Are singularities part of the manifold?

[TrickyDicky]

Mod note: Posts split off from https://www.physicsforums.com/showthread.php?p=4468795

Dun dun duuuun :)

Hi, WN, might the OP be referring to GR instead of SR, more specifically to the expanding FRW universe in which it is impossible to even consider the notion of exansion without agreeing about an "everywhere at once" notion?, after all the current opinion in physics is that our universe is globally described by GR (which implies locally by SR of course) rather than by Minkowski flat spacetime where relativity of simultaneity truly belongs.

 

[Nugatory]

Hi, WN, might the OP be referring to GR instead of SR

He might be, but so what? The answer is the same either way: "at the same time" is observer-dependent.

 

[TrickyDicky]

He might be, but so what? The answer is the same either way: "at the same time" is observer-dependent.

You mean expansion is observer-dependent?
I mean if you apply the relativity of simultaneity strictly to extended objects like the universe there is no possible BB model.

 

[Nugatory]

If you examine Einstein's Equivalence Principle, it shows what Einstein was talking about... something physically, absolutely moved in Einstein's "external world.

Einstein's equivalence principle is about acceleration not motion; you'd have to replace the word "moved" with "accelerated" to make what you said above correct.

 

[WannabeNewton]

For the FLRW universes, everything is described in terms of the observers comoving with the Hubble flow. The global simultaneity slices are with respect to the congruence defined by these observers so again the global simultaneity is only for said family of observers. A different family of observers in the same space-time that define a congruence with non-vanishing twist won't even have global simultaneity slices.

 

[TrickyDicky]

For the FLRW universes, everything is described in terms of the observers comoving with the Hubble flow. The global simultaneity slices are with respect to the congruence defined by these observers so again the global simultaneity is only for said family of observers. A different family of observers in the same space-time that define a congruence with non-vanishing twist won't even have global simultaneity slices.

Certainly, but the relativity of simultaneity argument you guys are using to answer the OP forbids the existence of such family of observers when strictly applied (and even more something physical like a last scattering surface and the CMBR everyone in the universe, any family of observers must agree about). This was my point when making the distinction SR/GR.

 

[TrickyDicky]

A different family of observers in the same space-time that define a congruence with non-vanishing twist won't even have global simultaneity slices.

That family of observers is free to define congruences as they see fit, but the mainstream theory says they should still observe the CMB and be able to relate the doppler shift they observe it with to the family of observers that define global simultaneity slices (global "now instants") and that allow the notion of a homogeneous universe at each instant and a density change with respect to those instants (spatial expansion).

 

[Dale]

There is no absolute simultaneity implied by GR either. You can take the FLRW spacetime, do a coordinate transform with a different global simultaneity convention, and still be able to correctly describe all physics. The GR vs SR distinction doesn't make a difference here. Simultaneity is relative either way.

 

[king vitamin]

DaleSpam: If you're referring to my post, I agree with you completely on the relativity of simultaneity. My qualifier was on my statement that there always exists a global synchronization procedure. I don't feel comfortable generalizing this to GR (does it simply depend on whether the manifold admits a global coordinate system?).

 

[TrickyDicky]

There is no absolute simultaneity implied by GR either. You can take the FLRW spacetime, do a coordinate transform with a different global simultaneity convention, and still be able to correctly describe all physics. The GR vs SR distinction doesn't make a difference here. Simultaneity is relative either way.

You woul need to define what you mean by "absolute" here, I don't think it is a scientific term.
Also I'm not sure what you mean by a coordinate transformation with global simultaneity convention, what global convention? you surely know there is no global coordinates in GR.
And exactly relative to what, is the simultaneity of the CMB?

 

[PeterDonis]

the expanding FRW universe in which it is impossible to even consider the notion of expansion without agreeing about an "everywhere at once" notion?

This is not correct; the expansion of the congruence of "comoving" observers in FRW spacetime is independent of coordinates and independent of any choice of simultaneity convention.

You mean expansion is observer-dependent?

No. See above.

I mean if you apply the relativity of simultaneity strictly to extended objects like the universe there is no possible BB model.

Sure there is; the spacetime geometry described by the BB model is independent of coordinates and independent of any choice of simultaneity convention, just like any spacetime geometry in GR.

the relativity of simultaneity argument you guys are using to answer the OP forbids the existence of such family of observers when strictly applied (and even more something physical like a last scattering surface and the CMBR everyone in the universe, any family of observers must agree about).

No, it doesn't. The "last scattering" surface is independent of coordinates and independent of any choice of simultaneity convention. So is the CMBR.

