Some doubts concerning the mathematical bases of GR

[TrickyDicky]

Yes, a topological manifold has a topological structure just by being a manifold, this in no way contradicts the fact that when a manifold has a metric, this metric induces the topology.

You are right that the issues I'm bringing up are not mentioned in the usual GR texts, that is why I'm asking for some mathematical proofs, not just convenient assumptions.

 

[Dale]

Yes, a topological manifold has a topological structure just by being a manifold, this in no way contradicts the fact that when a manifold has a metric, this metric induces the topology.

It sounds like a contradiction to me. If the manifold already has a topology then if you introduce a metric which has a topology then it seems that you could get situations where the manifold topology and the metric topology contradict each other. In fact, I think that is exactly this contradiction that is causing your concerns. The topology you are trying to induce via the metric includes all points on the light cone as indistinguishable from a given point, while the topolgy of the manifold distinguishes them. The latter defines the topology of a pseudo-Riemannian manifold, not the former.

On the other hand, if you believe (which I don't) that both the manifold and the metric induce separate topologies and that there is no contradiction in that fact, then simply use the manifold topology rather than the metric topology for defining the manifold as Hausdorff. Presumably, if you have both then you can use either wherever convenient.

 

[PAllen]

You have the sequence wrong, at least according to Hawking and Ellis, whom I trust more than the wikipedia.
All your arguments are based on authority like percentages of authors and definitions without mathematical proof, that is ok in itself but ignores that in the OP I was asking for mathematical rigor rather than authority or physical convenience arguments.

Proof is not relevant at this point. These are definitions. If one defines natural numbers as integers greater than zero, there is no such thing as proving natural numbers are positive.

It is true that there are two definitions of manifold, the common one and the more general one. Most books on GR base it on the common one. It is a definitional choice.

I gave wikipedia links because they are easy to find. However, my GR books that use manifolds all start from the common definition.

This link clarifies some things, and mentions Hawking and Ellis less common usage:

http://mathworld.wolfram.com/TopologicalManifold.html

This link clarifies the usual usage (e.g. that manifold assumes T2-space = hausdorff space property). See the description 'all manifolds'. This clearly means 'under the common definition', otherwise it would be wrong (as opposed to just being shorthand).

http://mathworld.wolfram.com/ParacompactSpace.html

Obviously, for their investigations, Hawking and Ellis have chosen to start from less common definitions. They deliberately start from a manifold that is not necessarily a topological manifold. As I don't have their book, I can't say much more on this.

So, trying to rephrase what the OP is possibly getting at:

If one uses the definitional scheme of Hawking and Ellis, it is then meaningful to ask about proving the Hausdorff property under some particular conditions. Other questions which are true by definition in the common definitional framework also become interesting.

At this point, having clarified that the OP specifically refers the Hawking and Ellis sheme, it would useful for a re-statement of the specific questions the OP wants to discuss.

Unfortunately, I can't contribute further, as I have only studied the more common framework and don't have a copy of Hawking and Ellis.

 

[PAllen]

This excerpt might help clarify some confusion that keeps being posted about metrics and metric tensors:

From the wikipedia entry on Metric:
" Important cases of generalized metrics In differential geometry, one considers metric tensors, which can be thought of as "infinitesimal" metric functions. They are defined as inner products on the tangent space with an appropriate differentiability requirement. While these are not metric functions as defined in this article, they induce metric functions by integration. A manifold with a metric tensor is called a Riemannian manifold. If one drops the positive definiteness requirement of inner product spaces, then one obtains a pseudo-Riemannian metric tensor, which integrates to a pseudo-semimetric . These are used in the geometric study of the theory of relativity, where the tensor is also called the "invariant distance"."

Note that this talks about integrating the pseudo-riemannian metric into a pseudo-semimetric not a pseudometric. Semi-metric is defined in the same article, and pseudo-semimetric is obvious by context. This validates my argument that a pseudometric structure cannot (generally) be imposed (by integration) on pseudo-riemannian manifold, because the pseudo-metric tensor does not integrate into pseudometric. Even for integrating to a pseudo-semimetric, one would have to add some definitions to specify the integration (e.g. minimum interval (any type - timelike, spacelike, or null) over all geodesics [parallel transport definition] connecting two points). It is clear this definition (which is not unique) would, indeed, produce a pseuo-semimetric but not either a semimetric or a pseudometric.

Thus, it is true, as I claimed, that there simply no connection between pseudo-Riemannian manifolds and psuedometric spaces. This is contrast to the fact that any connected Riemannian manifold can be treated as a metric space via natural, unique, integration.

 

[Matterwave]

The topology on a manifold should be induced from the mappings to the Euclidean space. I.e. open sets in the Euclidean space are mapped to open sets in the manifold.

I can always introduce a metric (or more precisely a distance function) to this manifold (since it's a set of points after all). I can also always introduce a distance function that would induce a topology that contradicts the topology induced by the mappings. I can introduce the trivial distance function d(m,n)=0 for all m and n, and this will certainly induce a non-hausdorff topology; however, this is not how we define a topology on a manifold. At least, I have always seen the manifold topology defined the first way and not this second, more convoluted way.

