudge factors" like the Christoffel symbols to write the laws of physics in "the same form", whereas in the PoR, the use of such fudge factors is not allowed.

 

[DrGreg]

It seems to me that the question to be addressed isn't "Can the laws of physics be expressed the same in all coordinate systems?" but "Are there some coordinate systems in which all the laws of physics take a simpler form than in an arbitrary coordinate system?"

Before relativity, it was thought the answer was yes for the "aether frame" only. Special relativity asserts that the answer is yes for inertial frames. In general relativity the answer is no, but the equivalence principle says the answer is "locally yes as an approximation" for "locally inertial" frames.

 

[PeterDonis]

what is invertible is the function that maps open sets containing a point of the manifold to open sets at the tangent space, the inverse map (from the tangent space at a point in M to M itself) would be the exponential map and is given by the affine connection in GR.

Yes, no problem here. You can do this for the tangent space at any point in the manifold, and it doesn't even involve a chart, strictly speaking.

The map between charts fails to be invertible but it is injective, it turns out that that is all you need for vacuum solutions and for conformally flat ones (like FRW metrics).

Then I'm not sure I understand what you mean by "the map between charts". Take a case where the "double cover" issue in isotropic coordinates on Schwarzschild spacetime doesn't come up: suppose I want a map between ingoing Painleve coordinates and ingoing Eddington-Finkelstein coordinates on Schwarzschild spacetime. Both of these charts cover the exact same portion of the maximally extended manifold (regions I and II, as they're usually labeled on the Kruskal chart), and both of these charts assign unique coordinate values to every point of the manifold in the region they cover. So it seems to me that the map between them is obviously invertible and bijective. What am I missing? Or is it just that we are using different terminology, so you mean something else by "the map between charts" than what I just described? (And if so, what?)

The examples are any coordinate transformation in GR of the above mentioned kind of solutions, that is what I mentioned as the solution to the Einstein's "hole argument". He realized that: "All our spacetime verifications invariably amount to a determination of spacetime coincidences." (Einstein, 1916, p.117)

I understand and agree with the Einstein quote; the real content of our physical models is in observables like "the worldlines of objects A and B intersect at event E".

 

[lugita15]

This is a key point. Often not well explained in GR textbooks.

In fact, you can even define curvature as the extent to which global spacetime symmetries are violated.

 

[zonde]

Yes, if statement 2 taken to be a special form of statement 1 (general covariance), then it places no restriction on what theories are possible, ie. it is not the PoR.

In order for something with the same words as statement 2 to be the PoR, it cannot be a special form of statement 1. In particular, it must use a definition of "laws of physics have the same form" in which Christoffel symbols are not allowed.

Basically statement 1 is less restrictive than the PoR because it allows use "fudge factors" like the Christoffel symbols to write the laws of physics in "the same form", whereas in the PoR, the use of such fudge factors is not allowed.

The difference between statement 2 and PoR is that PoR requires certain symmetry in laws of physics (they are transformed but look the same as a group but not individually) while statement 2 does not require any symmetry in laws of physics (each law individually have exactly the same form after transformation i.e. they are not transformed at all).

 

[lugita15]

The difference between statement 2 and PoR is that PoR requires certain symmetry in laws of physics (they are transformed but look the same as a group but not individually) while statement 2 does not require any symmetry in laws of physics (each law individually have exactly the same form after transformation i.e. they are not transformed at all).

What exactly do you mean by the laws of physics looking the same as a group vs looking the same individually? Also, what is your answer to the question: statement 2 is to the PoR as statement 1 is to ... what? That is to say, what is the physically meaningful statement that bears the same relation to statement 1 that the principle of relativity bears to statement 2? Would it be the statement that the laws of physics look the same "as a group" (whatever you mean by that) in all coordinate systems?

 

[zonde]

What exactly do you mean by the laws of physics looking the same as a group vs looking the same individually?

Say there are physical laws A and B and there is such a transformation that law A after transformation looks like original law B and law B looks like original A. So that group consisting of A and B looks the same after transformation.

Also, what is your answer to the question: statement 2 is to the PoR as statement 1 is to ... what? That is to say, what is the physically meaningful statement that bears the same relation to statement 1 that the principle of relativity bears to statement 2? Would it be the statement that the laws of physics look the same "as a group" (whatever you mean by that) in all coordinate systems?

From physical standpoint statements 1 and 2 are equal IMO. So that I doubt there is meaningful answer to your question.

