Some doubts concerning the mathematical bases of GR

[atyy]

Given any manifold, it is not always possible to put an indefinite-signature metric tensor on it. There are topological obstructions. For example, a torus can have a Lorentzian metric tensor, but a sphere cannot.

So when people give an example of AdS/CFT and say eg. it's string theory on AdS5 X S5, there isn't a Lorentzian metric on S5?

 

[PAllen]

So when people give an example of AdS/CFT and say eg. it's string theory on AdS5 X S5, there isn't a Lorentzian metric on S5?

I am guessing Ben meant a 2-sphere, and that for higher dimensions, answer is different.

A little searching turns up the following claim to a proof that an n-sphere admits a Lorentzian metric if n is odd:

http://mathoverflow.net/questions/4...ossible-that-sn-can-have-a-lorentz-metric-why

See the checked answer.

 

[Ben Niehoff]

So when people give an example of AdS/CFT and say eg. it's string theory on AdS5 X S5, there isn't a Lorentzian metric on S5?

No, the S5 part of AdS5 x S5 has a Riemannian metric on it (i.e., the timelike part of the total metric lies entirely within the AdS5 factor).

However, as Pallen points out, the odd-dimensional spheres can be given Lorentzian metrics. This is because S^(2n+1) can be decomposed as an S^1 fibered over CP^n. This S^1 can then be chosen as the timelike direction.

 

[atyy]

@Ben Niehoff & PAllen, got it - thanks!

 

[TrickyDicky]

That is the only distinction.

Yes, that's my point. That is the only difference and the one that gives rise to many of the properties of GR, like the different type of geodesics (null, timelike..), the causal structure, the geodesic incompleteness and the singularities, etc...
But if that difference is obviated by the smooth manifold structure that turns pseudosemimetrics spaces into metric spaces how do we justify mathematically all those properties associated to the pseudosemimetricity that are such important features of GR?

 

[Ben Niehoff]

Yes, that's my point. That is the only difference and the one that gives rise to many of the properties of GR, like the different type of geodesics (null, timelike..), the causal structure, the geodesic incompleteness and the singularities, etc...
But if that difference is obviated by the smooth manifold structure that turns pseudosemimetrics spaces into metric spaces how do we justify mathematically all those properties associated to the pseudosemimetricity that are such important features of GR?

I'm not really sure what you're getting at. None of this causes a problem for GR. GR only cares about the metric tensor; whether it integrates into any global structure is bonus.

When we say some space is a "manifold", all we mean is that it has certain topological properties. Topology cares about how the various points in a set are connected. It is completely agnostic as to the concept of those points having any "distance" defined between them.

A manifold is just some space that can be covered by open sets, each of which looks just like an open set of R^n. By "looks just like", I mean that the points in the open set U on the manifold are connected to each other the same way as the points in the open set V of R^n. At no point do I care about the Euclidean distance which is possible to define in R^n; it's irrelevant.

Now, an added fact is that I can use the natural Euclidean distance in R^n to define a notion of distance on the manifold. One uses the usual tangent space construction to define a Riemannian metric tensor (turning our manifold into a Riemannian manifold). This metric tensor can be integrated to obtain a global distance function, turning our Riemannian manifold into a metric space. (One can also define metric spaces which are not manifolds, so in fact this object is both a Riemannian manifold and a metric space, those being independent properties).

Metric spaces have the additional property that the distance function can be used to define a topology. That is, open sets can be defined as the interiors of metric balls. It happens that when we do this to a Riemannian manifold, the topology induced by the metric structure agrees with the topology we already had from the manifold structure. This is not hard to prove.

But these concepts do not carry over to the pseudo-Riemannian case. As you have pointed out, a pseudo-Riemannian metric tensor does not integrate to a distance function, for one. Whatever object a pseudo-Riemannian metric integrates to, it must fail to satisfy the distinguishability axiom,

d(x,y) = 0 \; \text{iff} \; x = y,
and hence, the topology induced by such a distance function will not agree with the topology we already have from the manifold structure. This is easy to see in flat Minkowski space, whose topology is that of R^4.

