How GR Resolves the Conservation of Momentum and Energy

[TrickyDicky]

Correct; there is no diffeomorphism that converts Schwarzschild into Kerr spacetime, for example. But I can certainly define diffeomorphisms that convert Schwarzschild under one coordinate chart (say the Schwarzschild chart) into Schwarzschild under another coordinate chart (say the Painleve chart). And both are solutions of the EFE. I realize that that seems like a tautology, since both charts describe the same geometry; but the point is that formally, the metric written in Schwarzschild coordinates solves the EFE written in those coordinates, and the metric written in Painleve coordinates solves the EFE written in those coordinates. The metric and the coordinate chart transform together to keep the underlying geometry invariant. So GR is diffeomorphism invariant in this sense.

This was my line of reasoning too, but then what happens with the Kerr metric? isn't it also a solution of the EFE?

 

[TrickyDicky]

See what I mean?

Hmm, I'm afraid we re either talking about different things or else I detect several confusing issues in your statements.

Absolutely not! Forget about GR for a second. It is not even true about classical mechanics.

For instance, laws of physics (eg special relativity) are invariant under translations! However a particular solution typically is NOT invariant. For instance, if you are in an everywhere empty universe except for one room with a wall, that particular solution explicitly breaks Lorentz invariance in one direction.

Laws of physics accordint to what specific theory? GR? I thought you said forget about GR, SR? Minkowski spacetimes are strictly empty, the moment you introduce Lorentz invariance breakings is no longer SR. That is precisely what happens in GR for translations, due to the intrinsic curvature of the spacetime. Lorentz invariance is only local there.

Another example more pertinent the real world. In general, the particular solution of the EFE that we live in is decidedly not invariant under all diffeomorphisms. It is not even invariant under changes of scale (which is a subgroup of the diffeomorphism group) due to the presence of massive particles (which explicitly break conformal symmetry).

I don't think changes of scale (dilations) are usually included in the diffeomorphism group, at least as understood in GR, as the group of coordinate transformations. A dilation is not a diffeomorphism. See this thread post #4 for reference: https://www.physicsforums.com/showthread.php?t=572492

Another example the Minkowski metric breaks an infinite amount of diffeomorphisms, and only leaves a finite amount of isometries unbroken (four rotations, three translations and three boosts)

I don't even know what you mean by "breaks an infinite amount of diffeomorphisms", it seems a wrong statement.

In fact the only tensor that is invariant under all diffeomorphisms is the trivial metric with all coefficients zero, but this fails to be a metric b/c it is not invertible.

Please, go thru your notion of diffeomorphism again, we might not be taking about the same thing. Not every transformation you might think of is a diffeomorphism, it is mostly restricted to coordinate charts changes AFAIK.

 

[PeterDonis]

This was my line of reasoning too, but then what happens with the Kerr metric? isn't it also a solution of the EFE?

Yes, and you can express it in different charts as well: Boyer-Lindquist, Doran, and Kerr-Schild all come to mind. All of those can be inter-converted via diffeomorphisms. But no diffeomorphism can change Kerr into Schwarzschild or vice versa.

I see what you are getting at: one could interpret the term "diffeomorphism invariance" as requiring that diffeomorphisms can convert between *any* pair of solutions. But that interpretation is obviously too strong, because different solutions can have different topologies. But the other obvious interpretation, limiting "diffeomorphism" to just passive diffeomorphisms, changes of coordinate chart on the same spacetime, seems too weak, since it leaves no scope for "active diffeomorphisms" at all. I'm struggling with that too; I don't see any obvious middle ground between the "too strong" version and the "too weak" version, because I can't come up with any examples of spacetimes which are non-trivially different but still have the same topology.

 

[George Jones]

Also, could you still do an active diffeomorphism taking just a portion of one manifold to a portion of another?

I can't come up with any examples of spacetimes which are non-trivially different but still have the same topology.

I don't have time to write much, as I'm out with my family now.

Knowing the manifold and topology isn't enough, because most manifolds admit inequivalent Lorentzian metrics. It even is possible to have non-zero Riemann curvature tensor tensor for (M,g) while (M,h) is completely (intrinsically) flat! Intrinsic curvature is not a property of the manifold and topology alone. Minkowski space with a point removed is the topological space S^3 x R, the underlying space for the manifold of closed Friedmann-Robertson-Walker universes, and Minkowski space with a straight line removed is S^2 x R^2, the underlying space for the manifold of extended Schwarzschild.