But I suppose you can say it the way you did: The laws of physics look the same "as a group" in all coordinate systems.

 

[lugita15]

Say there are physical laws A and B and there is such a transformation that law A after transformation looks like original law B and law B looks like original A. So that group consisting of A and B looks the same after transformation.

Could you give me an example of this? I've never heard of the phenomenon you describe.

 

[zonde]

Could you give me an example of this? I've never heard of the phenomenon you describe.

When observing distant stationary star from moving observatory we have aberration. When observing distant moving from stationary observatory we observe it at it's past position.

 

[TrickyDicky]

Then I'm not sure I understand what you mean by "the map between charts". Take a case where the "double cover" issue in isotropic coordinates on Schwarzschild spacetime doesn't come up: suppose I want a map between ingoing Painleve coordinates and ingoing Eddington-Finkelstein coordinates on Schwarzschild spacetime. Both of these charts cover the exact same portion of the maximally extended manifold (regions I and II, as they're usually labeled on the Kruskal chart), and both of these charts assign unique coordinate values to every point of the manifold in the region they cover. So it seems to me that the map between them is obviously invertible and bijective. What am I missing? Or is it just that we are using different terminology, so you mean something else by "the map between charts" than what I just described? (And if so, what?)

We might be not coinciding in our idea of what bijectivity implies for differential manifolds mappings, rather than by map between charts.
To me the case you describe involves injectivity(that is you cannot map two different points of one chart to the same point of the other),with the point in the manifold remaining "static", we are just changing the coordinates of the point, it is a one-to-one function, but it is not a one-to-one correspondence, for that, as was previously commented, you need to be able to "move around" the points of the manifold (active diffeomorphisms) , but in GR you are limited in doing that due to the local nature of the spacetime symmetries, except for rotations that is a symmetry we usually impose on the solutions of the EFE.
What you call the "double cover issue" is just an instance where the lack of one-to-one correspondence manifests.

 

[haushofer]

It's amazing how such a widely used concept (general covariance) can still be so tricky.

 

[TrickyDicky]

It's amazing how such a widely used concept (general covariance) can still be so tricky.

Indeed, see for instance:"General covariance and the foundations of general relativity: eight decades of dispute" by J. D. Norton. It is linked in the wiki entry for GC.

Also this issue seems not to be fully addressed in the usual GR textbooks (maybe because it is still controversial to a certain point), and many people are still not clear on certain subtleties of the mathematical implications of coordinate transformations in GR's pseudoRiemannian manifolds versus general coordinate transformation in Riemannian manifolds.
In practical terms there is a further reason why all this seems to bother very few relativists, since most results in GR only require to be valid locally or "minimal coupling" prescription, these theoretical implications about GC had little impact on the core of GR computations and experimental work.

 

[PeterDonis]

We might be not coinciding in our idea of what bijectivity implies for differential manifolds mappings, rather than by map between charts.

I didn't realize "bijective" had multiple meanings; AFAIK it always means "one-to-one".

To me the case you describe involves injectivity(that is you cannot map two different points of one chart to the same point of the other),

But it's that way in both directions, i.e., bijective, not just injective.

with the point in the manifold remaining "static", we are just changing the coordinates of the point, it is a one-to-one function, but it is not a one-to-one correspondence, for that, as was previously commented, you need to be able to "move around" the points of the manifold (active diffeomorphisms)

But that's a different kind of transformation from a "map between charts". Or at least, the term "map between charts" does not suggest an active diffeomorphism to me, only a passive one. If you're moving around the points of the manifold, and you're also changing the coordinates (how the points in the manifold are labeled), then what exactly are you *not* changing? And if everything is changing, how do you even define the transformation?

What you call the "double cover issue" is just an instance where the lack of one-to-one correspondence manifests.

But the double cover issue I was talking about involves a passive diffeomorphism only; I was using it to illustrate that a "passive map between charts" (to make it clear what kind of diffeomorphism I'm talking about) might not be the same as a one-to-one passive diffeomorphism, because two different patches in one chart might map to the same patch in the other chart.

 

[TrickyDicky]

I didn't realize "bijective" had multiple meanings; AFAIK it always means "one-to-one".

It doesn't, what can be ambiguous is the one-to-one part, both injection and bijection are one-to-one, but the former doesn't include surjection, so it is not necessarily invertible for the complete codomain.

