Is our universe a geodesically complete manifold?

[TrickyDicky]

It's always been my understanding that given the existence of BH singularities and the initial BB singularity our universe couldn't be geodesically complete. But then one of the premises of our cosmology models is that the universe is homogeneous, and all homogeneous manifolds are geodesically complete (at least according to wikipedia), so I'm a bit confused about this. Can someone clarify?
Thanks

 

[clamtrox]

The universe is certainly not a homogeneous manifold: there is quite a noticeable asymmetry in one of the four spacetime dimensions. I'll let you figure out which one.

What people usually mean when they say universe is homogeneous, is that when you split the spacetime into 3 spatial and 1 time directions, the spatial submanifold is homogeneous.

Also, if there exists a black hole somewhere, then clearly the universe is not exactly homogeneous.

 

[TrickyDicky]

Yes, I should have been more precise, I was referring only to the spatial submanifold, singularities are also spatial, so let me rephrase my question: is the spatial submanifold geodesically complete or not? (caveat:I'm not sure if the theorem about homogeneous manifolds and complete geodesics can also be used for submanifolds).

 

[clamtrox]

Particles can fall into black holes so the submanifold (submanifolds are manifolds too so no worries!) is not geodesically complete.

In this context, the word homogeneous is used very strictly. In this sense our universe is very far from homogeneous. I am here but not there, therefore the universe is manifestly not homogeneous.

 

[TrickyDicky]

In this context, the word homogeneous is used very strictly. In this sense our universe is very far from homogeneous. I am here but not there, therefore the universe is manifestly not homogeneous.

Yes I'm aware the spatial homogeneity is a question of scale, however in a cosmological context a space is either homogeneous or not, and the current cosmological model is based in a metric (FRW) that is manifestly spatially homogeneous, not in our scale of course but in the sense of determining if geodesics are complete or not.

 

[clamtrox]

Yes I'm aware the spatial homogeneity is a question of scale, however in a cosmological context a space is either homogeneous or not, and the current cosmological model is based in a metric (FRW) that is manifestly spatially homogeneous, not in our scale of course but in the sense of determining if geodesics are complete or not.

It's not a question of scale at all. Either the spatial hyperslices are homogeneous -- that is energy-momentum tensor does not depend on spatial coordinates -- or it's not. You cannot prove that a manifold which is only on average homogeneous is geodesically complete, and it's easy to think of counterexamples (say, a cubic grid of black holes).

In the usual context, homogeneity means just that the FRW metric describes large-scale evolution of the universe. For example, the cubic grid of black holes would evolve on large scales exactly like a FRW universe with pressureless dust.

 

[TrickyDicky]

It's not a question of scale at all. Either the spatial hyperslices are homogeneous -- that is energy-momentum tensor does not depend on spatial coordinates -- or it's not.

Thanks, that was precisely my point.

You cannot prove that a manifold which is only on average homogeneous is geodesically complete, and it's easy to think of counterexamples (say, a cubic grid of black holes).

In the usual context, homogeneity means just that the FRW metric describes large-scale evolution of the universe. For example, the cubic grid of black holes would evolve on large scales exactly like a FRW universe with pressureless dust.

is a manifold "only on average homogeneous" really homogeneous or not according to what you yourself said in the first quote? Either way I don't need to prove anything, there is already a theorem that says that if it is, then it must be geodesically complete.

Also, I don't know why you introduce "evolution" in this discussion, we are only talking about the spatial submanifold, and homogeneity in FRW cosmology refers only to the 3-space.

 

[TrickyDicky]

So we have agreed that our space submanifold is geodesically incomplete (gravitational singularities everywhere) and that this implies not being spatially homogeneous (because there is a theorem that states that all homogeneous manifolds are geodesically complete), but I don't know yet if someone thinks this is consistent or not with the spatial homogeneity of our cosmological model. Can someone shed light on this?