 

[TrickyDicky]

Yes, and you can express it in different charts as well: Boyer-Lindquist, Doran, and Kerr-Schild all come to mind. All of those can be inter-converted via diffeomorphisms. But no diffeomorphism can change Kerr into Schwarzschild or vice versa.

I see what you are getting at: one could interpret the term "diffeomorphism invariance" as requiring that diffeomorphisms can convert between *any* pair of solutions. But that interpretation is obviously too strong, because different solutions can have different topologies. But the other obvious interpretation, limiting "diffeomorphism" to just passive diffeomorphisms, changes of coordinate chart on the same spacetime, seems too weak, since it leaves no scope for "active diffeomorphisms" at all. I'm struggling with that too; I don't see any obvious middle ground between the "too strong" version and the "too weak" version, because I can't come up with any examples of spacetimes which are non-trivially different but still have the same topology.

Glad you see what I mean.
As I said diffeomorphisms in their mathematical use include both active and passive transformations (bijectivity), so they only leave room for the "strong version interpretation" that is not compatible with different topologies.

After some reading it is obvious to me this is still controversial after 98 years of GR.

 

[PeterDonis]

Knowing the manifold and topology isn't enough, because most manifolds admit inequivalent Lorentzian metrics. It even is possible to have non-zero Riemann curvature tensor tensor for (M,g) while (M,h) is completely (intrinsically) flat! Intrinsic curvature is not a property of the manifold and topology alone.

George, thanks, this helps to clarify things and to narrow the focus of the issue that I'm struggling with. Consider your examples:

Minkowski space with a point removed is the topological space S^3 x R, the underlying space for the manifold of closed Friedmann-Robertson-Walker universes

The latter are solutions of the EFE, but is Minkowski space with a point removed a solution of the EFE? I know Minkowski space itself is, but wouldn't removing a point from it mess up the solution? The removed point would act like a singularity, but a solution of the EFE can't have a singularity in an otherwise completely flat spacetime, can it?

, and Minkowski space with a straight line removed is S^2 x R^2, the underlying space for the manifold of extended Schwarzschild.

Same question as above, or more generally: are there any other solutions of the EFE, besides extended Schwarzschild, which have S^2 x R^2 as the underlying space?

 

[PeterDonis]

Same question as above, or more generally: are there any other solutions of the EFE, besides extended Schwarzschild, which have S^2 x R^2 as the underlying space?

After thinking about this some more, it seems to me that Reissner-Nordstrom spacetime should have this topology. If so, a diffeomorphism between Schwarzschild and R-N spacetime would be an "active" diffeomorphism.

It also seems like FRW spacetime with k = 0 (i.e., flat spatial slices) should have topology R^4, as Minkowski spacetime does, so there should be an "active" diffeomorphism between those two as well.

 

[TrickyDicky]

is Minkowski space with a point removed a solution of the EFE? I know Minkowski space itself is, but wouldn't removing a point from it mess up the solution? The removed point would act like a singularity, but a solution of the EFE can't have a singularity in an otherwise completely flat spacetime, can it?

But if removing a point turns it into a FRW solution it certainly must not be a completely flat spacetime anymore, and it surely is a solution of the EFE.


One thing we didn't mention before is that different boundary conditions or geometric constraints like cilindrical or spherical symmetry for the EFE obviously give rise to geometrically inequivalent solutions, that helps explain why the Kerr solution is not diffeomorphic or even homeomorphic to solutions that are demanded to be spherically symmetric.

 

[Dale]

One thing we didn't mention before is that different boundary conditions or geometric constraints like cilindrical or spherical symmetry for the EFE obviously give rise to geometrically inequivalent solutions, that helps explain why the Kerr solution is not diffeomorphic or even homeomorphic to solutions that are demanded to be spherically symmetric.

But as far as I know coffee mugs and donuts have the same topological group and are therefore diffeomorphic, even though they have different symmetries. So I am not sure that the symmetry constraints are that strong.