There is no reason to expect a pseudo-Riemannian metric tensor to induce a topology that agrees with the one already present, because the theorem in the Riemannian case relies upon the details of all the axioms. Just because "pseudo-Riemannian metric tensor" contains the words "Riemannian metric tensor" does not mean you can borrow theorems and expect them to still be true.

 

[TrickyDicky]

I'm not really sure what you're getting at. None of this causes a problem for GR. GR only cares about the metric tensor; whether it integrates into any global structure is bonus.

When we say some space is a "manifold", all we mean is that it has certain topological properties. Topology cares about how the various points in a set are connected. It is completely agnostic as to the concept of those points having any "distance" defined between them.

But these concepts do not carry over to the pseudo-Riemannian case. As you have pointed out, a pseudo-Riemannian metric tensor does not integrate to a distance function, for one. Whatever object a pseudo-Riemannian metric integrates to, it must fail to satisfy the distinguishability axiom,

d(x,y) = 0 \; \text{iff} \; x = y,
and hence, the topology induced by such a distance function will not agree with the topology we already have from the manifold structure.

Ok, so from this I interpret that Riemannian and Pseudo-Riemannian manifolds are topologically indistinguishable, no?

OTOH, when you say that GR only cares about the metric tensor, I am not sure how to make this statement compatible with the fact that in GR preserving lengths, that is, distance, is fundamental, as it is the Levi-Civita connection that being torsion-free assures integrability of the metric tensor.

What I'm getting at is that the GR features derived from its metric tensor indefiniteness seem to be overridden by the smooth manifold topology, so that at least at the large scale (not at the infinitesimal level of the metric tensor) there seems to be no difference between Riemannian and Pseudo-Riemannian manifolds, it could only have R^4 topology in the case of GR.

And all lengths in Pseudo-Riemannian manifolds must equal the Riemannian manifold case since the pseudo-Riemannian metric tensor can only integrate to a metric in a smooth manifold.

 

[Ben Niehoff]

Spacetimes are not pseudometric spaces. They are manifolds on which we have defined a pseudo-Riemannian metric tensor.

Do you agree that various points along a null geodesic are distinct points?

 

[TrickyDicky]

Spacetimes are not pseudometric spaces. They are manifolds on which we have defined a pseudo-Riemannian metric tensor.

Right.

Do you agree that various points along a null geodesic are distinct points?

Yes.

I asked a related question before that was left unanswered. How can we separate distinct points on the null light cone if their distance can be zero? Now, given the fact that due to the fact that the smooth manifold structure underlying Pseudo-Riemannian manifolds assures the spacetime is not pseudosemimetric but metric, we should be able to adjudicate a non-zero distance to distinct points on the null cone but I don't know how when their metric tensor is vanishing, ds=0.

 

[Ben Niehoff]

I asked a related question before that was left unanswered. How can we separate distinct points on the null light cone if their distance can be zero?

"Distance" is not part of the subject of topology, period.

Points along a null geodesic are separate because they have different values of affine parameter. This is the whole point of saying the manifold can be covered by open sets that map continuously to open sets of R^n.

A pseudo-Riemannian metric tensor does not induce a topology that agrees with the underlying manifold structure. We do not use, nor care, what topology the pseudo-Riemannian metric tensor does induce, precisely because it disagrees with the underlying manifold structure.

Just because a structure can be defined does not mean that it is useful or physically reasonable. GR uses differential geometry, which is done on manifolds, and hence it is the manifold structure we require. Physically, it is reasonable that null geodesics be a series of distinct points, rather than a single point, because it is our physical observation that light rays travel.

Note also, that on any manifold, we can define a Riemannian metric tensor. But in GR we choose not to, because a Riemannian metric tensor is incompatible with the physical requirement of local Lorentz symmetry. So this is another example of a structure that is possible to define, but is left unused in the context of GR.