 

[JamesOrland]

Well, first you have to remember that current accepted models don't posit real singularities. Singularities are meaningless infinities resulting from failures in the models, they're not actual parts of the models. Most people think any good theory of quantum gravity will explain away those meaningless infinities.

But furthermore, the cosmological model states that the cosmos is homogenous at the ultra-large scale only. That is, at the ultra-large scale, the effects of matter and energy are equivalent to those one would observe if it was actually spaced as thin homogenous dust everywhere.

And at the very small scales, space is anything but homogenous; with all that quantum foam floating about, it should look more like a grainy rugged space than a homogenous smooth one.

 

[TrickyDicky]

Well, first you have to remember that current accepted models don't posit real singularities. Singularities are meaningless infinities resulting from failures in the models, they're not actual parts of the models. Most people think any good theory of quantum gravity will explain away those meaningless infinities.

This is not correct, black holes are considered quite real and are part of the model according to GR.

But furthermore, the cosmological model states that the cosmos is homogenous at the ultra-large scale only. That is, at the ultra-large scale, the effects of matter and energy are equivalent to those one would observe if it was actually spaced as thin homogenous dust everywhere.

And at the very small scales, space is anything but homogenous; with all that quantum foam floating about, it should look more like a grainy rugged space than a homogenous smooth one.

This was already discussed, see previous posts. Scale has nothing to do with the mathematical definition of homogeneity:Wikipedia: " a space X is homogeneous if intuitively X looks locally the same everywhere", or with the cosmological:Wikipedia:"Homogeneity means that the same observational evidence is available to observers at different locations in the universe".
Clamtrox affirmed that homogeneity in cosmology is a different concept than the strict mathematical one. I'm trying to confirm if that is the case, but in any case the cosmologycal meaning quoted above is not compatible with singularities either, since there is not the same observational evidence from the inside of a BH as there from other observational points.

 

[JamesOrland]

This is not correct, black holes are considered quite real and are part of the model according to GR.

Didn't say anything about black holes. I was talking about singularities. We all know current physics breaks down beyond the event horizon, and that means exactly what it says on the tin: current physics breaks down. Of course black holes exist. But just because current physics can't explain what's going on inside it doesn't mean it's unexplainable. It means the theory is wrong. That's all.

This was already discussed, see previous posts. Scale has nothing to do with the mathematical definition of homogeneity:Wikipedia: " a space X is homogeneous if intuitively X looks locally the same everywhere", or with the cosmological:Wikipedia:"Homogeneity means that the same observational evidence is available to observers at different locations in the universe".
Clamtrox affirmed that homogeneity in cosmology is a different concept than the strict mathematical one. I'm trying to confirm if that is the case, but in any case the cosmologycal meaning quoted above is not compatible with singularities either, since there is not the same observational evidence from the inside of a BH as there from other observational points.

That is the case. The homogeneity in cosmology is not the mathematical concept of homogeneity. But you cannot say anything definite about the inside of a BH because current accepted models do not have an answer for what happens in there without invoking meaningless singularities that break physics.

 

[TrickyDicky]

Didn't say anything about black holes. I was talking about singularities. We all know current physics breaks down beyond the event horizon, and that means exactly what it says on the tin: current physics breaks down. Of course black holes exist. But just because current physics can't explain what's going on inside it doesn't mean it's unexplainable. It means the theory is wrong. That's all.



That is the case. The homogeneity in cosmology is not the mathematical concept of homogeneity. But you cannot say anything definite about the inside of a BH because current accepted models do not have an answer for what happens in there without invoking meaningless singularities that break physics.

But you are not answering, your just saying the theory is wrong since is based in "meaningless singularities" , and I don't even think most agree about that.

 

[JamesOrland]

All other previous theories that contained singularities were eventually replaced by theories that explained the magical singular effects. Check Wikipedia, if you like.

But even if there is a true singularity there, we cannot really affirm that yet, because we don't have a quantum theory of gravity, so we don't know what would actually happen around a GR singularity.

