Curvature of Flat Lorentz manifolds

[lavinia]

While Minkowski space and Euclidean space both have identically zero curvature tensors it seems that a flat Lorentz manifold in general, may not admit a flat Riemannian metric. Such a manifold is the quotient of Minkowski space by the action of a properly discontinuous group of Lorentz isometries. This group in general is not a subgroup of the group of rigid motions of Euclidean space and so can not have a flat Riemannian metric.

What can one say about the curvature of Riemannian metrics on flat Lorentz manifolds?

 

[lavinia]

One idea might be to look at the metric that corresponds to the flat Lorentz metric, the matrix obtained by setting all the negative self inner products to positive.

 

[RockyMarciano]

While Minkowski space and Euclidean space both have identically zero curvature tensors it seems that a flat Lorentz manifold in general, may not admit a flat Riemannian metric. Such a manifold is the quotient of Minkowski space by the action of a properly discontinuous group of Lorentz isometries. This group in general is not a subgroup of the group of rigid motions of Euclidean space and so can not have a flat Riemannian metric.

What can one say about the curvature of Riemannian metrics on flat Lorentz manifolds?

For some time I've wondered about this myself, and since this mostly comes up in particle physics (Minkowski spacetime is the base manifold used there), I have tried to understand if that question ever shows up. Apparently it doesn't. There is a certain added complexity in the fact that Minkowski space is usually defined in math textbooks as a flat affine 4-manifold (R4), and not as the quotient you described which would seem more logical physically.
On the other hand a metric Levi-Civita connection is used in physics and its curvatures are derived from this connection. I'm not sure how interested you are in the physics side of this so I will leave it here.

 

[lavinia]

For some time I've wondered about this myself, and since this mostly comes up in particle physics (Minkowski spacetime is the base manifold used there), I have tried to understand if that question ever shows up. Apparently it doesn't. There is a certain added complexity in the fact that Minkowski space is usually defined in math textbooks as a flat affine 4-manifold (R4), and not as the quotient you described which would seem more logical physically.
On the other hand a metric Levi-Civita connection is used in physics and its curvatures are derived from this connection. I'm not sure how interested you are in the physics side of this so I will leave it here.

I have found some mathematical articles on the structure of flat Lorentz manifolds that are covered by Minkowski space. MIlnor and other have studied the structure of the fundamental group. I can send you links.

I would love to know the Physics of these manifolds. I have only seen Ricci flat manifolds as fibers of a bundle over Minkowski space.

Flat Lorentz manifolds that are covered by Minkowski space by a non-trivial group of covering transformations cannot admit a flat Riemannian metric. They have non-zero Riemannian curvature for any Levi-Civita connection. But I don't know anything about them nor of any examples.

The question arose because it seems that in the General Relativity Forum that flat Minkowski space is thought of as Euclidean. But this cannot be true.

It seems that every Lorentz metric should naturally determine a Riemannian metric by choosing a time like non-zero vector field and redefining itself inner product to have positive sign.

 

[RockyMarciano]

I have found some mathematical articles on the structure of flat Lorentz manifolds that are covered by Minkowski space. MIlnor and other have studied the structure of the fundamental group. I can send you links.

That would be great. The usual mathematically formal definitions of Minkowski space I have read such as:"a real 4-dimensional affine space, with an associated 4-dimensional real vector space V on which is defined a nondegenerate, symmetric, bilinear form g of index 1." or similar ones are the only ones I've seen, never saw it defined as a quotient manifold.

I would love to know the Physics of these manifolds. I have only seen Ricci flat manifolds as fibers of a bundle over Minkowski space.

Can you give me an example of that? I think I recall seeing it in the context of vacuum solutions in GR with Fubini-Study metric but I'm not sure you refer to this. On the other hand if you have seen anything about Yang-Mills fields then you should have seen something about curvatures of bundles over Minkowski space.

Flat Lorentz manifolds that are covered by Minkowski space by a non-trivial group of covering transformations cannot admit a flat Riemannian metric. They have non-zero Riemannian curvature for any Levi-Civita connection. But I don't know anything about them nor of any examples.

I guess this would be the case I mentioned above about Yang-Mills fields, at least in the physics context. If you are thinking about something different please specify.

The question arose because it seems that in the General Relativity Forum that flat Minkowski space is thought of as Euclidean. But this cannot be true.