 

[micromass]

I don't see how a pseudometric could ever come into play here.

Every Riemannian manifold or pseudo-Riemannian manifold is by definition a manifold. A manifold is almost always taken to be a locally Euclidean, second countable Hausdorff space.

By Whitney's embedding theorem, a smooth manifold can always be embedded in \mathbb{R}^n for a suitable n. As such, a smooth manifold is metrizable. Thus every smooth manifold can be given the structure of a metric space.

If we define a pseudometric on a smooth manifold, then this pseudometric is always a metric. Indeed, a pseudometric space is a metric space if and only if it is Hausdorff.

It is true that there are non-Hausdorff manifolds, these are topological spaces that are locally Euclidean and second countable. But these are usually not regarded as topological manifolds.

Note that a non-Hausdorff manifold is not even necessarily pseudometrizable. For example, the line with two origins is perhaps the most famous example of a non-Hausdorff manifold. But this line with two origins is T_1 and not Hausdorff, therefore it cannot be pseudometrizable.

We can even go further, a non-Hausdorff manifold is always a T_1. Indeed: given two point x and y in our manifold M. Take a Euclidean neighborhood U of x. If y is not in our neighborhood, then U is a neighborhood of x that does not contain y and as such the T_1 axiom is satisfied. If y is in our neighborhood, then (since our neighborhood is locally Euclidean), we can find a smaller neighborhood around x which does not contain y. Again, then T_1-axiom is satisfied.

So a non-Hausdorff manifold is always T_1 and non-Hausdorff. As such, a non-Hausdorff manifold is never pseudo-metrizable. If it were pseudometrizable, then it would be either Hausdorff or not T_1.

So talking about pseudometrizable non-Hausdorff manifolds is useless, since there are no such things.

 

[TrickyDicky]

Thanks Dalespam, Pallen an matterwave for your interesting contributions, they're surely helpful.
Micromass, that is a great , really informative post, thanks , I value it even more coming from a mathematician rather than a physicist or relativist
that might be contaminated by old habits in their thinking about GR.

 

[TrickyDicky]

Note that this talks about integrating the pseudo-riemannian metric into a pseudo-semimetric not a pseudometric. Semi-metric is defined in the same article, and pseudo-semimetric is obvious by context. This validates my argument that a pseudometric structure cannot (generally) be imposed (by integration) on pseudo-riemannian manifold, because the pseudo-metric tensor does not integrate into pseudometric. Even for integrating to a pseudo-semimetric, one would have to add some definitions to specify the integration (e.g. minimum interval (any type - timelike, spacelike, or null) over all geodesics [parallel transport definition] connecting two points). It is clear this definition (which is not unique) would, indeed, produce a pseuo-semimetric but not either a semimetric or a pseudometric.

This is right, the non-positive definite metric tensor integrates to give a both pseudo- and semi-metric, this means it doesn't only relax the point separation axiom of metric spaces, but the triangle inequality axiom. Pseudosemimetric spaces are also tipically non-Hausdorff, but see the post by micromass.
We must bear in mind that the change of signature of the metric tensor (and therefore its distance function upon integration) is the only difference between a Riemannian and a Pseudo-Riemannian manifold. So anything that eliminates that difference makes them indistinguishable.

This is contrast to the fact that any connected Riemannian manifold can be treated as a metric space via natural, unique, integration.

See micromass answer.

 

[TrickyDicky]

...every smooth manifold can be given the structure of a metric space.

If we define a pseudometric on a smooth manifold, then this pseudometric is always a metric. Indeed, a pseudometric space is a metric space if and only if it is Hausdorff.
So talking about pseudometrizable non-Hausdorff manifolds is useless, since there are no such things.

These are both great points.
In the interest of rigor and as PAllen points out let's call the distance function in question a pseudosemimetric for the Lorentzian manifold used in GR case.
Now if defining a pseudosemimetric in a smooth manifold we actually define a metric, I guess because the smooth manifold topology is the one that rules, mathematically (at least topologically) how do we make a distinction between Riemannian and Pseudo-Riemannian manifolds if their only difference is in the metric tensor that in one case integrates to a metric and in the other to a pseudosemimetric?

 

[Ben Niehoff]

how do we make a distinction between Riemannian and Pseudo-Riemannian manifolds if their only difference is in the metric tensor that in one case integrates to a metric and in the other to a pseudosemimetric?

That is the only distinction. They are both manifolds. One has been given a positive-definite metric tensor; the other has been given an indefinite metric tensor.

Given any (connected) manifold, it is always possible to put a positive-definite metric tensor on it, which can always be integrated to a metric (taking the infimum of multiple geodesics, if need be).

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.

 

[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.

 

[TrickyDicky]

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.

Both Riemannians and Pseudoriemannian manifolds use the Levi-Civita connection so in this particular case parallel transport does require s to be the parameter that locally extremizes the path. Even for null geodesics. As long as we want to respect the underlying manifold topology. A pseudosemimetric space for instance doesn't have this requirement, but PseudoRiemannian manifolds are not pseudosemimetric spaces.