What you should be saying is that there is only one set of coordinates and one simultaneity convention in which the last scattering surface is a surface of constant coordinate time, and in which the isotropy of the CMBR is manifest in the coordinates. But that statement doesn't justify the other claims you're making.

That family of observers is free to define congruences as they see fit, but the mainstream theory says they should still observe the CMB and be able to relate the doppler shift they observe it with to the family of observers that define global simultaneity slices (global "now instants") and that allow the notion of a homogeneous universe at each instant and a density change with respect to those instants (spatial expansion).

And all of this is indeed what such a family of observers will be able to do. Why do you think it wouldn't be?

 

[Dale]

You woul need to define what you mean by "absolute" here, I don't think it is a scientific term.

You are correct. By "absolute simultaneity" I simply mean the converse of "relative simultaneity" where simultaneity is a matter of convention and different conventions lead to different notions of simultaneity.

Also I'm not sure what you mean by a coordinate transformation with global simultaneity convention, what global convention? you surely know there is no global coordinates in GR.

That isn't true, in general. Many space times do admit global coordinates. However, to make my statement applicable to general space times you can weaken it to "non-local" rather than "global".

And exactly relative to what, is the simultaneity of the CMB?

The CMB doesn't have an intrinsic simultaneity. You will have to explain what you mean by simultaneity of the CMB.

 

[TrickyDicky]

Peter, you are making Exactly the same point I am, I don't know how you manage to make it look like you are arguing with me. ;)

 

[TrickyDicky]

That isn't true, in general. Many space times do admit global coordinates. However, to make my statement applicable to general space times you can weaken it to "non-local" rather than "global".

No, not in general, I said in GR so I was referring to curved spacetimes.

The CMB doesn't have an intrinsic simultaneity. You will have to explain what you mean by simultaneity of the CMB.

Sure I mean the relativity of simultaneity of observers very far apart but at rest with the CMB.

 

[PeterDonis]

Peter, you are making Exactly the same point I am, I don't know how you manage to make it look like you are arguing with me. ;)

Because it seems like you're saying the opposite. This has happened before.

 

[Dale]

Peter, you are making Exactly the same point I am, I don't know how you manage to make it look like you are arguing with me. ;)

It seems to me that you are making opposing points.

 

[PeterDonis]

I mean the relativity of simultaneity of observers very far apart but at rest with the CMB.

Relativity of simultaneity according to what simultaneity convention? Remember that the global simultaneity convention of the "comoving" FRW chart (in which each surface of constant coordinate time is a surface of simultaneity for each "comoving" observer) "lines up" with the local simultaneity convention of each "comoving" observer--i.e., within the local inertial frame of each "comoving" observer, events which are simultaneous with respect to the global FRW coordinates are also simultaneous with respect to the LIF.

 

[Dale]

No, not in general, I said in GR so I was referring to curved spacetimes.

Me too, specifically I was referring to the FLRW spacetime.

The FLRW spacetime admits a traditional coordinate chart which defines the simultaneity convention you are focused on, but it also admits many other coordinate charts. The notion of simultaneity defined by those alternate coordinate charts is every bit as "global" as the notion of simultaneity on the traditional chart, although they disagree. All of the physics of the FLRW chart is the same regardless of the simultaneity convention adopted. Therefore, simultaneity is relative in the FLRW spacetime also, the CMB notwithstanding.

 

[TrickyDicky]

My only point has been that relativity of simultaneity is a local notion, how is this making opposing points?

 

[TrickyDicky]

Me too, specifically I was referring to the FLRW spacetime.

I guess we are referring to different things when talking about global coordinates, I mean the standard sense in which global means covering the whole manifold. I'm sure now you will agree with me. ;-)

 

[TrickyDicky]

Relativity of simultaneity according to what simultaneity convention? Remember that the global simultaneity convention of the "comoving" FRW chart (in which each surface of constant coordinate time is a surface of simultaneity for each "comoving" observer) "lines up" with the local simultaneity convention of each "comoving" observer--i.e., within the local inertial frame of each "comoving" observer, events which are simultaneous with respect to the global FRW coordinates are also simultaneous with respect to the LIF.

Again there seems to be some confusión here about what are global coordinates, FRW coordinates are not global in the same sense Minkowski coordinates are in Minkowski spacetime, and this is key to understand the difference between a local and a global simultaneity convention.

