Mathematical rigor of GR spacetime manifold assumptions

[PeterDonis]

The fact is that null geodesics in Minkowski spacetime are the asymptotical limit of timelike geodesics.

Well, Ben Niehoff posted that this depends on the topology you adopt for the space of paths, and I think I agree with him (but see the question I posed in my latest post before this one). And I gave at least one example of a set of timelike paths which do *not* approach a null path in any limit; so the claim you are making here can't be true exactly as you state it, since you didn't give any qualifications on how the "asymptotical limit" is to be taken.

 

[Ben Niehoff]

I see the intuition that leads to the limit idea: in any particular inertial frame, if I look at timelike geodesics from the origin, (t, x) = (0, 0), to another surface of simultaneity, i.e., to various endpoints (t, x) with t always the same, their length gets shorter and shorter as their relative velocity approaches c (meaning x approaches t). So it's natural to think of a null geodesic lying on the light cone from the origin as a limit of those timelike geodesics as v -> c.

In this case, it is easy to define a topology on the space of paths. We can just borrow the topology from the surface of simultaneity where all the path endpoints lie. Two paths in this 1-parameter space are "nearby" if their endpoints are "nearby" in the surface of simultaneity. Open sets on the space of paths can be put on a 1-to-1 correspondence with open sets on the surface of endpoints.

Then a sequence of paths whose endpoints converge to (1,1) will converge to the lightlike path between (0,0) and (1,1).

However, this set of geodesics only looks natural in that particular frame.

Whether it looks natural in a given frame is irrelevant. In any frame, there is 1-parameter space of paths whose endpoints lie on some spacelike surface, and in that space there is a sequence of timelike paths that converges to a null path.

There is, of course, an invariant "natural" set of timelike geodesics corresponding to any particular one we pick: the set whose endpoints lie on the hyperbola t^2 - x^2 = tau^2, where tau is the proper time along the geodesic we pick. But of course these geodesics all have the *same* length; they do *not* approach any null geodesic as a limit, though they do appear to "point" closer and closer to the light cone in the particular frame we picked. But again, we can transform to any other frame and change the way all of the geodesics appear to "point".

In this case, you are talking about a different space of paths: the 1-parameter space of paths starting at (0,0) and having a fixed Minkowski length. In this case, the null paths on the lightcone are strictly outside this space. However, since this space of paths is a subset of all the paths (through the origin) in Minkowski space, the natural topology to use is the one inherited from Minkowski space.

In this case, a sequence of paths with endpoints on the hyperbola will converge pointwise to a lightlike path, but will not converge uniformly to it. Look up the difference between pointwise and uniform convergence.

Note again that your comments on Lorentz transforms are irrelevant: While any particular element in the sequence and be transformed to point straight up, the sequence has infinitely many elements, and there are always elements that lie close to the limiting path, in every frame.

 

[PeterDonis]

We can just borrow the topology from the surface of simultaneity where all the path endpoints lie.
...
Then a sequence of paths whose endpoints converge to (1,1) will converge to the lightlike path between (0,0) and (1,1).

Yes, I see that, and it seems like a good way of making mathematically precise the intuition that stevendaryl and TrickyDicky were expressing.

Whether it looks natural in a given frame is irrelevant. In any frame, there is 1-parameter space of paths whose endpoints lie on some spacelike surface, and in that space there is a sequence of timelike paths that converges to a null path.

"Looks natural" was probably a bad choice of words. All I was trying to point out was that the space of paths defined in this way is frame-dependent; but the space of paths defined as having fixed Minkowski length (see below) is frame-invariant. So there is a difference between the two kinds of "path spaces".

In this case, you are talking about a different space of paths: the 1-parameter space of paths starting at (0,0) and having a fixed Minkowski length. In this case, the null paths on the lightcone are strictly outside this space. However, since this space of paths is a subset of all the paths (through the origin) in Minkowski space, the natural topology to use is the one inherited from Minkowski space.

This confuses me a bit. Wouldn't the "natural" topology be the one on the hyperbola itself as a parametrized curve? Or is that the same as the topology inherited from Minkowski space?

In this case, a sequence of paths with endpoints on the hyperbola will converge pointwise to a lightlike path, but will not converge uniformly to it. Look up the difference between pointwise and uniform convergence.