Certainly Minkowski space is not Euclidean space or it wouldn't have the properties physicists routinely use. However if you use the Levi-Civita connection to compute its vanishing curvature tensor it must at least admit a flat torsion-free connection. This is a starting point in the more general principal bundle construction of curvatures in the context of particle physics, and it would seem to suggest it must admit a flat riemannian metric but there are certain ambiguities that should be taken care of before.

It seems that every Lorentz metric should naturally determine a Riemannian metric by choosing a time like non-zero vector field and redefining itself inner product to have positive sign.

But such redefinition would turn it into Euclidean space and make it loose its physical properties no?

 

[dextercioby]

Lavinia, can you, please, throw in some references to the literature? Indeed, physicists don't worry about the differential geom. aspects of spacetimes, especially the one used by special relativity.

 

[lavinia]

Lavinia, can you, please, throw in some references to the literature? Indeed, physicists don't worry about the differential geom. aspects of spacetimes, especially the one used by special relativity.

what references are you looking for - math research on flat Lorentz manifolds?

 

[dextercioby]

what references are you looking for - math research on flat Lorentz manifolds?

Yes, and curved as well, should you have these, too. Thank you!

 

[lavinia]

Yes, and curved as well, should you have these, too. Thank you!

http://www2.math.umd.edu/~wmg/iso.pdf

https://pdfs.semanticscholar.org/a13e/90e9f7b538cef2963534646a3b053b267289.pdf

https://www.ma.utexas.edu/users/jdanciger/pdfs/RNC-slides.pdf

https://borel.slu.edu/pub/lscc.pdf

more to come

 

[RockyMarciano]

http://www2.math.umd.edu/~wmg/iso.pdf

https://pdfs.semanticscholar.org/a13e/90e9f7b538cef2963534646a3b053b267289.pdf

https://www.ma.utexas.edu/users/jdanciger/pdfs/RNC-slides.pdf

https://borel.slu.edu/pub/lscc.pdf

more to come

Thanks for the links.
After a brief inspection it seems to me the first and the fourth pdf's (in the order you wrote them) are the only ones referring to what is known as Minkowski space(just in three dimensions). Only the objects subject of the second and third documents are the quotients that you mention in the OP, and are actually compact Lorentz flat space forms(that certainly don't admit flat riemannian metrics), these quotients are not the most general Lorentzian flat manifolds(in the sense that for instance the hyperbolic manifolds quotients of hyperbolic space are not the most general negatively curved Riemannian manifolds but Hn itself), the most general flat Lorentzian manifold(also a space form) is Minkowski space itself which is not compact, does this help?

 

[lavinia]

Thanks for the links.
After a brief inspection it seems to me the first and the fourth pdf's (in the order you wrote them) are the only ones referring to what is known as Minkowski space(just in three dimensions). Only the objects subject of the second and third documents are the quotients that you mention in the OP, and are actually compact Lorentz flat space forms(that certainly don't admit flat riemannian metrics), these quotients are not the most general Lorentzian flat manifolds(in the sense that for instance the hyperbolic manifolds quotients of hyperbolic space are not the most general negatively curved Riemannian manifolds but Hn itself), the most general flat Lorentzian manifold(also a space form) is Minkowski space itself which is not compact, does this help?

Thanks for answering and trying to help.
I know that these manifolds are compact and that they are Eilenberg Maclane spaces. In fact they are Kπ1's and so have no homotopy groups above dimension 1. But their Riemannian geometry is not explained, nor their topology. For instance, are there compact flat lorentz 4 manifolds that are not cobordant to zero?

I don't see why Minkowski space is the most general flat Lorentz manifold. What do you mean by "general"?

The fundamental theorem in Riemannian geometry is that a flat Riemannian manifold is covered by a torsion free group of isometries of flat Euclidean space. These groups have a distinct structure that is clearly different from the fundamental groups described in the links above. This shows that these Lorentz flat manifolds can not be Riemannian flat. So the natural question is what can be said about the curvature.

 

[RockyMarciano]

Thanks for answering and trying to help.
I know that these manifolds are compact and that they are Eilenberg Maclane spaces. In fact they are Kπ1's and so have no homotopy groups above dimension 1. But their Riemannian geometry is not explained, nor their topology. For instance, are there compact flat lorentz 4 manifolds that are not cobordant to zero?