 

[TrickyDicky]

"Distance" is not part of the subject of topology, period.

Points along a null geodesic are separate because they have different values of affine parameter. This is the whole point of saying the manifold can be covered by open sets that map continuously to open sets of R^n.

A pseudo-Riemannian metric tensor does not induce a topology that agrees with the underlying manifold structure. We do not use, nor care, what topology the pseudo-Riemannian metric tensor does induce, precisely because it disagrees with the underlying manifold structure.

Just because a structure can be defined does not mean that it is useful or physically reasonable. GR uses differential geometry, which is done on manifolds, and hence it is the manifold structure we require. Physically, it is reasonable that null geodesics be a series of distinct points, rather than a single point, because it is our physical observation that light rays travel.

Note also, that on any manifold, we can define a Riemannian metric tensor. But in GR we choose not to, because a Riemannian metric tensor is incompatible with the physical requirement of local Lorentz symmetry. So this is another example of a structure that is possible to define, but is left unused in the context of GR.

Ok, I was not trying to link topology and distance, I happen to have questions about both but independently.

My question about topology that has not yet been addressed was if Riemannian and Pseudo-Riemannian manifolds have the same topology?

My question about distance was how do we separate points on null geodesic in Lorentzian manifolds?
If as you say we do it relying on the geodesic affine parametrization, how is this different from a timelike geodesic? IOW, it looks like a Lorentzian manifold has the same notion of distance that the Riemannian ones have, even if infinitesimally one has ds=0, ds>0 or ds<0.

Also, why exactly is local lorentz symmetry incompatible with a Riemannian metric?

 

[Ben Niehoff]

My question about topology that has not yet been addressed was if Riemannian and Pseudo-Riemannian manifolds have the same topology?

That question doesn't make sense as written.

A 2-sphere and a 2-torus have different topologies, although both will accept Riemannian metrics.

On a 2-torus you can put either a Riemannian metric or a pseudo-Riemannian one. In both cases, the underlying topology is the same: a 2-torus.

If what you mean is "local topology", then the answer is yes. A manifold (be it Riemannian or pseudo-Riemannian) looks locally like a piece of R^n (not R^{n-1,1}).

In fact, R^{n-1,1} is topologically the same as R^n; it just has a pseudo-Riemannian metric defined on it.

My question about distance was how do we separate points on null geodesic in Lorentzian manifolds?
If as you say we do it relying on the geodesic affine parametrization, how is this different from a timelike geodesic? IOW, it looks like a Lorentzian manifold has the same notion of distance that the Riemannian ones have, even if infinitesimally one has ds=0, ds>0 or ds<0.

You need to be very careful what you mean when you say "distance".

Let's just take standard, 2-dimensional Minkowski space. The points (0,0) and (1,1) are distinct points because they have different coordinates.

Also, why exactly is local lorentz symmetry incompatible with a Riemannian metric?

Because Riemannian manifolds have local Euclidean symmetry.

 

[TrickyDicky]

That question doesn't make sense as written.

A 2-sphere and a 2-torus have different topologies, although both will accept Riemannian metrics.

On a 2-torus you can put either a Riemannian metric or a pseudo-Riemannian one. In both cases, the underlying topology is the same: a 2-torus.

If what you mean is "local topology", then the answer is yes. A manifold (be it Riemannian or pseudo-Riemannian) looks locally like a piece of R^n (not R^{n-1,1}).

In fact, R^{n-1,1} is topologically the same as R^n; it just has a pseudo-Riemannian metric defined on it.

I guess I should have been more formal when formulating the question but I thought the context was clear enough.

Let's see I am aware that any smooth manifold admits infinite Riemannian metrics, this is not related to what I meant with my question.

Your second example is closer to where I was getting at, given a certain topology given by the smooth manifold structure, topologically makes no difference if the metric we add on top that structure is Riemannian or Pseudo-Riemannian, agree?

The thing is that unlike geometry topology has only global structure, that is why topology is usually considered something global, and therefore the local topology distinction you are making is confusing to me, if a differentiable manifold has "local" topology R^4 then that is also its (global) topology, no?