But it's that way in both directions, i.e., bijective, not just injective.

It may be for the neighbourhood of the point but not for the entire manifold.

But that's a different kind of transformation from a "map between charts". Or at least, the term "map between charts" does not suggest an active diffeomorphism to me, only a passive one. If you're moving around the points of the manifold, and you're also changing the coordinates (how the points in the manifold are labeled), then what exactly are you *not* changing? And if everything is changing, how do you even define the transformation?

What you are describing is known as (global)isometry , a type of active diffeomorphism that preserves the metric, and defines spacetime symmetries when the manifold is invariant to them. The thing to keep in mind in GR is that these isometries are local, not global, with the exception of those that might be imposed on the solutions, like isotropy.

But the double cover issue I was talking about involves a passive diffeomorphism only; I was using it to illustrate that a "passive map between charts" (to make it clear what kind of diffeomorphism I'm talking about) might not be the same as a one-to-one passive diffeomorphism, because two different patches in one chart might map to the same patch in the other chart.

They are the same thing, as I said it is an example where the lack of bijectivity of passive transformations shows up.

 

[PeterDonis]

It doesn't, what can be ambiguous is the one-to-one part, both injection and bijection are one-to-one, but the former doesn't include surjection, so it is not necessarily invertible for the complete codomain.

Yes, I see now, I was mixing up definitions in my head. Basically, injective is "one to one", surjective is "onto", and bijective = injective + surjective.

It may be for the neighbourhood of the point but not for the entire manifold.

Huh? The mapping between ingoing Painleve and ingoing Eddington-Finkelstein *is* bijective over the entire manifold, or at least the entire region of the manifold that they both cover. I suppose you could say that since that region is not the entire (maximally extended) manifold, the mapping can't "count" as bijective because there are points of the manifold that aren't mapped, but that applies equally well to both domain and codomain, and there doesn't really seem to be a word for a function that is "one to one", but doesn't map every element of its "domain" (since the definition of "domain" implicitly includes only points for which the function is defined).

What you are describing is known as (global)isometry , a type of active diffeomorphism that preserves the metric, and defines spacetime symmetries when the manifold is invariant to them.

Huh? In an isometry of the type you describe, the chart is held constant. More precisely, an isometry can be described without using a chart at all, so no issue of "mapping between charts" even arises. You just define equivalence classes of points in the spacetime with respect to the isometry. So I don't see how a "map between charts" even comes into play.

It would really be helpful if you would give a specific example; for example, take the "map between charts" that I gave (ingoing Painleve to ingoing Eddington-Finkelstein) and show explicitly how it relates to an isometry.

 

[TrickyDicky]

Huh? The mapping between ingoing Painleve and ingoing Eddington-Finkelstein *is* bijective over the entire manifold, or at least the entire region of the manifold that they both cover. I suppose you could say that since that region is not the entire (maximally extended) manifold, the mapping can't "count" as bijective because there are points of the manifold that aren't mapped...

Yes, that is what I'd say.

Huh? In an isometry of the type you describe, the chart is held constant. More precisely, an isometry can be described without using a chart at all, so no issue of "mapping between charts" even arises. You just define equivalence classes of points in the spacetime with respect to the isometry. So I don't see how a "map between charts" even comes into play.

Precisely isometry invariance is the property of Riemannian manifolds that allows us to do without coordinates, because in Riemannian geometry coordinate transformations are isometries(bijective).
In GR this is limited to its local counterpart, local isometries.

It would really be helpful if you would give a specific example; for example, take the "map between charts" that I gave (ingoing Painleve to ingoing Eddington-Finkelstein) and show explicitly how it relates to an isometry.

They are local isometries.

 

[PeterDonis]

Precisely isometry invariance is the property of Riemannian manifolds that allows us to do without coordinates, because in Riemannian geometry coordinate transformations are isometries(bijective).

Now you're using the word "isometry" in a different sense than I understand it. As I understand the term "isometry", it is what is generated by a Killing vector field. A KVF, and therefore an isometry, can certainly be defined in coordinate-free terms, but that doesn't mean coordinate transformations are isometries. An "isometry" is what you have been calling an "active diffeomorphism" (at least, if I understand your usage of *that* term right); for example, a rotation of a 2-sphere about any axis is an isometry, because it leaves the intrinsic geometry of the 2-sphere invariant. But that has nothing to do with coordinate transformations.