I don't see why Minkowski space is the most general flat Lorentz manifold. What to you mean by "general"?

The fundamental theorem in Riemannian geometry is that a flat Riemannian manifold is covered by a torsion free group of isometries of flat Euclidean space. These groups have a distinct structure that is clearly different from the fundamental groups described in the links above. This shows that these Lorentz flat manifolds can not be Riemannian flat. So the natural question is what can be said about the curvature.

Ok. When I wrote "general" I was using the term informally, I had in mind one previous reply of yours saying that your question arose in a physical (relativity) context. In mainstream physics in general the only flat Lorentz space used is noncompact Minkowski n-space, for instance one reason to avoid compact manifolds is related to the appearance of closed timelike curves(but there might be of course more speculative theories using other Lorentz flat manifolds).
I guess you are more interested in the purely mathematical problem concerning the flat compact Lorentz manifolds quotients of Minkowski space and their possible Riemannian curvatures, in which case I'm afraid I can be of little help.
If you are also interested in discussing curvature in bundles with Minkowski base space in physics or comment on any of the more physical points I referred to in #5 let me know.

 

[lavinia]

Ok. When I wrote "general" I was using the term informally, I had in mind one previous reply of yours saying that your question arose in a physical (relativity) context. In mainstream physics in general the only flat Lorentz space used is noncompact Minkowski n-space, for instance one reason to avoid compact manifolds is related to the appearance of closed timelike curves(but there might be of course more speculative theories using other Lorentz flat manifolds).
I guess you are more interested in the purely mathematical problem concerning the flat compact Lorentz manifolds quotients of Minkowski space and their possible Riemannian curvatures, in which case I'm afraid I can be of little help.
If you are also interested in discussing curvature in bundles with Minkowski base space in physics or comment on any of the more physical points I referred to in #5 let me know.

Thanks.

I thought that you were referring to physics of flat Lorentz manifolds other than Minkowski space. My mistake.

I understand what a gauge field is and also that the field strength is the curvature of the gauge field. Mathematically I know about connections on principal bundles for Levi-Civita and more general connections where a metric is not involved. I have seen the interpretation of Maxwell's equations on a U(1) bundle. As I understand it a gauge field is the pull back of a connection 1 form to the base manifold over a coordinate domain. The field strength is the pull back of the matrix of curvature 2 forms.

I have seen what a Yang-Mills action is as well. But I know little about the physics.

 

[RockyMarciano]

I understand what a gauge field is and also that the field strength is the curvature of the gauge field. Mathematically I know about connections on principal bundles for Levi-Civita and more general connections where a metric is not involved. I have seen the interpretation of Maxwell's equations on a U(1) bundle. As I understand it a gauge field is the pull back of a connection 1 form to the base manifold over a coordinate domain. The field strength is the pull back of the matrix of curvature 2 forms.

I have seen what a Yang-Mills action is as well. But I know little about the physics.

I guess the advantage of the gauge theory approach making use of the principal bundle framework is the flexibility to combine different Riemannian curvatures whereas if restricted to Riemannian manifold theory with a Levi-Civita affine connection you have less flexibility.
For instance to connect with your question: what can one say about the curvature of Riemannian metrics on flat Lorentz manifolds? At least for the specific case you refer to(compact flat Lorentz space forms), wouldn't the compactness force torsion since the connection is flat, which seems to exclude any Riemannian metrics( not just flat Riemannian metrics)?

 

[lavinia]

I guess the advantage of the gauge theory approach making use of the principal bundle framework is the flexibility to combine different Riemannian curvatures whereas if restricted to Riemannian manifold theory with a Levi-Civita affine connection you have less flexibility.

If by "Riemannian curvature" you mean the curvature associated to a connection that is compatible with a Riemannian metric, then the idea of connection on a principal bundle is far more general. One can have a connection that is not compatible with any metric. The curvature in such a case does not correspond to a Riemann curvature tensor. Curvature is a more primitive idea than angle and length measurement.

For instance to connect with your question: what can one say about the curvature of Riemannian metrics on flat Lorentz manifolds? At least for the specific case you refer to(compact flat Lorentz space forms), wouldn't the compactness force torsion since the connection is flat, which seems to exclude any Riemannian metrics( not just flat Riemannian metrics)?