You need to be very careful what you mean when you say "distance".

Let's just take standard, 2-dimensional Minkowski space. The points (0,0) and (1,1) are distinct points because they have different coordinates.

Yes, I guess I should have been more precise here too, when I say Pseudo-Riemannian manifold and since what we are finally always dealing here with is GR, I always mean a connected, curved, Lorentzian manifold.
Minkowski spacetime is flat and therefore it can be covered by a single coordinate system, a curved manifold will need more than one chart and there will be overlap between charts, so using coordinates to separate distinct points is not always possible.

Because Riemannian manifolds have local Euclidean symmetry.

Infinitesimally, how is Euclidean symmetry different from Lorentzian symmetry?
I thought the whole point of manifolds was that they were locally isomorphic to a Euclidean space.

 

[TrickyDicky]

To make more precise what I mean by distance, if (connected) Pseudo-Riemannian manifolds have no pseudosemimetric space structure, must I suppose they have metric space structure like (connected) Riemannian manifolds do?

 

[Ben Niehoff]

To make more precise what I mean by distance, if (connected) Pseudo-Riemannian manifolds have no pseudosemimetric space structure, must I suppose they have metric space structure like (connected) Riemannian manifolds do?

If you haven't defined a metric (i.e. global, positive distance function), then there is no metric space structure.

I really suggest you pick up a more mathematical text, rather than learning from the traditional GR sources, if you are concerned about the precise mathematical underpinnings of differential geometry. I think Nakahara is good, Micromass might have some other suggestions.

 

[Ben Niehoff]

It seems to me you are making some exploration of "What kind of mathematical structures can I define, and how are they related?", which is good and interesting from a pure mathematical point of view. I suggest you keep exploring that, you'll gain some better understanding of what's really going on.

In the context of GR (or any physical theory, really), you need to ask an additional question: "Does this structure model the physics I want to model?" This question is answered both by intuition about what you think the physical theory should be like, and also by contact with experiment.

 

[TrickyDicky]

It seems to me you are making some exploration of "What kind of mathematical structures can I define, and how are they related?", which is good and interesting from a pure mathematical point of view. I suggest you keep exploring that, you'll gain some better understanding of what's really going on.

In the context of GR (or any physical theory, really), you need to ask an additional question: "Does this structure model the physics I want to model?" This question is answered both by intuition about what you think the physical theory should be like, and also by contact with experiment.

Ben, thanks for your wise advice which I'll follow.
Meanwhile, could you comment on my last questions?

 

[Ben Niehoff]

Your second example is closer to where I was getting at, given a certain topology given by the smooth manifold structure, topologically makes no difference if the metric we add on top that structure is Riemannian or Pseudo-Riemannian, agree?

Yes, with one caveat: some manifolds do not allow us to define a pseudo-Riemannian metric tensor. For example, the 2-sphere cannot have a p-R structure; whereas the 2-torus can. As explained in Pallen's earlier link, it is possible to define a p-R structure if and only if there is a non-vanishing vector field; i.e. if the Euler characteristic is zero.

The thing is that unlike geometry topology has only global structure, that is why topology is usually considered something global, and therefore the local topology distinction you are making is confusing to me, if a differentiable manifold has "local" topology R^4 then that is also its (global) topology, no?

Topology concerns homeomorphisms; that is, continuous one-to-one maps. Local topology concerns local homeomorphisms; i.e., homeomorphisms between open patches. The local topology of an n-fold is the same as the local topology of R^n.

There is more to topology than counting invariants like handles, holes, etc. Topology is the study of how points are connected, which is another way of expressing what it means for something to be "continuous" (since topology concerns continuous maps). A homeomorphism is any map that leaves all the "connectedness" information intact.

Consider graph theory. The connectedness of all the vertices in a graph can be represented by a matrix. Two graphs are isomorphic if and only if their adjacency matrices are related by a permutation; i.e., the graphs are topologically the same if they have the same connectedness information.