[Edit: I also don't understand why you appear to equate "isometry" with "bijective". I would agree that an isometry must be bijective, but I would not agree that every bijective transformation must be an isometry.]

 

[TrickyDicky]

. As I understand the term "isometry", it is what is generated by a Killing vector field.

Correct.

A KVF, and therefore an isometry, can certainly be defined in coordinate-free terms, but that doesn't mean coordinate transformations are isometries.

Right, it is the other way around, isometries are a subgroup of generalized coordinate transformations, consider this textbook definition:
"Isometry: a coordinate transformation x′^μ = x^μ + ζ^μ(x), which we think of as infinitesimal. The term isometry applies to any transformation that leaves the metric of the same form. The metric is form invariant under such a transformation. We will, however, only consider continuous symmetries."
Simply put an isometry is a (active)diffeomorphism that preserves the metric.

for example, a rotation of a 2-sphere about any axis is an isometry, because it leaves the intrinsic geometry of the 2-sphere invariant. But that has nothing to do with coordinate transformations.

Are you sure rotations are not a type of coordinate transformations?

[Edit: I also don't understand why you appear to equate "isometry" with "bijective". I would agree that an isometry must be bijective, but I would not agree that every bijective transformation must be an isometry.

I don't equate them, it just happens that every isometry just by being a diffeomorphism is bijective. This is not the case with local isometries(that is local diffeomorphisms that preserve the metric), which are injective.


Maybe you should try and compare what I'm saying with a GR/differential geometry textbook.

 

[PeterDonis]

Simply put an isometry is a (active)diffeomorphism that preserves the metric.

I'm fine with that.

Are you sure rotations are not a type of coordinate transformations?

Not the way I think of "coordinate transformations", no. Coordinate transformations ought to, it seems to me, involve coordinates. A rotation does not involve coordinates; it can be defined without ever talking about coordinates at all. See further comments below.

I don't equate them, it just happens that every isometry just by being a diffeomorphism is bijective. This is not the case with local isometries(that is local diffeomorphisms that preserve the metric), which are injective.

Once again, it would be really helpful if you could give a specific example. So far every time I've asked you to do that, you've just stated that an example I gave applies. If that were enough to resolve my confusion, I wouldn't have needed to ask you for an example. I am asking *you* to explicitly exhibit an example of a global isometry and a local isometry and show how they are different, and why the former must be bijective while the latter may only be injective (i.e., not surjective). All the examples I can come up with to fit the term "local isometry" are either bijective, or not even injective (e.g., the mapping between isotropic and Schwarzschild coordinates, if we consider both patches of isotropic coordinates mapping to a single patch of Schwarzschild coordinates as a single "mapping", is not even injective).

Maybe you should try and compare what I'm saying with a GR/differential geometry textbook.

Different textbooks appear to use different terminology as well, so that's not necessarily helpful. I'm not confused about the underlying concepts; I'm confused about which terms you are using to refer to which underlying concepts. I don't have any particular attachment to any particular terminology; I have preferences, but I'm perfectly willing to put them aside and adopt your terminology (or anyone else's) for the sake of having a clear discussion. But I have to be able to understand *what* your terminology is to do that. Even a simple statement like "I'm using the same terminology as textbook X" would help, but just saying "textbooks" or "a textbook" without saying which one is not helpful, because, as I said, they use different conventions for terminology.

Anyway, this subthread seems to me to be getting away from the main topic, which is general covariance. As far as that is concerned, I basically agree with the position Ben Niehoff stated in the second post in this thread: *any* physical theory can be written in "coordinate-independent" form, so general covariance as it's usually stated is trivial; it's just a reminder to write theories in coordinate-independent form. I.e., general covariance has nothing much to say about the *content* of a physical theory; and once we start talking about isometries and other properties of solutions, we are talking about content, not form.

 

[TrickyDicky]

Not the way I think of "coordinate transformations", no. Coordinate transformations ought to, it seems to me, involve coordinates. A rotation does not involve coordinates; it can be defined without ever talking about coordinates at all. See further comments below.

Ok, I think I know what you mean.
All comes from the two different senses of "coordinate transformation". When you say a rotation (as a global symmetry) can be defined in a coordinate-free way, you are of course right, this transformation is a diffeomorphism and therefore a bijection, and it defines two coordinate transformations in the case we want use coordinates to describe a fixed point in the manifold. These diffeomorphisms are sometimes called coordinate transformations by mathematicians but they are clearly not what physicists usually consider coordinate transformations, which are what are called passive transformations, and I call in the case of GR local isometries. I think these are the ones general covariance in GR refer to when talking about coordinate transformation invariance.