Will do. It's been a long time since I dug into this subject, so it will do me good.

Note again that your comments on Lorentz transforms are irrelevant: While any particular element in the sequence and be transformed to point straight up, the sequence has infinitely many elements, and there are always elements that lie close to the limiting path, in every frame.

Yes, this is true. See above for some clarification on what I was trying to get at with my comments about Lorentz transforms.

 

[stevendaryl]

Defining a limit on a space of paths requires you to define a topology on the space of paths. This is what topology is all about.

I think there are perfectly sensible topologies on the space of paths in which a null path is the limit of a sequence of timelike paths. Or spacelike paths, for that matter. These topologies, like the underlying manifold topology, are not induced from the pseudo-Riemannian metric.

However, as you point out, the proper time along the path will approach zero as the path approaches the limit. Therefore proper time is not a good parameter along null paths (which we already knew).

Something that I realized that I don't know how to do is calculus of variations in the case where the derivative of the Lagrangian is undefined.

If we are looking for a parametrized path $P(s)$ that extremizes the proper time, then we can cast this as a Lagrangian dynamics problem:

$A = \int{L ds}$

where the action $A$ is interpreted as the proper time $\tau$, and the lagrangian $L$ is interpreted as the expression $\sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}}$

Using the Euler-Lagrange equations of motion gives for the extremizing path:

$\dfrac{d}{ds} \dfrac{\partial L}{\partial U^{\mu}} - \dfrac{\partial L}{\partial x^{\mu}} = 0$

where $U^{\mu} = \dfrac{d x^{\mu}}{ds}$

For the particular choice of $L = \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}}$, the equations of motion become:

$\dfrac{\partial_{\lambda} g_{\mu \nu} U^{\lambda} U^{\nu}}{L} + \dfrac{g_{\mu \nu} \dfrac{dU^{\nu}}{ds}}{L}<br /> - \dfrac{g_{\mu \nu} U^{\nu} \dfrac{dL}{ds}}{L^{2}} - \dfrac{1}{2 L} \partial_{\mu} g_{\nu \lambda} U^{\nu} U^{\lambda} = 0$

If we multiply through by $L$, this becomes:

$\partial_{\lambda} g_{\mu \nu} U^{\lambda} U^{\nu} + g_{\mu \nu} \dfrac{dU^{\nu}}{ds}<br /> - g_{\mu \nu} U^{\nu} \dfrac{d \ log (L)}{ds} - \dfrac{1}{2} \partial_{\mu} g_{\nu \lambda} U^{\nu} U^{\lambda} = 0$

Finally, if we assume an affine parametrization, so that $L$ is constant along the path, then this simplifies to:

$\partial_{\lambda} g_{\mu \nu} U^{\lambda} U^{\nu} + g_{\mu \nu} \dfrac{dU^{\nu}}{ds}<br /> - \dfrac{1}{2} \partial_{\mu} g_{\nu \lambda} U^{\nu} U^{\lambda} = 0$

which is equivalent to the usual geodesic equation. The problem, mathematically, is that the manipulations only make sense if $L$ is nonzero. If $L$ is zero, then all the equations (with $L$ in the denominator) are undefined.

For this reason, it seems to me that one needs to assume during the derivation that $L$ is small, but nonzero. So I don't see how you can actually get a null geodesic this way, except possibly as a limiting case.

 

[PAllen]

I think it is generally best to use a parallel transport definition of geodesic for semi-riemannian manifolds - the path the parallel transports its tangent vector. All the issues you refer to disappear, and the derivation of all types of geodesics is trivial.

 

[stevendaryl]

I think it is generally best to use a parallel transport definition of geodesic for semi-riemannian manifolds - the path the parallel transports its tangent vector. All the issues you refer to disappear, and the derivation of all types of geodesics is trivial.

I'm a little confused about the two methods. If you start by extremizing proper time, you get a term, $g_{\mu \nu} U^{\nu} \dfrac{d \ log(L)}{ds}$ that can be made to vanish by choosing an affine parametrization. If instead you use parallel transport, such a term doesn't come up (I don't think). So are the two methods only equivalent in the case of an affine parametrization of the path?