Yes, I guess I should have been more precise here too, when I say Pseudo-Riemannian manifold and since what we are finally always dealing here with is GR, I always mean a connected, curved, Lorentzian manifold.
Minkowski spacetime is flat and therefore it can be covered by a single coordinate system, a curved manifold will need more than one chart and there will be overlap between charts, so using coordinates to separate distinct points is not always possible.

If two charts do not overlap, then they contain distinct points. If they do overlap, you can use the transition functions to compare coordinates in one chart with coordinates in the other. Hence you can always decide whether two points are distinct.

Infinitesimally, how is Euclidean symmetry different from Lorentzian symmetry?
I thought the whole point of manifolds was that they were locally isomorphic to a Euclidean space.

Manifolds are locally homeomorphic to R^n, but you must forget any notion of "distance" on R^n. Topology does not care about distance. It only care about how points are connected to each other.

Riemannian manifolds have an extra structure defined on them: the metric tensor. Riemannian manifolds have local Euclidean symmetry because each tangent space is isomorphic to Euclidean space, with the usual notion of Euclidean distance. Euclidean distance is preserved by Euclidean symmetry.

Pseudo-Riemannian manifolds also have extra structure defined on them: a pseudo-Riemannian metric tensor. P-R manifolds have local Lorentz (actually Poincare) symmetry, because each tangent space is isomorphic to Minkowski space, with the Minkowski product. The spacetime interval is preserved by Poincare symmetry.

 

[TrickyDicky]

Thanx, time to give it some more thought and read.

 

[TrickyDicky]

A couple of related interesting links:

https://www.physicsforums.com/showthread.php?t=495816
http://mathpages.com/rr/s9-01/9-01.htm

A certain clash between GR Lorentzian metric and rigorous definitions of smooth manifolds seems to be hinted at in those links. That is, extra conditions like strong causality have to be added to the causal structure of the Lorentzian metric in order to make conform its Alexandrov topology to the smooth manifolds topology, and that only in the flat Minkowski case.
When curvature is present only in certain cases are p-r metrics admitted like those mentioned by Ben Niehoff and PAllen by virtue of using the S¹, CP space decomposition that allows closed timelike curves. On the other hand CTC's are not usually considered physical.
Do, say, non-compact curved 4-manifolds admit pseudoriemannian metrics? Knowing that would surely help constrain GR's possible topologies.

The problem I see is that many of the physical features associated to GR, like those related to singularities(BH's, horizons, BB, KS space...) fail to fulfill the extra conditions needed to qualify as a smooth manifold like the strong causality condition. Has that ever been seen as a problem by physicists?

 

[stevendaryl]

When curvature is present only in certain cases are p-r metrics admitted like those mentioned by Ben Niehoff and PAllen by virtue of using the S¹, CP space decomposition that allows closed timelike curves...

The problem I see is that many of the physical features associated to GR, like those related to singularities(BH's, horizons, BB, KS space...) fail to fulfill the extra conditions needed to qualify as a smooth manifold like the strong causality condition. Has that ever be seen as a problem by physicists?

I haven't read the relevant papers, but I would think that the difficulty would be: given a manifold, it may not always be possible to come up with a metric with signature (+---) on that manifold. However, in cases such as the Schwarzschild black hole, that's not what is done. Instead, one starts with a hypothetical metric (or rather, equivalently, a quadratic form for ds² that has the right signature) and then uses that to define the manifold. There is no question about whether it is possible to give it the right type of metric, since you started with the right type of metric in the first place.

The approach that I'm talking about has its own problems, of course. One of them is the fact that a metric that is defined in terms of a particular set of coordinates only describes a "patch" of the manifold, and it may not be clear if and how it can be extended to a complete manifold. The other problem is that if you start with the metric, then you may not have a realistic stress-energy tensor. But the problem that you seem to be worried about doesn't really come up---we never start with an arbitrary manifold, and then ask what metric can we put on top of it.