Once again, it would be really helpful if you could give a specific example. So far every time I've asked you to do that, you've just stated that an example I gave applies. If that were enough to resolve my confusion, I wouldn't have needed to ask you for an example. I am asking *you* to explicitly exhibit an example of a global isometry and a local isometry and show how they are different, and why the former must be bijective while the latter may only be injective (i.e., not surjective). All the examples I can come up with to fit the term "local isometry" are either bijective, or not even injective (e.g., the mapping between isotropic and Schwarzschild coordinates, if we consider both patches of isotropic coordinates mapping to a single patch of Schwarzschild coordinates as a single "mapping", is not even injective).

Peter, you know I'm just an interested layman, the farthest from a physicist or a mathematician, I'm trying to help but I might not be the most qualified to do that, I was hoping some of the pros would jump in. In the meantime to me for instance rotations in the Schwarzschild spacetime are global isometries, and any passive coordinate transformation from a point in Schwarzschild coordinates to a different chart is a local isometry.

Anyway, this subthread seems to me to be getting away from the main topic, which is general covariance. As far as that is concerned, I basically agree with the position Ben Niehoff stated in the second post in this thread: *any* physical theory can be written in "coordinate-independent" form, so general covariance as it's usually stated is trivial; it's just a reminder to write theories in coordinate-independent form. I.e., general covariance has nothing much to say about the *content* of a physical theory; and once we start talking about isometries and other properties of solutions, we are talking about content, not form.

I also agree.

 

[PeterDonis]

These diffeomorphisms are sometimes called coordinate transformations by mathematicians but they are clearly not what physicists usually consider coordinate transformations, which are what are called passive transformations, and I call in the case of GR local isometries. I think these are the ones general covariance in GR refer to when talking about coordinate transformation invariance.

I agree.

Peter, you know I'm just an interested layman

So am I.

rotations in the Schwarzschild spacetime are global isometries, and any passive coordinate transformation from a point in Schwarzschild coordinates to a different chart is a local isometry.

I'll have to take some time to work through these examples with the definitions.

I do have one rather lengthy comment: transformations between charts aren't always viewed as "local". Some are, for example transformations between Fermi normal coordinates for two observers in relative motion at a particular event. But others are not, for example the transformation between Painleve and Eddington-Finkelstein coordinates; that transformation applies at every point in the manifold that is covered by both charts.

However, the latter transformation is "local" in another sense, that it maps the *same* point from one chart to the other, so it can be viewed as an infinite "family" of transformations, each one mapping a single point only. A rotation (or in general any active transformation) can't be viewed this way; it intrinsically is a mapping from the entire manifold into itself (or into another manifold, in the case of a more general active transformation), and can't be "decomposed" into a family of local transformations that each affect only a single point.

(I think you basically said this in an earlier post, but I wasn't really grokking it until I stepped back and walked through things in more detail.)

 

[haushofer]

I still don't get it really, reflecting the fact that some of my earlier posts here confuse different things.

I think most people now agree that general covariance, the ability to write the equations of motion in a gct-covariant way, is physically void. But I also sometimes see the statement that the real deal of GR is "background independency" or the lack of "a priory geometry": the geometry of spacetime, uniquely determined by the metric, is dynamical and obeying EOM called the Einstein equations. But Newton-Cartan theory is in the same way "background independent": both the metrics (spatial and temporal) are dynamically determined by equations analogous to the Einstein equations of GR. The connection is not uniquely determined by both metrics; one obtains an extra vector field, but that does not change the matter. I could even apply the hole argument for Newton-Cartan theory in the same way as for General Relativity, because the EOM are gct-invariant.

The solution should thus be found in the fact that GR is really a non-linear self-interacting theory of massless spin-2 particles which becomes clear after gauge-fixing, while for Newton-Cartan this cannot be said: after gauge-fixing one obtains a spin (spin is here wrt to the Galilei group!) 0 theory, which is static and non self-interacting.

Does this make sense? And what does "background independence" really mean then?

 

[TrickyDicky]

what does "background independence" really mean then?