 

[TrickyDicky]

But as far as I know coffee mugs and donuts have the same topological group and are therefore diffeomorphic,

They share the same topology, yes, so restricting to their topology we say that they are homeomorphic, being diffeomorphic alludes to their putative additional differential (smooth) structure.

even though they have different symmetries.

Not sure exactly what specific symmetries you are referring to, and also the 2-dimensional case is very particular among manifolds and very different from the 4-dimensional case. A donut and a coffee mug both share axis symmetry. Which is the one that matters topologically in this case.

So I am not sure that the symmetry constraints are that strong.

Certain global symmetries are that strong, and in the case we were discussing for instance the difference between a spherical symmetry and a cylindrical symmetry is strong in a topological level. The usual example is that of a cylinder versus a sphere when they are spread out in a plane, the sphere cannot be smooth out without tearing while the cylinder can.

My conclusion is that whenever one compares solutions of the EFE one must consider also the specific geometric conditions that are used, because if they don't share certain different global symmetries, they will hardly be solutions topologically equivalent, which is a condition to be difeomorphic.
This renders the demand for diffeomorphism invariance a bit empty of content IMO,(it would have only the weak interpretion mentioned by PeterDonis) being only an almost trivial constraint that is not peculiar to GR but that can be applied to any physical theory in as much as it is agreed that coordinates are not physical.

 

[Haelfix]

Laws of physics accordint to what specific theory? GR? I thought you said forget about GR, SR? Minkowski spacetimes are strictly empty, the moment you introduce Lorentz invariance breakings is no longer SR.

No not true! Again, the equations must be invariant, but a particular solution or model need not (due to either explicit or spontaneous symmetry breaking).

Another example: Consider Newton and Maxwells laws. They are invariant under time reversal t--> -t, the laws seem to be reversible. However a particular solution that corresponds to the real world need not be. So for instance an icecube that is melting in the sun is irreversible. It never unmelts! This is a consequence of explicit symmetry breaking by initial conditions (in this case, the low entropy configuration preferentially picks out a direction or arrow of time). Note that we don't say that this model violates Newtonian physics. I recommend reading the Feynman lectures volume 1, there is a chapter on symmetries in physics.

I don't think changes of scale (dilations) are usually included in the diffeomorphism group, at least as understood in GR, as the group of coordinate transformations. A dilation is not a diffeomorphism. See this thread post #4 for reference: https://www.physicsforums.com/showthread.php?t=572492

A conformal transformation (for the present purposes sometimes called a conformal isometry) is a diffeomorphism that preserves the metric up to a scale. Eg schematically: F*g = (e^2sigma) g. See Nakahara for the precise definitions.

Now there is a bit of a subtlety b/c whether a conformal transformation is viewed as a symmetry or a diffeomorphism depends on what you keep fixed and what you allow to be transformed, so it's always important to keep that in mind. However the mathematics that comes out are guarenteed to be isomorphic in the classical case of GR, provided you take the appropriate pullbacks when needed.

 

[Haelfix]

I don't have time to write much, as I'm out with my family now.

Knowing the manifold and topology isn't enough, because most manifolds admit inequivalent Lorentzian metrics. It even is possible to have non-zero Riemann curvature tensor tensor for (M,g) while (M,h) is completely (intrinsically) flat! Intrinsic curvature is not a property of the manifold and topology alone. Minkowski space with a point removed is the topological space S^3 x R, the underlying space for the manifold of closed Friedmann-Robertson-Walker universes, and Minkowski space with a straight line removed is S^2 x R^2, the underlying space for the manifold of extended Schwarzschild.

Yes! You also need to specify the differential structure as well. For instance, the space S^7 has 27 different differential structures.

 

[TrickyDicky]

No not true! Again, the equations must be invariant, but a particular solution or model need not (due to either explicit or spontaneous symmetry breaking).
Another example: Consider Newton and Maxwells laws. They are invariant under time reversal t--> -t, the laws seem to be reversible. However a particular solution that corresponds to the real world need not be. So for instance an icecube that is melting in the sun is irreversible. It never unmelts! This is a consequence of explicit symmetry breaking by initial conditions (in this case, the low entropy configuration preferentially picks out a direction or arrow of time). Note that we don't say that this model violates Newtonian physics. I recommend reading the Feynman lectures volume 1, there is a chapter on symmetries in physics.