 

[Dale]

I guess we are referring to different things when talking about global coordinates, I mean the standard sense in which global means covering the whole manifold. I'm sure now you will agree with me. ;-)

In any spacetime which admits a global chart (standard sense) you can do a coordinate transform to obtain a different global chart with a different notion of simultaneity. So no, I don't agree with you. Simultaneity is relative in GR also including curved spacetimes. Relativity of simultaneity is not merely a local concept.

 

[TrickyDicky]

In any spacetime which admits a global chart (standard sense) you can do a coordinate transform to obtain a different global chart with a different notion of simultaneity. So no, I don't agree with you. Simultaneity is relative in GR also including curved spacetimes. Relativity of simultaneity is not merely a local concept.

To be sure, you are saying you consider FRW coordinates global in the same sense cartesian coordinates are in Euclidean space?

 

[PeterDonis]

FRW coordinates are not global in the same sense Minkowski coordinates are in Minkowski spacetime

I don't understand what you're saying here. FRW coordinates cover the entire spacetime, just as Minkowski coordinates do, and that's the sense of "global" you said you were using.

 

[Dale]

To be sure, you are saying you consider FRW coordinates global in the same sense cartesian coordinates are in Euclidean space?

I didn't say that. I think that the answer to that depends on the topology of the universe.

What I am saying is that simultaneity is relative, regardless of whether you are talking about SR or GR.

 

[TrickyDicky]

I didn't say that.

Good, then we are agreeing on that.

What I am saying is that simultaneity is relative, regardless of whether you are talking about SR or GR.

And you'd have to quote me saying that simultaneity is not relative. Locally in GR and globally in SR.

 

[WannabeNewton]

In what sense is global simultaneity not relative? Two different time-like congruences in the same space-time don't necessarily pick out the same one-parameter family of orthogonal hypersurfaces foliating the space-time (assuming that such families actually exist for the two congruences). In other words, the global simultaneity slices of two different families of observers in a given space-time (assuming such slices can actually be defined) need not agree.

 

[TrickyDicky]

In what sense is global simultaneity not relative?

Mate, I just invited anyone to quote me saying simultaneity is not relative, I never wrote such thing, I merely made a distinction between the concept in GR and SR, it is local in the former and global in the latter but in both simultaneity is relative. Is my english that bad or what?

 

[Dale]

Mate, I just invited anyone to quote me saying simultaneity is not relative, I never wrote such thing, I merely made a distinction between the concept in GR and SR, it is local in the former and global in the latter but in both simultaneity is relative. Is my english that bad or what?

Yes. At least three of us got a different impression from your posts. Specifically, this quote makes it seem like you believe relativity of simultaneity does not belong in GR:

our universe is globally described by GR ... rather than by Minkowski flat spacetime where relativity of simultaneity truly belongs.

In any case, the relativity of simultaneity is not merely local in GR.

 

[TrickyDicky]

Yes.

Yes to my bad english?!

At least three of us got a different impression from your posts.

Fine, then, we all know this things are democratic so yes I wrote somewhere in the thread "simultaneity is not relative".

Specifically, this quote makes it seem like you believe relativity of simultaneity does not belong in GR:

Make it seem is opinable, Simultaneity of relativity is an SR concpt historically and from any point of view you might want to choose. It so happens that GR is locally Minkowskian and that makes it part of GR.

In any case, the relativity of simultaneity is not merely local in GR.

This sounds as if it made you mad that SR is merely local in GR. But if it were global it would be SR not GR don't you think?

 

[PeterDonis]

It looks to me like everyone is using different definitions of "simultaneity" and "relativity of simultaneity". Maybe we should taboo those terms in this thread, so that everyone has to explicitly define what they mean by them.

 

[Dale]

This sounds as if it made you mad that SR is merely local in GR. But if it were global it would be SR not GR don't you think?

No, I disagree. If you have a global coordinate chart in GR it doesn't suddenly become SR.

 

[TrickyDicky]

No, I disagree. If you have a global coordinate chart in GR it doesn't suddenly become SR.

Er...there are no global coordinates (defined as it is standard as those that cover the whole manifold) in curved manifolds, it is one of the most basic facts of differential geometry.

 

[WannabeNewton]

No, you're thinking of global inertial frames. There's a difference. As an example, Kruskal coordinates for the Schwarzschild space-time cover the entire manifold.

 

[PeterDonis]

Er...there are no global coordinates (defined as it is standard as those that cover the whole manifold) in curved manifolds, it is one of the most basic facts of differential geometry.