I think it is actually a misnomer and somewhat ill-defined but anyway I always understood it as something like the difference between the EM theory of Maxwell and the GRT of Einstein, in the sense that the former equations refer to fields that act in " a background space" and so are "background dependent" while the latter equations refer to a field that "is" the spacetime in itself and therefore "background independent".
I say it is a misnomer and superfluous term because at least since Riemann we know manifolds don't need to be embedded in any "background space" to be defined.

 

[PeterDonis]

I say it is a misnomer and superfluous term because at least since Riemann we know manifolds don't need to be embedded in any "background space" to be defined.

True, but there is a difference between the manifold being predetermined (as in EM) and it being dynamic (as in GR). I agree that "background independence" is a bad term for the latter case, though; why not "dynamic manifold" or something like that? After all, the key difference is that in GR the manifold (spacetime) appears in the dynamical equations of the theory, where in EM it doesn't.

 

[haushofer]

True, but there is a difference between the manifold being predetermined (as in EM) and it being dynamic (as in GR). I agree that "background independence" is a bad term for the latter case, though; why not "dynamic manifold" or something like that? After all, the key difference is that in GR the manifold (spacetime) appears in the dynamical equations of the theory, where in EM it doesn't.

I agree with you on the terminology, but bad terminology seems to dominate this whole discussion anyway ;) So in the sense of BI Newton-Cartan theory and GR don't differ. But why do people like Rovelli then keep hammering on the importance of BI, if clearly even Newtonian gravity can be made BI? Clearly, it doesn't say that much.

I think ultimately, the fact that GR is BI is not a defining property of the theory; what is the defining property is what is left of your theory after you have fixed gauges to uncover physical degrees of freedom (in the case of GR, this comes down to a perturbative analysis and noticing that one is really dealing with massless self-interacting spin-2).

Do you agree that the hole argument is just as applicable to Newton-Cartan theory as GR?

 

[PeterDonis]

So in that sense Newton-Cartan theory and GR don't differ.

I'm not sure that's true. Newton-Cartan theory has absolute time and an absolute slicing of the complete manifold into spacelike slices, so the spacetime isn't completely dynamic as it is in GR. (Btw, you said in an earlier post that N-C theory determines the "temporal metric" dynamically; I'm not sure that's true either. There is no gravitational time dilation in N-C theory.)

I think ultimately, the fact that GR is BI is not a defining property of the theory

I think the question here is, is GR the *only* possible theory that is BI in the way GR is? (I.e., with a *completely* dynamic spacetime metric.) I don't think anybody really knows the answer to that.

 

[Ben Niehoff]

I think the question here is, is GR the *only* possible theory that is BI in the way GR is? (I.e., with a *completely* dynamic spacetime metric.) I don't think anybody really knows the answer to that.

The answer is obviously "no". GR is defined by the Einstein-Hilbert action

S = \int (R(g) - 2 \Lambda) \, \sqrt{|g|} d^4x
Clearly I can put together any curvature invariants I feel like into a Lagrangian and I will have another theory where the metric is dynamic. "Background independent" is not overly restrictive.

 

[haushofer]

I'm not sure that's true. Newton-Cartan theory has absolute time and an absolute slicing of the complete manifold into spacelike slices, so the spacetime isn't completely dynamic as it is in GR. (Btw, you said in an earlier post that N-C theory determines the "temporal metric" dynamically; I'm not sure that's true either. There is no gravitational time dilation in N-C theory.)

In NC one has a spatial and temporal metric, which are metric-compatibel defining a connection up to a two-form K. The temporal metric with lower indices is determined by its metric-compatibility. Inverses of these metrics are defined via projective relations, and the temporal metric with upper indices is not fixed by the metric compatibility conditions. One can then impose field equations as one likes in terms of the Riemann/Ricci tensor (the question if these equations can be derived via an action principle is a different matter), and the usual Newton-Cartan field equations are chosen such that all the dynamical metric components and components of the two-form K can be gathered into a Galilei-scalar, known as the Newton potential, and all the other metric components become constant. This last fact is the flat-space content of Newton-Cartan, and is an explicit choice; one could also choose other dynamics such that space is not flat, giving a Galilean theory of gravity with curved space (i.e. the transformations in the tangent space are the Galilei transformations).

Of course, because the field equations of NC just reproduce Newtonian gravity, there will be no time dilation. It also depends on what one calls "dynamics"; usually metric compatibility is not considered to be dynamics.