But most physicists agree that the prediction of a singularity in a Black Hole is a failure of the General Relativity Theory, especially taking into account the fact that Q.M. says that nothing can occupy a space smaller than its wavelength. So, either Q.M. or G.R. is wrong, and most people nowadays guess the answer is 'both', which is why everyone is so trying to find a TOE that unifies Q.M. with G.R. in a satisfactory manner.

The answer to the question would be that it appears the Universe is not geodesically complete, because it's not mathematically homogenous. You can just check Wikipedia's article on that: when you click the word 'homogenous' you are linked to a page talking about translation invariance, within an article called Homogeneity (physics). So no, the use of the word 'homogenous' in cosmological models is not the same as the one used in a homogenous space concept.

 

[TrickyDicky]

The answer to the question would be that it appears the Universe is not geodesically complete, because it's not mathematically homogenous.

Well, maybe, but I fail to understand your position here, first you tell me that singularities are meaningless and that this is the reason GR is wrong, and now you answer that the universe must be geodesically incomplete and therefore inhomogeneous (or the other way around?). But you do realize that the only reason to consider it geodesically incomplete is the fact that is supposed to have singularities, right? If there were no singularities, as you (and according to you also wikipedia) claim, then it would be geodesically complete.
Also I wonder what kind of a black hole is one without singularity, an event horizon alone has no physical consequences.

So no, the use of the word 'homogenous' in cosmological models is not the same as the one used in a homogenous space concept.

The problem is that the FRW metric is based in the mathematical use of homogeneity, and it surely admits perturbative approximations that may vary the scale of homogeneity, but can never admit singularities, because if a not (mathematically)homogeneous 3-space is used in the metric (and there are only three choices :euclidean, elliptic and hyperbolic), then expansion(or contraction) is not mathematically feasible, and in addition it wouldn't fulfill the cosmological principle.

 

[JamesOrland]

Well, maybe, but I fail to understand your position here, first you tell me that singularities are meaningless and that this is the reason GR is wrong, and now you answer that the universe must be geodesically incomplete and therefore inhomogeneous (or the other way around?). But you do realize that the only reason to consider it geodesically incomplete is the fact that is supposed to have singularities, right? If there were no singularities, as you (and according to you also wikipedia) claim, then it would be geodesically complete.

(...)

The problem is that the FRW metric is based in the mathematical use of homogeneity, and it surely admits perturbative approximations that may vary the scale of homogeneity, but can never admit singularities, because if a not (mathematically)homogeneous 3-space is used in the metric (and there are only three choices :euclidean, elliptic and hyperbolic), then expansion(or contraction) is not mathematically feasible, and in addition it wouldn't fulfill the cosmological principle.

I'm sorry, my bad. You are right, indeed the current models state that the metric of space is geodesically complete. What I really should have said was that we currently lack enough information to state whether or not it is indeed geodesically complete, but our models currently assume it is. So, until we have more information, and unless the current model is replaced by another non-homogenous one, it's safe to say that yes, our universe is a geodesically complete manifold.

Also I wonder what kind of a black hole is one without singularity, an event horizon alone has no physical consequences.

I think you don't quite understand what I mean by 'there are no singularities.' I mean that singularities as GR puts them - points/rings with zero volume and infinite density, the things that would break homogeneity - do not exist, but are rather things that behave like singularities.

I'm saying that the infinities predicted by GR are meaningless, and that a better understanding of Quantum Gravity will probably lead to a more accurate description of what's really going on, plus will accept a singularity as a good approximation at larger scales (the same way classical theory is a good approximation of the real world).

Also, an event horizon is a direct consequence of a singularity-like structure, so without a singularity-like structure there really is no event horizon to be spoken of.

 

[Mark M]

But you are not answering, your just saying the theory is wrong since is based in "meaningless singularities" , and I don't even think most agree about that.