I don't even know where to start disentangling this, you seem to be conflating classic GR with different or more general systems that include spontaneous symmetry breaking in which the laws are invariant but the system isn't because the "background" of the system is non-invariant. I'm perfectly aware of those kind of systems that are frequent in certain models in QFT and hep. But the issue we are treating is restricted to classical GR and it was in that context that I made my claims, so If they are not true it is not because of the reasons you bring up.
By the way, according to the wiki page on symmetry breaking your example is incorrect, when the symmetry breaking is explicit the laws themselves aren't invariant, which is not the case you describe, you must be referring here to spontaneous symmetry breaking.

A conformal transformation (for the present purposes sometimes called a conformal isometry) is a diffeomorphism that preserves the metric up to a scale. Eg schematically: F*g = (e^2sigma) g. See Nakahara for the precise definitions.

Now there is a bit of a subtlety b/c whether a conformal transformation is viewed as a symmetry or a diffeomorphism depends on what you keep fixed and what you allow to be transformed, so it's always important to keep that in mind. However the mathematics that comes out are guarenteed to be isomorphic in the classical case of GR, provided you take the appropriate pullbacks when needed.

Whether a certain transformation is a diffeomorphism in a determined manifold depends on certain features of the manifold. That is the only subtlety I see here, for instance a scale transformation is a diffeomorphism if the manifold has no intrinsic curvature, but in the case we are dealing with (GR) the manifold is demanded to have intrinsic curvature, in which case once again a scale transformation is not a diffeomorphism in such manifolds.

 

[Haelfix]

I don't even know where to start disentangling this, you seem to be conflating classic GR with different or more general systems that include spontaneous symmetry breaking in which the laws are invariant but the system isn't because the "background" of the system is non-invariant. By the way, according to the wiki page on symmetry breaking your example is incorrect, when the symmetry breaking is explicit the laws themselves aren't invariant, which is not the case you describe, you must be referring here to spontaneous symmetry breaking.
.

This is not quite right, but it's getting astray. My point is valid for any physical system, and GR is no exception.

Whether a certain transformation is a diffeomorphism in a determined manifold depends on certain features of the manifold. That is the only subtlety I see here, for instance a scale transformation is a diffeomorphism if the manifold has no intrinsic curvature, but in the case we are dealing with (GR) the manifold is demanded to have intrinsic curvature, in which case once again a scale transformation is not a diffeomorphism in such manifolds..

This is confusing multiple things, both from this thread and from the one you linked too! A particular map is a diffeomorphism, if it satisfies the definition of a diffeomorphism! The definition I gave above for a conformal isometry automatically implies that it is a diffeomorphism (again see Nakahara). A related idea, is the concept of a Weyl rescaling (which some authors confusingly call a scale transformation). These are different things!

 

[TrickyDicky]

This is not quite right, but it's getting astray. My point is valid for any physical system, and GR is no exception.

It is indeed a rather orthogonal discussion to the one of the OP. Out of curiosity can you give me an example of a solution of the EFE that shows spontaneous symmetry breaking?

This is confusing multiple things, both from this thread and from the one you linked too! A particular map is a diffeomorphism, if it satisfies the definition of a diffeomorphism! The definition I gave above for a conformal isometry automatically implies that it is a diffeomorphism (again see Nakahara). A related idea, is the concept of a Weyl rescaling (which some authors confusingly call a scale transformation). These are different things!where the laws are invariant but the system isn't because the background of the system, its vacuum, is non-invariant.

First of all that definition is of a conformal transformation, not exactly the same as a scale transformation, a conformal transf. is a localized scale transformation so please don't generate unwarranted confusion.
A conformal tr. is just one that preserves the angles, a scale transformation doesn't when the manifold has curvature.

Anyway please read post #16 in the previously linked thread and tell me if you disagree with it.

 

[Haelfix]

It is indeed a rather orthogonal discussion to the one of the OP. Out of curiosity can you give me an example of a solution of the EFE that shows spontaneous symmetry breaking?

Sure but it depends what symmetry you are talking about! In general the presence of any background metric, spontaneously breaks the diffeomorphism gauge 'symmetry' of GR down to a subgroup of isometries. If you are, on the other hand, talking about some specific symmetry (for instance one generated by a killing field), then its a little less obvious however it does exist (you will need to use the initial value formulation of GR). For instance, the static Einstein universe is unstable, and although its not often presented that way, you can think of it as undergoing a phase transition where the cosmological constant acts like an order parameter. Very much like balancing a pencil on its tip!