I think the question here is, is GR the *only* possible theory that is BI in the way GR is? (I.e., with a *completely* dynamic spacetime metric.) I don't think anybody really knows the answer to that.

One could formulate Newton-Cartan theory without the flat space condition, giving an honest BI theory with metrics which are even after gauge-fixing dynamical. The question is if such a theory is always some limiting case of GR. One can also define stringy versions of Newton-Cartan, based on strings or even branes instead of point particles, see

http://arxiv.org/abs/1206.5176

These theories are not the usual Newtonian limits of GR, so in that sense GR (with the possible additional terms to the Einstein Hilbert action as Bien Niehoff mentions) doesn't seem to be the only BI theory.

 

[haushofer]

Anyway, I have the ambition to, once I thoroughly understand all this stuff, put it in some notes without all the usual explicit (often coordinate-free) mathematical mumbo-jumbo and vague terminology obscuring for me personally what's really going on. Somehow I still haven't found a nice and clear overview of the meaning of covariance, the meaning of and relation between active and passive coordinate transformations, etc. Even good books like Carroll couldn't really satisfy my needs. I sometimes have the feeling that a lot of physicists don't really care, and a lot of "philosophers of physics" make the discussion so obscure that it makes me wanting to run back to the "shut up and calculate"- mentality :D

 

[PAllen]

Anyway, I have the ambition to, once I thoroughly understand all this stuff, put it in some notes without all the usual explicit (often coordinate-free) mathematical mumbo-jumbo and vague terminology obscuring for me personally what's really going on. Somehow I still haven't found a nice and clear overview of the meaning of covariance, the meaning of and relation between active and passive coordinate transformations, etc. Even good books like Carroll couldn't really satisfy my needs. I sometimes have the feeling that a lot of physicists don't really care, and a lot of "philosophers of physics" make the discussion so obscure that it makes me wanting to run back to the "shut up and calculate"- mentality :D

Did you try to read Ben Crowell's reference:

http://arxiv.org/abs/gr-qc/0603087

This makes a better attempt than most I've seen to try formalize what distinguishes GR from e.g. Newton-Cartan (for example). Unfortunately, its overall conclusion is that the matter is not yet resolved, after all these years; that ultimately, background independence, no prior geometry, no absolute structures, etc. is not yet subject to any rigorous, problem free definition.

 

[atyy]

But Newton-Cartan theory is in the same way "background independent": both the metrics (spatial and temporal) are dynamically determined by equations analogous to the Einstein equations of GR. The connection is not uniquely determined by both metrics; one obtains an extra vector field, but that does not change the matter. I could even apply the hole argument for Newton-Cartan theory in the same way as for General Relativity, because the EOM are gct-invariant.

The solution should thus be found in the fact that GR is really a non-linear self-interacting theory of massless spin-2 particles which becomes clear after gauge-fixing, while for Newton-Cartan this cannot be said: after gauge-fixing one obtains a spin (spin is here wrt to the Galilei group!) 0 theory, which is static and non self-interacting.

Does this make sense? And what does "background independence" really mean then?

Yes, I think the quantum spin-2 way is the best, since if one formulates GR as field on flat spacetime, then there is a flat spacetime which is clearly not background independent. Also, because of the comments in the paper PAllen mentions in #93.

What I'm not sure about is: does the field on flat spacetime contain cosmology? No need to include the "big bang singularity", but just the physically relevant bits that present observations constrain?

 

[haushofer]

Did you try to read Ben Crowell's reference:

http://arxiv.org/abs/gr-qc/0603087

This makes a better attempt than most I've seen to try formalize what distinguishes GR from e.g. Newton-Cartan (for example). Unfortunately, its overall conclusion is that the matter is not yet resolved, after all these years; that ultimately, background independence, no prior geometry, no absolute structures, etc. is not yet subject to any rigorous, problem free definition.

No, I still haven't due to other obligations, but I hope to read it carefully when I'm able to. It rather strikes me that all these terms are still not that well understood, and I like the fact that some people do make an effort to shed more light on it. The discussions here also help a lot, so thanks for that!

One of the reasons why I got so interested in this whole notion of "background independence" was because I heard the claim from some string theory critics that "any good theory of quantum gravity should be BI" and "string theory is not BI". But the meaning of this becomes, after these discussions, a bit blurry to say the least. (The primary reason was actually that for my master thesis I had to read Wald's article on "black hole entropy is Noether charge". It then occurred to me I'd never really understood this whole business.)