You're kidding, right? Some examples of coordinates on curved manifolds that cover the entire manifold:

Kruskal coordinates on maximally extended Schwarzschild spacetime;

FRW coordinates on FRW spacetime (I asked you before about this--are you claiming that FRW coordinates do *not* cover all of FRW spacetime? If so, please show me, explicitly, what part of FRW spacetime FRW coordinates do not cover.)

Any of several standard charts on de Sitter spacetime (any of the ones mentioned in the Wikipedia page would work).

And, of course, a Penrose chart on *any* of the spacetimes I mentioned; Penrose charts are specifically constructed to make sure they cover the entire manifold--and what's more, they do so with a finite range of all coordinates.

 

[TrickyDicky]

You're kidding, right? Some examples of coordinates on curved manifolds that cover the entire manifold:

Kruskal coordinates on maximally extended Schwarzschild spacetime;

FRW coordinates on FRW spacetime (I asked you before about this--are you claiming that FRW coordinates do *not* cover all of FRW spacetime? If so, please show me, explicitly, what part of FRW spacetime FRW coordinates do not cover.)

Any of several standard charts on de Sitter spacetime (any of the ones mentioned in the Wikipedia page would work).

And, of course, a Penrose chart on *any* of the spacetimes I mentioned; Penrose charts are specifically constructed to make sure they cover the entire manifold--and what's more, they do so with a finite range of all coordinates.

My dear Peter, I'm afraid you are using a rather loose concept of "the whole manifold", it is obvious that all of those charts leave out some point, namely the singularity.
Think about the sphere to make this simpler, according to your use of the term global coordinates you are saying that the sphere can be covered with only one set of global coordinates, after all it only leaves out one point(one of the poles) right? Well I'm afraid not.

 

[WannabeNewton]

The Schwarzschild singularity is not a part of the space-time manifold. It is an excise, you complement it out. $S^2$ is a completely different situation; the north and south poles are not geometric singularities.

 

[PeterDonis]

My dear Peter, I'm afraid you are using a rather loose concept of "the whole manifold", it is obvious that all of those charts leave out some point, namely the singularity.

The singularity is not part of the manifold in any of the cases I gave. That's why your analogy with a sphere does not hold (not that it matters anyway--see below): a sphere's poles are part of the manifold.

Think about the sphere to make this simpler, according to your use of the term global coordinates you are saying that the sphere can be covered with only one set of global coordinates

I said no such thing. *You* said "there are no global coordinates in curved manifolds". I responded by saying "there are global coordinates in the following curved manifolds", which is sufficient to refute your claim. I did *not* say "there are global coordinates in *all* curved manifolds", which would, as you say, be false, since a sphere is an obvious counterexample.

You asked earlier in this thread if your use of English was really that bad. Given that you have repeatedly had this same problem, with multiple people, perhaps you should consider the possibility that yes, it is; and that you should take more care to make sure you are actually saying explicitly what you mean, instead of saying something else and expecting us to read your mind and translate what you said into what you actually meant. This would also have the benefit of focusing disagreement much faster; the only thing we appear to disagree about in this particular subthread is that you think singularities are part of the manifold, but it's taken longer than it needed to for us to get to that point.

 

[TrickyDicky]

The singularity is not part of the manifold in any of the cases I gave.

Take the closed FRW metric, it is Minkowski spacetime plus the singularity point, if it had no singularity it would be flat, ergo the singularity must belong to the manifold in order to be curved, do you get it now?

Hope my english is understandable here.

 

[TrickyDicky]

No, you're thinking of global inertial frames. There's a difference. As an example, Kruskal coordinates for the Schwarzschild space-time cover the entire manifold.

No, and I was expecting your usually rigorous use of the math would allow you to see this. Kruskal coordinates don't cover the singularity. Do you also think that the singularity has nothing to do with the manifold?

 

[PeterDonis]

Take the closed FRW metric, it is Minkowski spacetime plus the singularity point

It is no such thing. The closed FRW metric has topology S3 X R; neither the past nor the future singularity are part of the manifold. And there is no relationship that I'm aware of between the closed FRW metric and Minkowski spacetime.

There is an "empty" FRW metric (the Milne model) that is isometric to one "wedge" of Minkowski spacetime (with an unusual coordinate chart), but the "singularity" (which is just the origin of the Minkowski spacetime) is *not* part of the FRW manifold in that case (and it *is* part of the Minkowski spacetime, not an extra point added on to it, so even if it were part of the FRW manifold it wouldn't match your description).

if it had no singularity it would be flat, ergo the singularity must belong to the manifold in order to be curved, do you get it now?

I get that you are either confused or still not expressing yourself very well.