A conformal tr. is just one that preserves the angles, a scale transformation doesn't when the manifold has curvature.
Anyway please read post #16 in the previously linked thread and tell me if you disagree with it.

I don't disagree with it, but he is talking about what I just called a Weyl rescaling. In general, and particular when ever someone talks about general covariance, diffeomorphism invariance and active and passive transformations, it is vital to stick to one textbook. Names get mixed up all over the literature. I recommend Wald or Nakahara.

 

[TrickyDicky]

Sure but it depends what symmetry you are talking about! In general the presence of any background metric, spontaneously breaks the diffeomorphism gauge 'symmetry' of GR down to a subgroup of isometries. If you are, on the other hand, talking about some specific symmetry (for instance one generated by a killing field), then its a little less obvious however it does exist (you will need to use the initial value formulation of GR). For instance, the static Einstein universe is unstable, and although its not often presented that way, you can think of it as undergoing a phase transition where the cosmological constant acts like an order parameter. Very much like balancing a pencil on its tip!

Yes the Einstein universe solution is unstable, I'm not sure it works as an example of spontaneous SB (wrt what symmetry and how that particular solution is not invariant to that symmetry?) but that is perhaps a theme for a different thread.

I don't disagree with it, but he is talking about what I just called a Weyl rescaling.

You were the first to mention scale transformations as an example but let's drop it, no point arguing about it.

A question to recover the original argument: do you then think there are solutions of the EFE that are not diffeomorphism invariant?

 

[PeterDonis]

But if removing a point turns it into a FRW solution it certainly must not be a completely flat spacetime anymore, and it surely is a solution of the EFE.

Removing a point from Minkowski spacetime doesn't turn it into an FRW solution. It just creates a spacetime which happens to have the same underlying topology as an FRW solution which is spatially closed. But Minkowski spacetime with a point removed still has the Minkowski metric on it, and I don't think *that* is a solution of the EFE.

 

[TrickyDicky]

Removing a point from Minkowski spacetime doesn't turn it into an FRW solution. It just creates a spacetime which happens to have the same underlying topology as an FRW solution which is spatially closed.

Yes, my answer was too rushed, I should have been more explicit.
The spacetime that results from removing a point is in any case diffeomorphic to the FRW closed solution if the latter is a patch of the former, do you agree?

But Minkowski spacetime with a point removed still has the Minkowski metric on it, and I don't think *that* is a solution of the EFE.

Here I disagree, removing a point of a space changes its topology and that can lead to change the metric from flat to curved.
We know the closed FRW solution is a solution of the EFE, and if you agree with what I said above about it being diffeomorphic to the space that results from removing a point from Minkowski space then the latter must also be a solution, right?
If not having singularities was a condition to be an EFE solution then the FRW spacetimes wouldn't qualify, don't you think?

 

[PeterDonis]

The spacetime that results from removing a point is in any case diffeomorphic to the FRW closed solution if the latter is a patch of the former, do you agree?

The FRW solution with closed spatial slices isn't a "patch" of Minkowski spacetime with a point removed. They are different Lorentzian metrics that can be put on the same underlying topological space, S^3 x R. But not every Lorentzian metric is a solution of the EFE; the flat Minkowski metric on S^3 x R (i.e., Minkowski space with a point removed, but with metric unchanged), to the best of my knowledge, is not. Given that it's not, I'm not sure whether it would be diffeomorphic to S^3 x R with the spatially closed FRW metric on it, since the latter of course is a solution of the EFE.

Here I disagree, removing a point of a space changes its topology and that can lead to change the metric from flat to curved.

Topology and metric are separate things. You can change the topology without changing the metric, and vice versa. Whether a particular combination of topology and metric is a solution of the EFE is a different question.

If not having singularities was a condition to be an EFE solution then the FRW spacetimes wouldn't qualify, don't you think?

I didn't say a solution of the EFE couldn't have singularities, period. I said I didn't think a solution of the EFE could have a singularity at one point but have a flat metric everywhere else.

