Are singularities part of the manifold?

[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).

 

[TrickyDicky]

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)

I can see all those references agree to consider the singularity is not part of the manifold because they are trying to do physics and singularities are outside the reach of physics.

 

[TrickyDicky]

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

See post above.
But using logic alone, if you say the singularity is not part of the manifold, you are saing that the manifold is singularity-free. See Hawking and Penrose singularity theorems.

 

[PeterDonis]

But using logic alone, if you say the singularity is not part of the manifold, you are saing that the manifold is singularity-free.

Only if you define "singularity-free" in this way. That's not how the term is usually used; see below.

See Hawking and Penrose singularity theorems.

Which say that geodesics in such a manifold can't be extended to arbitrary values of their affine parameters; physical invariants along them increase without bound as some finite value of the affine parameter is approached. This does not require, and the singularity theorems do not state, that the point at which the affine parameter actually *achieves* the finite value that indicates the singularity (infinite values of physical invariants) is part of the manifold. It only requires that the singularity can be approached arbitrarily closely (where "closely" is defined by the value of the affine parameter along the geodesic of approach) while staying within the manifold.

The term "singularity-free", then, doesn't just mean there are no singularities in the manifold; it means something stronger, that there are no singularities in the manifold *and* that all geodesics can be extended to arbitrary values of their affine parameter with all physical invariants along them remaining finite. I believe Hawking and Ellis explicitly give this definition.

 

[TrickyDicky]

I mentioned the theorems to indicate that GR manifolds seem to have singularities, otherwise why worry about them?

 

[PeterDonis]

I mentioned the theorems to indicate that GR manifolds seem to have singularities, otherwise why worry about them?

If "singularities" is defined appropriately, I agree. But the appropriate definition does not imply that there are actual points contained in the manifold that are singularities. It only implies what I said in my last post, that a manifold with singularities has geodesics that can't be extended to arbitrary values of their affine parameters.

 

[TrickyDicky]

If "singularities" is defined appropriately, I agree. But the appropriate definition does not imply that there are actual points contained in the manifold that are singularities.

The fact is there is no definition whatsoever of singularity in GR(only of singular spacetime, which to my surprise according to many is the one that doesn't contain a singularity) unlike in mathematics, so I wonder how it can be appropriately defined at all.

 

[WannabeNewton]

Hawking and Ellis address your exact misconception in section 8.1

 

[Nugatory]

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.

Totally completely against my better judgement I'm going to step into this thread, just long enough to say that the best definition I've been able to come up with is:

Two events are simultaneous if they have the same time coordinate. This definition makessit clear that that simultaneity is a convention based on the choice of coordinates; and it also allows for the SR definition of simultaneity in which every observer chooses to use the coordinate system in which he is at rest.

Relativity of simultaneity is just the observation that different observers using different coordinate systems will will assign different values of the time coordinate to events, and therefore may disagree about which events have the same time coordinate.

 

[stevendaryl]

The fact is there is no definition whatsoever of singularity in GR(only of singular spacetime, which to my surprise according to many is the one that doesn't contain a singularity) unlike in mathematics, so I wonder how it can be appropriately defined at all.

The sort of singularity that is associated with a black hole can be characterized as the existence of a geodesic such that the curvature increases without bound as you follow the geodesic.

 

[PeterDonis]

The sort of singularity that is associated with a black hole can be characterized as the existence of a geodesic such that the curvature increases without bound as you follow the geodesic.

You also have to include that the curvature increases without bound as a finite value of the affine parameter along the geodesic is approached.

 

[PeterDonis]

Relativity of simultaneity is just the observation that different observers using different coordinate systems will will assign different values of the time coordinate to events, and therefore may disagree about which events have the same time coordinate.

This is one way of looking at it, yes. But it's not the only one. You could define simultaneity the way Einstein originally did: two events A and B are simultaneous for a given observer O if (1) the distance from O to A is the same as the distance from O to B; and (2) a light ray from event A reaches O at the same event on O's worldline as a light ray from event B.

Simultaneity is still relative, i.e., observer-dependent, on this definition; but this definition does not require defining any coordinates.

 

[TrickyDicky]

Having understood in what sense it was claimed that several spacetimes in GR can be covered by a single global coordinate system by considering singular points not to be part of the manifold, a doubt remains for me that I would appreciate that it was addressed. GR's general covariance amounts to saying that physical laws should not be affected by arbitrary changes of coordinates. According to this it should be possible to transform the single global coordinate chart of a singular spacetime to any other coordinate system like for instance an inertial coordinate system, but it is not possible in general for a curved manifold to be covered by a single catesian coordinate system.