 

[TrickyDicky]

I haven't read the relevant papers, but I would think that the difficulty would be: given a manifold, it may not always be possible to come up with a metric with signature (+---) on that manifold. However, in cases such as the Schwarzschild black hole, that's not what is done. Instead, one starts with a hypothetical metric (or rather, equivalently, a quadratic form for ds² that has the right signature) and then uses that to define the manifold. There is no question about whether it is possible to give it the right type of metric, since you started with the right type of metric in the first place.

I guess by "the right metric in the first place" you must mean that it is after all one of GR's EFE solutions, and since most of us think GR is the right theory, we are confident that there must be some smooth manifold that admits that metric. The problem is that mathematically (and even physically as many GR solutions don't make much sense physically), that heuristic is not very rigorous, it basically reverses the hierarchy usually employed in mathematics since the manifold cannot be defined from the metric in many cases, like those where their topologies don't agree, the manifold always comes first, or at least that is what I've been told throughout this thread.

The approach that I'm talking about has its own problems, of course. One of them is the fact that a metric that is defined in terms of a particular set of coordinates only describes a "patch" of the manifold, and it may not be clear if and how it can be extended to a complete manifold.

Exactly, that is what I mean. But not only that, it is not even assured that a manifold exists that admits that metric.

The other problem is that if you start with the metric, then you may not have a realistic stress-energy tensor.

That's part of the unphysical solutions issue I mentioned above.

But the problem that you seem to be worried about doesn't really come up---we never start with an arbitrary manifold, and then ask what metric can we put on top of it.

Right, I can see how that wouldn't even come up with the approach you are describing, that I presume is the usual one among physicists, but as we agreed that can lead to some serious problems and IMO to certain physical claims that might contradict the math.

 

[PAllen]

A couple of related interesting links:

https://www.physicsforums.com/showthread.php?t=495816
http://mathpages.com/rr/s9-01/9-01.htm

A certain clash between GR Lorentzian metric and rigorous definitions of smooth manifolds seems to be hinted at in those links. That is, extra conditions like strong causality have to be added to the causal structure of the Lorentzian metric in order to make conform its Alexandrov topology to the smooth manifolds topology, and that only in the flat Minkowski case.
When curvature is present only in certain cases are p-r metrics admitted like those mentioned by Ben Niehoff and PAllen by virtue of using the S¹, CP space decomposition that allows closed timelike curves. On the other hand CTC's are not usually considered physical.
Do, say, non-compact curved 4-manifolds admit pseudoriemannian metrics? Knowing that would surely help constrain GR's possible topologies.

The problem I see is that many of the physical features associated to GR, like those related to singularities(BH's, horizons, BB, KS space...) fail to fulfill the extra conditions needed to qualify as a smooth manifold like the strong causality condition. Has that ever been seen as a problem by physicists?

While I highly respect and usually like Keven Brown's expositions (author of mathpages), I have the opposite bias as to what is the physically reasonable topology of spacetime. Specifically, no observer would have any tendency to consider the source and emission events for a light pulse to be topologically indistinguishable (as would follow from using a pseudo-semimetric toplology). In fact, the most uniform explanation of redshift in GR involves parallel transporting vectors along null paths, something which clearly distinguishes all the points along the null path. To my mind, the role of the Lorentzian metric is purely about chronometry, i.e. geometry, not topology. Topology is determined by boundary conditions or assumption (i.e. 'what would be the physical consequences per GR if we assume some overall manifold topology').

A more interesting question is the physical utility of distinguishing semi-riemannian manifolds for which the Alexandrov topology [a specific way of inducing a topology from the Lorentzian metric that is, IMO, much more satisfactory than integrating to a pseudosemimetrc] matches the manifold topology. Exploring the consequences of such a requirement is in the same spirit as energy conditions.