The compact Lorentz space form has identically zero curvature for the Levi-Civita connection that it inherits from Minkowski space. This connection is torsion free.

Compactness does not exclude Riemannian metrics. Every compact smooth manifold has a Riemannian metric. One pieces together local metrics using a partition of unity with respect to a "good cover." The fundamental theorem says that each metric has a unique torsion free connection that is compatible with it.

 

[RockyMarciano]

If by "Riemannian curvature" you mean the curvature associated to a connection that is compatible with a Riemannian metric, then the idea of connection on a principal bundle is far more general. One can have a connection that is not compatible with any metric. The curvature in such a case does not correspond to a Riemann curvature tensor. Curvature is a more primitive idea than angle and length measurement.

Yes, in the physics case of gauge theory " Riemannian curvature" refers to a slightly more general case when the metric is not necessarily Riemannian(i.e. positive definite) but Lorentzian and it is translatable to the curvature forms on principal bundles..

The compact Lorentz space form has identically zero curvature for the Levi-Civita connection that it inherits from Minkowski space. This connection is torsion free.

Correct.

Compactness does not exclude Riemannian metrics. Every compact smooth manifold has a Riemannian metric. One pieces together local metrics using a partition of unity with respect to a "good cover." The fundamental theorem says that each metric has a unique torsion free connection that is compatible with it.

Certainly, but I'm of course not implying that compactness by itself excludes Riemannian metrics(sorry if I gave that impression). We have several elements here: the inherited LC connection from Minkowski space preserving the Lorentzian metric, the topology of the quotient space form, and the demand for the connection to be flat and torsion free.
Actually in my question I was making the unwarranted assumptions of simple connectednes and of isotropy of the connection, but this space is not simply connected so nevermind.

I guess I just can't see how either a Riemannian metric of constant positive or negative curvature can be endowed if the Levi-Civita connection is flat and is unique. Could you maybe elaborate on this?

 

[lavinia]

I guess I just can't see how either a Riemannian metric of constant positive or negative curvature can be endowed if the Levi-Civita connection is flat and is unique. Could you maybe elaborate on this?

Not sure what you mean by constant positive or negative curvature but a Lorentz manifold would seem to always have a non-zero time like vector field so its Euler characteristic is zero. (For a compact flat Lorentz manifold one can use the Generalized Gauss-Bonnet Theorem to conclude that the Euler class is zero.) If one puts a positive definite Riemannian metric on the manifold then the Generalized Gauss Bonnet theorem says that the Pfaffian of the curvature 2 form matrix integrates to the Euler Characteristc so if one writes this form as PVol where Vol is the volume form then P can not be only positive or only negative,

 

[RockyMarciano]

Not sure what you mean by constant positive or negative curvature

If a space form is a complete (pseudo)Riemannian manifold of constant sectional curvature K, and we have discarded a flat Riemannian metric I supposed that any Riemannian metric must have either positive or negative constant curvature, is this wrong?

but a Lorentz manifold would seem to always have a non-zero time like vector field so its Euler characteristic is zero. (For a compact flat Lorentz manifold one can use the Generalized Gauss-Bonnet Theorem to conclude that the Euler class is zero.) If one puts a positive definite Riemannian metric on the manifold then the Generalized Gauss Bonnet theorem says that the Pfaffian of the curvature 2 form matrix integrates to the Euler Characteristc so if one writes this form as PVol where Vol is the volume form then P can not be only positive or only negative,

Wich leaves us with what options for putting a Riemannian metric in this particular space?

 

[lavinia]

If a space form is a complete (pseudo)Riemannian manifold of constant sectional curvature K, and we have discarded a flat Riemannian metric I supposed that any Riemannian metric must have either positive or negative constant curvature, is this wrong?

I don't see why. What is your argument?

 

[RockyMarciano]

I don't see why. What is your argument?

It's maybe based on some basic confusion on my part, but to me the alternative is to have Riemannian metrics of variable curvature, but how would constant sectional curvature of a space form be compatible with variable curvature Riemann metrics? Are there other alternatives I can't think of? Or is constant curvature compatible with variable curvature?