 

[PeterDonis]

GR's general covariance amounts to saying that physical laws should not be affected by arbitrary changes of coordinates. According to this it should be possible to transform the single global coordinate chart of a singular spacetime to any other coordinate system like for instance an inertial coordinate system

That doesn't follow, because a global "inertial" coordinate system requires a specific spacetime geometry (Minkowski spacetime). If the spacetime geometry is something else, you can make any arbitrary coordinate transformation you want, but it won't get you an inertial coordinate system.

In other words, the presence of a global inertial coordinate system is a feature of one particular solution to the Einstein Field Equation; it's not a general property that any solution must have. But the physical law, the EFE, applies to any solution; and general covariance just means that I can do any arbitrary coordinate transformation I want, without affecting the validity of the EFE or changing the geometry of any particular solution.

 

[TrickyDicky]

That doesn't follow, because a global "inertial" coordinate system requires a specific spacetime geometry (Minkowski spacetime). If the spacetime geometry is something else, you can make any arbitrary coordinate transformation you want, but it won't get you an inertial coordinate system.

In other words, the presence of a global inertial coordinate system is a feature of one particular solution to the Einstein Field Equation; it's not a general property that any solution must have. But the physical law, the EFE, applies to any solution; and general covariance just means that I can do any arbitrary coordinate transformation I want, without affecting the validity of the EFE or changing the geometry of any particular solution.

And changing to an inertial coordinate is not an arbitrary coordinate transformation? It seems to be when dealing with local charts.

 

[TrickyDicky]

Consider the standard definition of a manifold's coordinate chart:
"A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure.[7] For a topological manifold, the simple space is some Euclidean space Rn and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. For a differentiable manifold the structure is preserved by diffeomorphisms between the manifold's subset and Euclidean space."

I just can't see how a diffeomorphism is possible between a single chart in GR covering the whole manifold and Euclidean space.

 

[stevendaryl]

Having understood in what sense it was claimed that several spacetimes in GR can be covered by a single global coordinate system by considering singular points not to be part of the manifold, a doubt remains for me that I would appreciate that it was addressed. GR's general covariance amounts to saying that physical laws should not be affected by arbitrary changes of coordinates. According to this it should be possible to transform the single global coordinate chart of a singular spacetime to any other coordinate system like for instance an inertial coordinate system, but it is not possible in general for a curved manifold to be covered by a single catesian coordinate system.

Saying that you can use any coordinates you like just means that

• if you have a description of spacetime in terms of coordinates $x,y,z,t$, and
• $X(x,y,z,t), Y(x,y,z,t), Z(x,y,z,t), T(x,y,z,t)$ are four functions that are smooth, and
• the map $\langle x,y,z,t \rangle \Rightarrow \langle X, Y, Z, T\rangle$ is invertible, then
• you can use $X,Y,Z,T$ just as well.

As you say, no change of coordinates of this form will transform curved spacetime into flat spacetime.

 

[TrickyDicky]

Saying that you can use any coordinates you like just means that

• if you have a description of spacetime in terms of coordinates $x,y,z,t$, and
• $X(x,y,z,t), Y(x,y,z,t), Z(x,y,z,t), T(x,y,z,t)$ are four functions that are smooth, and
• the map $\langle x,y,z,t \rangle \Rightarrow \langle X, Y, Z, T\rangle$ is invertible, then
• you can use $X,Y,Z,T$ just as well.

As you say, no change of coordinates of this form will transform curved spacetime into flat spacetime.

Exactly, thanks Steven. So if no diffeomorphism is possible between the global chart and R^n it is evident to me it doesn't correspond to the standard definition of a manifold's coordinate chart that I showed above and that I insisted all along that I was referring to.

 

[Dale]

Having understood in what sense it was claimed that several spacetimes in GR can be covered by a single global coordinate system by considering singular points not to be part of the manifold

Not all manifolds can be covered by a single chart, but that has little to do with singular points, it has to do with topology. The singularities only contribute insofar as they change the topology.

For example, a sphere is a 2D manifold which cannot be covered by a single chart. There are no curvature singularities on a sphere, but it cannot be covered in a single global chart. In contrast, a cone does have a curvature singularity, but it can be covered in a single global chart since it can be mapped to a plane.

 

[stevendaryl]

Exactly, thanks Steven. So if no diffeomorphism is possible between the global chart and R^n it is evident to me it doesn't correspond to the standard definition of a manifold's coordinate chart that I showed above and that I insisted all along that I was referring to.

I think that there might be some confusion about what it means to map something to $R^n$. If you have coordinates for a region of spacetime, then you ALREADY have a mapping from that region to $R^n$. That's what coordinates ARE. So every chart can be mapped to $R^n$. But these mappings don't mean that you get to use the usual metric on $R^n$.

 

[PeterDonis]

And changing to an inertial coordinate is not an arbitrary coordinate transformation? It seems to be when dealing with local charts.

Changing to a local inertial coordinate chart isn't a global transformation; the local inertial chart will only be valid on a small local patch of the manifold.