 

[TrickyDicky]

The FRW solution with closed spatial slices isn't a "patch" of Minkowski spacetime with a point removed. They are different Lorentzian metrics that can be put on the same underlying topological space, S^3 x R. But not every Lorentzian metric is a solution of the EFE; the flat Minkowski metric on S^3 x R (i.e., Minkowski space with a point removed, but with metric unchanged), to the best of my knowledge, is not. Given that it's not, I'm not sure whether it would be diffeomorphic to S^3 x R with the spatially closed FRW metric on it, since the latter of course is a solution of the EFE.

Since you were the one that asked GeorgeJones whether it is a solution, I take it you are at least not sure about it.

Topology and metric are separate things. You can change the topology without changing the metric, and vice versa. Whether a particular combination of topology and metric is a solution of the EFE is a different question.

I didn't say anything contradicting this, note I wrote "can lead to change" , not "necessarily changes". Do you not agree that a topological change (such as removing a point) can lead to a change in the metric?

I didn't say a solution of the EFE couldn't have singularities, period. I said I didn't think a solution of the EFE could have a singularity at one point but have a flat metric everywhere else.

Ah, ok. But why that bias towards flat metrics, if you accept that curved metrics like the FRW for instance do have singularities?

 

[TrickyDicky]

Oh,well , I'll throw this out on the stoop and see if the cat licks it up:

We are at all moments dealing with Riemannian (therefore differentiable) manifolds, and differentiable manifolds that share an underlying topology like the ones that we were considering in the previous post should be diffeomorphic by definition.
That means one can be obtained from the other thru an arbitrary coordinate transformation and we know that no coordinate transformation can change the intrinsic curvature of a manifold so they both must be curved since we know the closed FRW metric is curved and also we can deduce that the other manifold must be a solution of the EFE if it can be obtained by an arbitrary coordinate transformation from a solution.

Peter, I guess if you still think that the minkowski spacetime with a point removed must conserve its flat metric you can't agree with me, but I would like to know why you insist on it, can you base that reasoning on something? Maybe that would help us both.

 

[TrickyDicky]

differentiable manifolds that share an underlying topology should be diffeomorphic by definition.

Thinking it over I'm not sure this is true, could some expert confirm it or refute it?
Edit: It is possibly not true for dimension 4 or bigger, only for low dimensions.

 

[PeterDonis]

Since you were the one that asked GeorgeJones whether it is a solution, I take it you are at least not sure about it.

I wasn't sure about Minkowski space with a point removed being a solution, yes.

Do you not agree that a topological change (such as removing a point) can lead to a change in the metric?

Not really, because there's no connection between the two, so saying "can lead to" sounds misleading to me. At least, there's no connection mathematically; as I said before, you can choose the topological space and the metric independently if all you're doing is constructing a mathematical object.

If you're asking whether a change in the topology of spacetime would lead to a change in the metric *physically*, I can't really answer that, since I don't know what theoretical framework one would use to get a handle on it as a question of physics (as opposed to just mathematics).

Ah, ok. But why that bias towards flat metrics, if you accept that curved metrics like the FRW for instance do have singularities?

The only flat metric I'm aware of is the Minkowski metric. Are there others? And the only way I'm aware of that the flat Minkowski metric can be a solution of the EFE is as the most trivial vacuum solution: no matter, no cosmological constant, nothing. And that solution does not have any singularities: it's the flat Minkowski metric on R^4. So it seems to me that the flat Minkowski metric could not be a solution to the EFE when applied to any other topological space. But of course this argument is just heuristic; I haven't actually tried to prove anything mathematically.

 

[PeterDonis]

Peter, I guess if you still think that the minkowski spacetime with a point removed must conserve its flat metric you can't agree with me, but I would like to know why you insist on it, can you base that reasoning on something? Maybe that would help us both.

Just to amplify my previous post, "Minkowski spacetime with a point removed but with the same flat metric" is just a mathematical object. Call it M. It doesn't necessarily have any physical meaning. As a mathematical object, M is perfectly consistent and well-behaved, so I can consider it abstractly regardless of whether or not it is a solution of any particular equation.

"FRW spacetime with closed spatial slices" is another mathematical object. Call it F. Again, I can consider F abstractly; it doesn't have to have any physical meaning (although of course in this case it does). It just so happens that this mathematical object, F, shares some mathematical structure with M, above: both of them have the same underlying topological space, S^3 x R. But again, that's just a mathematical fact; it does not necessarily have any physical meaning.