 

[martinbn]

It's maybe based on some basic confusion on my part, but to me the alternative is to have Riemannian metrics of variable curvature, but how would constant sectional curvature of a space form be compatible with variable curvature Riemann metrics? Are there other alternatives I can't think of? Or is constant curvature compatible with variable curvature?

Take the Euclidean plain with the usual metric, it has curvature zero. Pick a different Riemannian metric at random and compute the curvature, it will not be constant.

 

[lavinia]

It's maybe based on some basic confusion on my part, but to me the alternative is to have Riemannian metrics of variable curvature, but how would constant sectional curvature of a space form be compatible with variable curvature Riemann metrics? Are there other alternatives I can't think of? Or is constant curvature compatible with variable curvature?

A manifold of constant positive sectional curvature is covered by a sphere. So a compact Lorentz space form can not be given a Riemannian metric of constant positive sectional curvature since it is covered by Minkowski space.

A compact hyperbolic manifold has non-zero Euler characteristic so it can not be a flat Lorentz space form.

 

[lavinia]

https://projecteuclid.org/download/pdf_1/euclid.bams/1183524262

 

[RockyMarciano]

Take the Euclidean plain with the usual metric, it has curvature zero. Pick a different Riemannian metric at random and compute the curvature, it will not be constant.

I'm not sure if by "pick a different Riemannian metric at random, it will not be constant" you mean that any other Riemannian metric on the plane different from the canonical will be of variable curvature. If so an obvious counterexample is the hyperbolic plane with constant negative curvature.

 

[RockyMarciano]

A manifold of constant positive sectional curvature is covered by a sphere. So a compact Lorentz space form can not be given a Riemannian metric of constant positive sectional curvature since it is covered by Minkowski space.

A compact hyperbolic manifold has non-zero Euler characteristic so it can not be a flat Lorentz space form.

Fine, I can see that it doesn't admit constant curvature riemannian metrics now. I guess I was confused by the definition of space form, it of course refers to the constant curvature of the Lorentzian metric in this case. So I see that the definition of space form doesn't mean it couldn't admit Riemannian metrics(with positive definite metric) of variable curvature.
Thanks for clarifying.

 

[martinbn]

I'm not sure if by "pick a different Riemannian metric at random, it will not be constant" you mean that any other Riemannian metric on the plane different from the canonical will be of variable curvature. If so an obvious counterexample is the hyperbolic plane with constant negative curvature.

No, I meant that if you actually wrote down a metric, chances are it will not have constant curvature. My comment was only because your post suggests that you think that a manifold with constant curvature cannot have a different metric (or just a connection) with non constant curvature.

 

[RockyMarciano]

No, I meant that if you actually wrote down a metric, chances are it will not have constant curvature. My comment was only because your post suggests that you think that a manifold with constant curvature cannot have a different metric (or just a connection) with non constant curvature.

Ok, I have certain difficulties to imagine this as is so common to visualize it with constant curvature . Can you give me an example of a metric of variable curvature on the plane?

 

[martinbn]

Ok, I have certain difficulties to imagine this as is so common to visualize it with constant curvature . Can you give me an example of a metric of variable curvature on the plane?

Just make up a metric and compute the curvature.

 

[RockyMarciano]

Just make up a metric and compute the curvature.

Well sure. Sorry, I was still stuck with the space form definition and got my mind mixed. Nevermind the silly questions. Thanks.

 

[lavinia]

Fine, I can see that it doesn't admit constant curvature riemannian metrics now. I guess I was confused by the definition of space form, it of course refers to the constant curvature of the Lorentzian metric in this case. So I see that the definition of space form doesn't mean it couldn't admit Riemannian metrics(with positive definite metric) of variable curvature.
Thanks for clarifying.

I like your idea that the Riemannian metric on the flat Lorentz manifold can be chosen to have a very uniform curvature tensor. Constant sectional curvature doesn't work but one would suspect that something very uniform does work.

 

[RockyMarciano]

I like your idea that the Riemannian metric on the flat Lorentz manifold can be chosen to have a very uniform curvature tensor. Constant sectional curvature doesn't work but one would suspect that something very uniform does work.

Very benevolent of you.

 

[RockyMarciano]

@lavinia, would Minkowski space itself, for the timelike vector case(where the Lorentz group acts on the interior of the future timelike oriented cone) not have the same issue with its Lorentz isometries acting properly discontinuously and not being a subgroup of rigid euclidean motions?