TrickyDicky, singularities do not exist, they're a phenomena that occur when you force a theory to describe something it cannot. For example, the inside of an event horizon. The regular solution to the Einstein Field Equations for a homogeneous non-rotating object is the Schwarzschild metric. However, the Schwarzschild metric breaks down at the event horizon, and produces nonsense infinite results. So, you must use another metric, such as Kruskal coordinates. But, at the center of a black hole, these will also break down and produce the singular result.

You can further see that a whole new theory is needed at this situation, because quantities that are diffeomorphism invariant become infinite, such as the Kretschmann scalar.

Also, singularities are incompatible with quantum mechanics, which states that a particle exist in a space smaller than its wavelength.

Einstein himself admitted that GR must be replaced in these situations. Today, we know quantum gravity should provide the solution and rid of the singularities.

 

[TrickyDicky]

Mark and James, first thanks for your imputs, I know what you are saying and might even share it(about the limited validity of the current model and the need to replace it).
Still I think there must be some
kind of explanation(other than saying the theory as of now is surely wrong) from standard cosmology for such a big inconsistency in the model.

 

[JamesOrland]

There actually isn't. Standard cosmology has one big gaping hole in it. But of course, it's not a big gaping hole until you actually look at it. It's just like classical physics: if you don't look too closely at tiny things or very big things, you won't ever notice that there's anything wrong with it. But once you do, you realize that it must be replaced.

Same thing with cosmology. As long as you don't look too closely at those things we call 'Black Holes', we don't even realize there's anything wrong with it.

But it's likely that the standard model is not wrong per se - what is wrong is our formulation of Black Holes, indeed. The standard model doesn't allow singularities; so, once they are explained away by a quantum theory of gravity, there won't be any singularities (just things that behave like singularities), and the inconsistency will be gone.

 

[TrickyDicky]

There actually isn't. Standard cosmology has one big gaping hole in it. But of course, it's not a big gaping hole until you actually look at it. It's just like classical physics: if you don't look too closely at tiny things or very big things, you won't ever notice that there's anything wrong with it. But once you do, you realize that it must be replaced.

Same thing with cosmology. As long as you don't look too closely at those things we call 'Black Holes', we don't even realize there's anything wrong with it.

But it's likely that the standard model is not wrong per se - what is wrong is our formulation of Black Holes, indeed. The standard model doesn't allow singularities; so, once they are explained away by a quantum theory of gravity, there won't be any singularities (just things that behave like singularities), and the inconsistency will be gone.

As a general reflection I find it ok (maybe a bit overoptimistic about the prospects of solving the problems with a new quantum gravity theory that has been promising that for several decades now without much success).
However I could almost assure you most of the usual posters in this subforum and in the relativity subforum wouldn't agree with your considerations about Black holes and singularities in general, oddly enough none of them seems to be interested in this discussion. who knows, maybe they've changed their minds.

 

[JamesOrland]

Well, to be honest, it was on a discussion with a few people here in this forum that it was pointed out to me that a singularity signaled the breakdown of a theory. So these people must be very quiet indeed :P

But regardless, I'm pretty sure physicists in general take singularities to mean the breakdown of a theory. I haven't yet seen someone seriously propose that a singularity must exist "out there". It just doesn't make sense, for something to occupy no space at all and possesses infinite density.

Now, don't get me wrong. I have long since abandoned hope that the Universe might be intuitivelly understandable (actually, I did so eagerly - I find the idea of a counterintuitive Universe much more exciting), so of course that no matter how little sense it makes to me, if it turns out that there really is such a thing as a zero-volume infinite density point, well, c'est la vie. But Quantum Mechanics does forbid that pretty vehemently, and no such infinities have ever been found in nature so far, so I will still hold onto hope that it will be explained away in a very elegant and mathematically beautiful way by a quantum theory of gravity.

 

[TrickyDicky]

Yeah, I hope so too.