I think the clearest exposition of how none of this is relevant to the mathematical soundness of GR is the following from the thread you linked:

https://www.physicsforums.com/showpost.php?p=3295723&postcount=13

Separate from mathematical foundations, are physical validity of the GR as mathematical model. Here, we have mostly open questions, without conclusive answers:

- Energy conditions that are both too strong and too weak (so far)
- Which causality conditions are enforced in a 'physically plausible' GR solution?
- In what ways does GR fail to match our actual universe at event horizons and singularities?

 

[TrickyDicky]

While I highly respect and usually like Keven Brown's expositions (author of mathpages), I have the opposite bias as to what is the physically reasonable topology of spacetime. Specifically, no observer would have any tendency to consider the source and emission events for a light pulse to be topologically indistinguishable (as would follow from using a pseudo-semimetric toplology). In fact, the most uniform explanation of redshift in GR involves parallel transporting vectors along null paths, something which clearly distinguishes all the points along the null path.

I tend to agree with you, but note that this observer-biased approach to null geodesics amounts to treating them as timelike geodesics, which is what we normally do when thinking about light as paths as rays traversing a certain distance in a certain time.

A more interesting question is the physical utility of distinguishing semi-riemannian manifolds for which the Alexandrov topology [a specific way of inducing a topology from the Lorentzian metric that is, IMO, much more satisfactory than integrating to a pseudosemimetrc] matches the manifold topology. Exploring the consequences of such a requirement is in the same spirit as energy conditions.

I think the clearest exposition of how none of this is relevant to the mathematical soundness of GR is the following from the thread you linked:

https://www.physicsforums.com/showpost.php?p=3295723&postcount=13

I used precisely this exposition in my post. I also specify that certain results of GR follow this conditions and certainly for those the mathematical soundness of GR is perfectly preserved, but that for a number of other GR results those conditions are apparently ignored.

Separate from mathematical foundations, are physical validity of the GR as mathematical model. Here, we have mostly open questions, without conclusive answers:

- Energy conditions that are both too strong and too weak (so far)
- Which causality conditions are enforced in a 'physically plausible' GR solution?
- In what ways does GR fail to match our actual universe at event horizons and singularities?

These are sensible questions, but I find the last one a bit confused when you attribute the "actual universe" precisely some of the predictions of GR, and then question how those two fail to match. IMO what needs to be compared is observation and GR predictions , not the "actual universe" as predicted by GR and GR itself.

 

[PAllen]

Separate from mathematical foundations, are physical validity of the GR as mathematical model. Here, we have mostly open questions, without conclusive answers:

- Energy conditions that are both too strong and too weak (so far)
- Which causality conditions are enforced in a 'physically plausible' GR solution?
- In what ways does GR fail to match our actual universe at event horizons and singularities?

These are sensible questions, but I find the last one a bit confused when you attribute the "actual universe" precisely some of the predictions of GR, and then question how those two fail to match. IMO what needs to be compared is observation and GR predictions , not the "actual universe" as predicted by GR and GR itself.

In all of these questions, I was implying comparison to observation.

I guess the difference is:

- With energy conditions or causality conditions, the goal to make improve the match between GR and observation by excluding nonsense. That is, exploring whether 'pure math' GR + <condition> better matches what is and isn't observed than 'pure math' GR. GR is thus improved or salvaged.

- With singularities and event horizons, it is believed that there is no simple way to remove these as predictions of GR. Thus, to the extent you believe observations would disagree with GR (pretty much universal for singularities; less so for horizons), you can explore where and how GR breaks down.

[edit: to clarify nonstandard usage: By pure math GR I mean GR where any manifold admitting a Lorentzian metric, along with such metric, is considered to be a possible prediction of GR. The EFE are then satisfied simply by using them to prescribe the stress energy tensor]

 

[TrickyDicky]

In all of these questions, I was implying comparison to observation.

I guess the difference is:

- With energy conditions or causality conditions, the goal to make improve the match between GR and observation by excluding nonsense. That is, exploring whether 'pure math' GR + <condition> better matches what is and isn't observed than 'pure math' GR. GR is thus improved or salvaged.