The question I was asking is whether mathematical object M is a solution of the EFE. I don't think it is, but that's based on physical reasoning (see my previous post), not mathematical reasoning. It's also a separate question, in principle, from the question of whether M is diffeomorphic with some other mathematical object, such as F. *If* M were a solution of the EFE, then I would agree that, since it shares the same underlying topological space with another solution F, there would have to be a diffeomorphism between M and F. But if M is *not* a solution of the EFE, as I think, then whether or not M and F are diffeomorphic has nothing to do with the diffeomorphism invariance of GR; the latter is only supposed to apply to solutions of the EFE, not to arbitrary Lorentzian metrics on arbitrary topological spaces, regardless of whether they solve the EFE.

 

[Ben Niehoff]

There is a lot of confusion in this thread, and it seems like I caused some of it, so I'll jump in.


It is important to deconstruct the layers of structure we're using in GR:

1. Topology. Topology cares only about connectedness and continuous maps between spaces. Manifolds are topological objects. For example, flat space has the base manifold R^4, and the Reissner-Nordstrom geometry has the base manifold S^2 x R^2.

A sphere and a cube are the same as topological manifolds. They are both the manifold S^2.

Topologically speaking, manifolds can be given an atlas of charts; that is, a set of continuous maps of open regions of the manifold to R^n, with continuous transition functions between them, such that the whole manifold is covered by the set of maps.

2. Differential structure. This gives a definition of "smoothness" on manifolds. Now the atlas of charts is required to be smooth, with smooth transition functions.

As differentiable manifolds, the sphere and the cube are not equivalent, because the cube has 8 points where it is not smooth.

Also, the (maximally-extended) Schwarzschild and Reissner-Nordstrom geometries are not equivalent as smooth manifolds (despite the fact that they are both S^2 x R^2 topologically), because Schwarzschild has two singularities and Reissner-Nordstrom has a countably infinite number of them.

3. (Pseudo)-Riemannian structure. This is where we give the manifold a local notion of distance and angle; i.e., a metric tensor.


Now, there are three kinds of maps between manifolds worth talking about, depending on how many layers of this structure are preserved:

Homeomorphisms preserve only layer 1. The sphere and the cube are homeomorphic.

Diffeomorphisms preserve up to layer 2. The sphere and the cube are not diffeomorphic; however, the sphere is diffeomorphic to any topological sphere that is stretched and distorted in any desired way, so long as it is still everywhere smooth. Diffeomorphisms do not care about size or shape.

Metric-preserving diffeomorphisms, which are maps \varphi : (M, g) \rightarrow (N, h), smooth with smooth inverse, such that g = \varphi^* h (i.e. the metric on M is the pullback along \varphi of the metric on N). Metric-preserving diffeomorphisms do care about size and shape, and are fully equivalent to coordinate transformations.


There is one claim I made in a previous thread which is wrong: I claimed that a sphere of radius A is not diffeomorphic to a sphere of radius B. Clearly this is wrong, because the smooth structure comes prior to the metric structure, and diffeomorphisms care only about the smooth structure. Any two round spheres, of any radius, are diffeomorphic. A sphere is also diffeomorphic to a sphere with a smooth bump on it, etc.

What is true is that a sphere of radius A and a sphere of radius B fail to be "metric-preserving diffeomorphic" (which unfortunately does not have a convenient word to describe it). This is because there is no diffeomorphism between them such that the metric on sphere A is the pullback of the one on sphere B. Hence the "metric-preserving" condition makes things quite rigid.

In general, local conformal transformations g \mapsto e^{2 \varphi} g are diffeomorphisms, provided that e^{2 \varphi} is everywhere finite and strictly positive. If these conditions are broken, then a local conformal transformation can change the topology, and thus fails to be even a homeomorphism (for example, all 2-dimensional orientable manifolds are locally conformal to each other).