 

[Haelfix]

Yes, I think the quantum spin-2 way is the best, since if one formulates GR as field on flat spacetime, then there is a flat spacetime which is clearly not background independent.

Again, it depends on definitions. If you are going by the definitions that Ben Niehoff for instance is using above, then the above *is* background independant.

Note the difference between the two following definitions...

1) Background dependence is tantamount to using the background field method for gravity, and ONLY for gravity (eg the metric tensor is split into a classical but arbitrary fixed background metric + a small perturbation). The approximation is valid up to some cutoff, whereupon the backreaction of the pertubation on the background can no longer be ignored.

2) Background independance is like asking whether the metric field is dynamical or not in the Lagrangian of the theory. In the sense that if you look at the variation in the action and consider (d/d&G), then you look for something that vanishes. So for instance, coupling a topological field theory to a theory with curvature invariants is clearly background independant in this definition. The terms with curvature invariants, owing to their general covariance, will integrate out any metric dependance, and terms that are topological have no metric dependancy at all. Contrast that with something like a Maxwell term, which when acted with the operator, will instantly pull out the nondynamical and absolute fixed structure.

Both definitions (as well as anyone that you can think off) are not going to generalize universally, or serve as a theory 'filter'. The first problem is that the word 'background' is often generalized in the literature to mean something more than just a classical solution of Einstein's equations. Second, its a little bit unclear what physical principle you are trying to capture that is so damn important, considering that even classical GR can be written in ways that make it look background dependant. (Consider writing GR like field theorists for the first case and consider the pure connection formalism for the 2nd)

 

[haushofer]

I finally read the paper by Giulini, and it is nice. He defines NC gravity being not background independent because of the appearence of "absolute structures": most of the metric components of NC gravity only have 1 solution (modulo gct's), whereas all the other components are gathered into the Newton potential.

I think I found some nice insights in the paper :) It's also nice to compare the general-covariantization of the Poisson equation (giulini does it for the Schrodinger equation, but the difference is only a time derivative) with the formulation of Newton-Cartan. The latter can be seen as a much less trivial general-covariantization of Newton.

 

[stevendaryl]

I think it is actually a misnomer and somewhat ill-defined but anyway I always understood it as something like the difference between the EM theory of Maxwell and the GRT of Einstein, in the sense that the former equations refer to fields that act in " a background space" and so are "background dependent" while the latter equations refer to a field that "is" the spacetime in itself and therefore "background independent".
I say it is a misnomer and superfluous term because at least since Riemann we know manifolds don't need to be embedded in any "background space" to be defined.

The way that I heard it described once is this:

You can formulate just about any theory of physics in a generally covariant form, which really means writing it in terms of geometric objects that can be defined independently of a choice of coordinates: Scalar fields, vector fields, tensor fields.

The theory is "background free" if there are no nondynamic scalar, vector or tensor fields. By "nondynamic", I mean a field that appears in the equations of physics (when written in generally covariant form) but which is not itself governed by the physics.

For example, in Newtonian physics, universal time is a scalar field that is nondynamic. In Special Relativity, the metric tensor is a tensor field that is nondynamic. General Relativity has no nondynamic fields.

 

[haushofer]

The way that I heard it described once is this:
The theory is "background free" if there are no nondynamic scalar, vector or tensor fields. By "nondynamic", I mean a field that appears in the equations of physics (when written in generally covariant form) but which is not itself governed by the physics.

For example, in Newtonian physics, universal time is a scalar field that is nondynamic. In Special Relativity, the metric tensor is a tensor field that is nondynamic. General Relativity has no nondynamic fields.

I'd say that is rather vague. E.g., in Newton-Cartan theory the Ricci tensor is determined by the matter density, and this Ricci tensor is defined by the connection, which on its turn depends on the temporal metric!

What is true, is that up to gct's this temporal metric (with lower indices) only has one solution. That's a hint that it is non-dynamical, and that the theory has been "Stückelberged".

All the metric components of NC-theory turn out to be non-dynamical this way, except for a combination of components which form the Newton potential. This potential does have more solutions up to gct's, and as such is the only dynamical field in the theory.

In your definition I can always postulate EOM for the non-dynamical field. The simplest example for this was already given; if one has a Minkowksi metric and partial derivatives in a theory, just general-covariantize this and impose as extra EOM that the Riemann tensor for this metric vanishes.