 

[stevendaryl]

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?

The local patches are just little sections of R⁴. There is really nothing much to distinguish one patch from another. In GR, locally everything looks like a little section of Minkowsky spacetime. The only issue really is how the patches are glued together. So I'm not sure what problem you're worried about.

 

[stevendaryl]

Both Riemannians and Pseudoriemannian manifolds use the Levi-Civita connection so in this particular case parallel transport does require s to be the parameter that locally extremizes the path. Even for null geodesics. As long as we want to respect the underlying manifold topology. A pseudosemimetric space for instance doesn't have this requirement, but PseudoRiemannian manifolds are not pseudosemimetric spaces.

I don't think what you are saying is correct. An extremal path (not necessarily the same thing as a geodesic, but it is in the case where the connection is compatible with the metric) from point A to point B is a parametrized path P(s) that extremizes the integral

\int \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds}\dfrac{dx^{\mu}}{ds}}

The parameter s does not need to be proper time; it can be any monotonically increasing parameter. The above integral is invariant under reparametrization.

 

[stevendaryl]

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.

I think you might be mixing necessary and sufficient conditions. You can directly verify in the case of a Schwarzschild black hole that the metric is defined everywhere except at the singularity, and that it always has the signature (+ - - -). So you don't need a theorem telling you under what circumstances it is possible to do that. If you had a theorem giving necessary conditions, and those conditions weren't met in the Schwarzschild case, that would be an indication of an inconsistency.

So are the conditions you are talking about necessary or sufficient?

 

[TrickyDicky]

I think you might be mixing necessary and sufficient conditions. You can directly verify in the case of a Schwarzschild black hole that the metric is defined everywhere except at the singularity, and that it always has the signature (+ - - -). So you don't need a theorem telling you under what circumstances it is possible to do that. If you had a theorem giving necessary conditions, and those conditions weren't met in the Schwarzschild case, that would be an indication of an inconsistency.

So are the conditions you are talking about necessary or sufficient?

Necessary, and the putative inconsistence would be derived from the problem you've mentioned several times that starts with the metric rather than with the manifold and simply expects that a global topology exists that admits that metric.

 

[TrickyDicky]

I don't think what you are saying is correct. An extremal path (not necessarily the same thing as a geodesic, but it is in the case where the connection is compatible with the metric) from point A to point B is a parametrized path P(s) that extremizes the integral

\int \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds}\dfrac{dx^{\mu}}{ds}}

The parameter s does not need to be proper time; it can be any monotonically increasing parameter.

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.
Any Optics text will show that light doesn't follow paths parametrized by any monotonically increasing parameter.

The above integral is invariant under reparametrization.

Different metrics will give different paths.
We are talking about the variation of that integral: \deltaS=0

 

[Ilmrak]

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.
Any Optics text will show that light doesn't follow paths parametrized by any monotonically increasing parameter.

Your book is lying xD
By definition a path is a continuos map \phi: \left[ 0, 1 \right] \rightarrow M\; .

Different metrics will give different paths.
We are talking about the variation of that integral: \deltaS=0

The integral is invariant under reparametrization, i.e. a change of the parameter s not of the metric.

Ilm

 

[stevendaryl]

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.

No, that's not correct. Proper time is defined via

d\tau = \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}} ds

The definition of proper time doesn't refer to extremal paths; if the path is non-extremal (which it will be if there are non-gravitational forces such as electromagnetism) the above still gives you proper time.

The definition of proper time is independent of your choice of the parameter ds. If you make a parameter change s&#039; = f(s) then d\tau is unchanged.

Now, for time-like paths, you can actually use the freedom to change parameters to choose s = \tau, but that's really only a convenience. It doesn't have any consequences, other than making the mathematics simpler.

 

[stevendaryl]

Necessary, and the putative inconsistence would be derived from the problem you've mentioned several times that starts with the metric rather than with the manifold and simply expects that a global topology exists that admits that metric.

But in the case of the Schwarzschild solution, it's possible to explicitly calculate the metric, and see that it has signature (+ - - -) everywhere except at the singularity. This is clearest using Kruskal-Szekeres coordinates, as described here: http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

I'm not sure what problem you are worried about.

 

[stevendaryl]

Now, for time-like paths, you can actually use the freedom to change parameters to choose s = \tau, but that's really only a convenience. It doesn't have any consequences, other than making the mathematics simpler.

...and if the path is lightlike, then you can't use proper time as the parameter, because the parameter has to be monotonically increasing along the path. But you can certainly use other parameters. For example, in 2-D spacetime, the light-like path x=ct can be parametrized by t itself, giving \dfrac{dx^0}{dt} = 1 and \dfrac{dx^1}{dt} = c. Then the proper time \tau is given by:

d\tau = \sqrt{g_{\mu \nu} \dfrac{dx^\mu}{dt} \dfrac{dx^\nu}{dt}} dt
= 0