Finally, there is the question of what kinds of transformations leave Einstein's equations invariant? First, look at vacuum solutions with cosmological constant:

R_{\mu\nu} - \frac12 R g_{\mu\nu} + \Lambda g_{\mu\nu} = 0.
Under conformal rescaling by a constant, g \mapsto a^2 g, we have R_{\mu\nu} \mapsto R_{\mu\nu} and R \mapsto R/a^2, and hence the equation is invariant if we also assume \Lambda \mapsto \Lambda/a^2. What about more general conformal rescaling? From here:

http://en.wikipedia.org/wiki/Ricci_curvature#Behavior_under_conformal_rescaling

one has for g \mapsto e^{2 \varphi} g

\tilde{\operatorname{Ric}}=\operatorname{Ric}+(2-n)[ \nabla d\varphi-d\varphi\otimes d\varphi]+[\Delta \varphi -(n-2)\|d\varphi\|^2],
which indicates that Einstein's equation cannot be invariant under general such transformations. Therefore it is clear that general diffeomorphisms are not a symmetry of Einstein's equations!


One could consider moving the extra terms to the right-hand-side and treating them as matter sources, but in a sense, the equation with sources

R_{\mu\nu} - \frac12 R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{1}{8 \pi G} T_{\mu\nu}
is trivial; we are simply taking some combination of curvatures and calling it "T_{\mu\nu}". We are only really interested in the kinds of T_{\mu\nu} that describe physically-reasonable distributions of matter, so to call it an "'invariance" of Einstein's equations where T_{\mu\nu} can be arbitrarily modified is kind of silly.


Therefore I conclude that Einstein's equations are not invariant under all diffeomorphisms, but only under certain kinds. Certainly the metric-preserving diffeomorphisms are included, since they are equivalent to coordinate transformations. I suspect that global rescaling by a constant is the only other possibility.

In any case, this means the whole argument over "active" vs. "passive" diffeomorphisms is a moot point. The only maps that need considering are the maps that are equivalent to coordinate transformations. And so GR's "diffeomorphism invariance" is really a trivial fact; any theory can be written in a parametrization-invariant way.

 

[TrickyDicky]

I edited my #83, and that was a necessary link of my reasoning so right now I haven't so much confidence that "M" is a solution of the EFE, although I don't completely rule it out.

 

[TrickyDicky]

Wow, thanks Ben, that really clears up a lot of my misunderstandings.
I'll study your post and come back if I have doubts.

 

[Ben Niehoff]

Peter, I guess if you still think that the minkowski spacetime with a point removed must conserve its flat metric you can't agree with me, but I would like to know why you insist on it, can you base that reasoning on something? Maybe that would help us both.

There is nothing wrong with "Minkowski space with a point removed, and flat metric" as a mathematical object, and yes, it is a solution to Einstein's equations, because Einsteins equations are local. The base topology is "R^4 with a point removed", and the metric tensor is the obvious one.

The feature this psuedo-Riemannian manifold is missing is that it is not "complete"; i.e., there are geodesics that leave the manifold at finite affine parameter (i.e., geodesics of finite length that leave the manifold). One may consider the "maximal extension" of this manifold, which is where you follow all the geodesics to infinite length, and fill the rest of the manifold in by analytic continuation. The result is that the missing point will be put back, giving standard Minkowski space.

A similar process is used to get the maximally-extended Schwarzschild, Reissner-Nordstrom, and Kerr spacetimes from their standard solutions (whose coordinates cover only an open patch of the total spacetime). Kruskal coordinates are an example of a coordinate chart that covers the total Schwarzschild spacetime; in the RN and Kerr cases, one still needs multiple coordinate charts.

 

[PeterDonis]

Also, the (maximally-extended) Schwarzschild and Reissner-Nordstrom geometries are not equivalent as smooth manifolds (despite the fact that they are both S^2 x R^2 topologically), because Schwarzschild has two singularities and Reissner-Nordstrom has a countably infinite number of them.

In other words, Schwarzschild and R-N are *not* diffeomorphic. Presumably the same would be true of Schwarzschild and Kerr--isn't Kerr also S^2 x R^2 topologically? (George Jones said they were different topological spaces earlier in this thread, but I suspect by "topological space" he meant to include the differential structure, your level 2. Since we were talking about diffeomorphisms, that makes sense.)

any theory can be written in a parametrization-invariant way.

I seem to remember some of the literature on loop quantum gravity saying something very like this--and then proceeding to talk about "active" diffeomorphisms as though they were a big deal. Which makes me suspect that the literature is not entirely clear on terminology.