- With singularities and event horizons, it is believed that there is no simple way to remove these as predictions of GR. Thus, to the extent you believe observations would disagree with GR (pretty much universal for singularities; less so for horizons), you can explore where and how GR breaks down.

[edit: to clarify nonstandard usage: By pure math GR I mean GR where any manifold admitting a Lorentzian metric, along with such metric, is considered to be a possible prediction of GR. The EFE are then satisfied simply by using them to prescribe the stress energy tensor]

I find all this quite reasonable.

 

[stevendaryl]

I guess by "the right metric in the first place" you must mean that it is after all one of GR's EFE solutions

No, I meant a metric having the right signature: (+ - - -).

The problem is that mathematically (and even physically as many GR solutions don't make much sense physically), that heuristic is not very rigorous, it basically reverses the hierarchy usually employed in mathematics since the manifold cannot be defined from the metric in many cases, like those where their topologies don't agree,

What do you mean that the "topologies don't agree"? Coordinates are only defined for a "patch"; a small simply connected region of a manifold. There is no topology implied. Now, as I said, there is a question of whether a collection of patches can be stitched together to form a complete manifold. So in addition to defining patches, and giving metrics for the patches, you also have to define how the patches are glued together, and show that the metrics are compatible on the overlap. But that is not difficult for a simple case such as the Schwarzschild solution. (Well, it's not difficult in retrospect.)

the manifold always comes first, or at least that is what I've been told throughout this thread.

Mathematics doesn't care about what order you do things. You can start with a complete manifold, and then try to come up with a metric for it, Or you can start with a metric for a patch, and try to glue the patches into a complete manifold. The end result is the same. But I was saying that I don't think the first approach is very common in practice.

... it is not even assured that a manifold exists that admits that metric.

I would not express the problem that way. It doesn't make sense to say that you have a metric, and are looking around for an appropriate manifold, because you can't really define a metric independently of having a manifold (or at least a section or patch of one). The problem, as I said, is that if I have a collection of patches, and each patch has a metric defined on it, there may be no way to glue them together into a manifold.

 

[stevendaryl]

I tend to agree with you, but note that this observer-biased approach to null geodesics amounts to treating them as timelike geodesics, which is what we normally do when thinking about light as paths as rays traversing a certain distance in a certain time.

The geometric object that makes sense for manifold is the parametrized path, P(s) which is a continuous function from reals to points in the manifold. If the path is timelike, then you can choose the parameter s to be proper time along the path, but that is not necessary to be able to talk about such things as parallel transport. s can be any monotonically increasing real-valued parameter. So there is no difficulty at all in doing parallel transport along lightlike paths. The only thing that is slightly annoying about it is that there is no natural best choice for parametrization in the case of null paths.

 

[samalkhaiat]

To model space time by differentiable manifold, the terms connected, unbounded, oriented, para-compact and Hausdorff need to be included in the term "manifold". For each of those properties ther is a well-founded physical argument. In mathematics, you can put any structure, any metric metric you please to put on the manifold, but only Lorentzian metric is meaning full in physics.

Sam

 

[TrickyDicky]

What do you mean that the "topologies don't agree"? Coordinates are only defined for a "patch"; a small simply connected region of a manifold. There is no topology implied. Now, as I said, there is a question of whether a collection of patches can be stitched together to form a complete manifold. So in addition to defining patches, and giving metrics for the patches, you also have to define how the patches are glued together, and show that the metrics are compatible on the overlap. But that is not difficult for a simple case such as the Schwarzschild solution. (Well, it's not difficult in retrospect.)

I mean precisely that, the topolog induced by a Lorentzian metric has to have several conditions added in order to agree with the smooth manifold topology, and I'm not referring only to the global topology you refer to with the "gluing of the paches". I mean the "local" topology that must be homeomorphic to R⁴ in order to agree with the smooth manifold natural topology. Why do you think coordinate patches don't have topological properties?
What you call a simple case, the KS space, doesn't actually have the required added conditions to have the smooth manifolds topology at the overlaps, for instance the strong causality condition.