Hope my english is understandable here.

I'm not sure, since if I take what you are saying at face value, as above, it is egregiously wrong. So I'm still thinking you meant to say something else; but it's possible that you actually have a seriously mistaken understanding of what you're talking about.

 

[TrickyDicky]

It is no such thing. The closed FRW metric has topology S3 X R; neither the past nor the future singularity are part of the manifold. And there is no relationship that I'm aware of between the closed FRW metric and Minkowski spacetime.

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

Maybe you believe George, it seems you put more effort discrediting what it is said depending on who says it rather than on the content.

 

[PeterDonis]

Maybe you believe George, it seems you put more effort discrediting what it is said depending on who says it that on the content.

No, it depends on saying it correctly, and on understanding what you're saying.

George said "Minkowski space with a point removed" gives the topology S3 X R, the topology of the closed FRW universe. I agree with that.

George did *not* say that the closed FRW universe (or any FRW universe) includes the singularity point. It doesn't.

You said "Minkowski space plus the singularity point", which is *not* what George said; it's wrong in two ways. See above.

 

[TrickyDicky]

No, it depends on saying it correctly, and on understanding what you're saying.

George said "Minkowski space with a point removed" gives the topology S3 X R, the topology of the closed FRW universe. I agree with that.

George did *not* say that the closed FRW universe (or any FRW universe) includes the singularity point. It doesn't.

You said "Minkowski space plus the singularity point", which is *not* what George said; it's wrong in two ways. See above.

Amusing. The point removed is the singularity point, you know this, don't you?

 

[PeterDonis]

Amusing. The point removed is the singularity point, you know this, don't you?

So what? The point is *removed*--that means it is *not* part of the manifold. Which is what George said, and what I said. And *not* what you said.

How long do you want to keep digging?

 

[TrickyDicky]

So what? The point is *removed*--that means it is *not* part of the manifold. Which is what George said, and what I said. And *not* what you said.

How long do you want to keep digging?

The funny thing is you understood what i meant all along. Well, it's sad actually.
Let's say that having a point removed is what keeps the manifold from being completely covered with a coordinate chart alone. do you like it more like this?
A singularity is the absence of a point(or points), the absence of a point seems to be a defining part of all the manifolds you mentioned, it is clear that in this sense the singularity(the absence of the point) is part of the manifold, is this clear enough?

 

[PeterDonis]

Let's say that having a point removed is what keeps the manifold from being completely covered with a coordinate chart alone. do you like it more like this?

[Edited]

Hmm. I see what you're saying [edited again: maybe not, in view of you're subsequent posts, but what follows is still valid], that the S3 part makes it uncoverable by a single chart; but the way you put it makes it sound like the only way to obtain the S3 X R manifold is by removing a point from R4. That's not really true; manifolds are topological spaces in their own right.

Also, none of this changes the fact that the singularity is not part of the manifold. Are you now agreeing with that?

 

[WannabeNewton]

http://books.google.com/books?id=Xw...skal extension covers entire manifold&f=false

http://books.google.com/books?id=k8...skal extension covers entire manifold&f=false

http://physics.stackexchange.com/qu...ition-of-a-timelike-and-spacelike-singularity

http://philsci-archive.pitt.edu/2980/1/The-singular-nature-of-space-time.pdf (page 5)

http://catdir.loc.gov/catdir/samples/cam031/72093671.pdf (page 7)

 

[TrickyDicky]

No. You can cover the entire manifold S3 X R with a single coordinate chart. That's what the standard FRW chart on closed FRW spacetime does.

It is evident it cannot cover a missing point, isn't it? A missing point that is a defining part of the manifold, otherwise it would be Minkowski.

 

[PeterDonis]

It is evident it cannot cover a missing point, isn't it? A missing point that is a defining part of the manifold, otherwise it would be Minkowski.

First, see my edit to that post; the S3 part of the manifold does introduce a complication.

Second, the missing point is not a "defining part" of the manifold; see my edited previous post. S3 X R is a topological space in its own right, independent of the fact that you can "obtain" it by removing a point from R4. (Note that it's R4, not "Minkowski space", because topologically Minkowski space is just R4; it's only when you introduce a metric on it that it becomes Minkowski space, as opposed to all the other geometries that are also topologically R4.)

Third, the fact that the FRW chart does not cover the "missing point" has nothing to do with it not being able to cover the entire manifold; the only reason it can't on closed FRW spacetime is the S3 part of the topology. The FRW charts on flat and open FRW spacetime do cover the entire manifold (which does *not* include the singularity).