 

[PAllen]

I'm a little confused about the two methods. If you start by extremizing proper time, you get a term, $g_{\mu \nu} U^{\nu} \dfrac{d \ log(L)}{ds}$ that can be made to vanish by choosing an affine parametrization. If instead you use parallel transport, such a term doesn't come up (I don't think). So are the two methods only equivalent in the case of an affine parametrization of the path?

You can always use the affine definition, which never has the extra term. This equation constrains (when you solve it) the parameter to be a linear function of proper time for time like geodesic, or some suitable affine parameter for null or spacelike geodesic. If, instead, you use a more general formula, you can get any admissable parameter - but then you can always do a simple parameter transform in which the geodesic equation takes the simplest form. This explained well on pages 70-71 of:

http://arxiv.org/pdf/gr-qc/9712019v1.pdf

 

[TrickyDicky]

Well, Ben Niehoff posted that this depends on the topology you adopt for the space of paths

That topology seems to be the one it is necessary adopting in order to consider it a smooth manifold, i.e. the one that coincides with the smooth manifold topology and as I said in the case of Minkowski space this topology comes from the globally hyperbolic causal structure.

 

[TrickyDicky]

Something that I realized that I don't know how to do is calculus of variations in the case where the derivative of the Lagrangian is undefined.

If we are looking for a parametrized path $P(s)$ that extremizes the proper time, then we can cast this as a Lagrangian dynamics problem:

$A = \int{L ds}$

where the action $A$ is interpreted as the proper time $\tau$, and the lagrangian $L$ is interpreted as the expression $\sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}}$

Using the Euler-Lagrange equations of motion gives for the extremizing path:

$\dfrac{d}{ds} \dfrac{\partial L}{\partial U^{\mu}} - \dfrac{\partial L}{\partial x^{\mu}} = 0$

where $U^{\mu} = \dfrac{d x^{\mu}}{ds}$

For the particular choice of $L = \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}}$, the equations of motion become:

$\dfrac{\partial_{\lambda} g_{\mu \nu} U^{\lambda} U^{\nu}}{L} + \dfrac{g_{\mu \nu} \dfrac{dU^{\nu}}{ds}}{L}<br /> - \dfrac{g_{\mu \nu} U^{\nu} \dfrac{dL}{ds}}{L^{2}} - \dfrac{1}{2 L} \partial_{\mu} g_{\nu \lambda} U^{\nu} U^{\lambda} = 0$

If we multiply through by $L$, this becomes:

$\partial_{\lambda} g_{\mu \nu} U^{\lambda} U^{\nu} + g_{\mu \nu} \dfrac{dU^{\nu}}{ds}<br /> - g_{\mu \nu} U^{\nu} \dfrac{d \ log (L)}{ds} - \dfrac{1}{2} \partial_{\mu} g_{\nu \lambda} U^{\nu} U^{\lambda} = 0$

Finally, if we assume an affine parametrization, so that $L$ is constant along the path, then this simplifies to:

$\partial_{\lambda} g_{\mu \nu} U^{\lambda} U^{\nu} + g_{\mu \nu} \dfrac{dU^{\nu}}{ds}<br /> - \dfrac{1}{2} \partial_{\mu} g_{\nu \lambda} U^{\nu} U^{\lambda} = 0$

which is equivalent to the usual geodesic equation. The problem, mathematically, is that the manipulations only make sense if $L$ is nonzero. If $L$ is zero, then all the equations (with $L$ in the denominator) are undefined.

For this reason, it seems to me that one needs to assume during the derivation that $L$ is small, but nonzero. So I don't see how you can actually get a null geodesic this way, except possibly as a limiting case.

Isn't this what we have been talking about all along and finally clarified?
It seems as you are now trying to do what you said can't be done:using proper time as parameter for null geodesics.
Sure it is a limit case, try reading the last page of the thread again.

 

[Ben Niehoff]

"Looks natural" was probably a bad choice of words. All I was trying to point out was that the space of paths defined in this way is frame-dependent; but the space of paths defined as having fixed Minkowski length (see below) is frame-invariant. So there is a difference between the two kinds of "path spaces".

Still irrelevant. Replace "Minkowski space" and "Lorentz-invariant" with "Euclidean space" and "rotation-invariant". Surely you would agree that objects exist which are not rotationally-invariant?

The fact that you can look at something from a different angle (cf. a different inertial frame) does not cause that thing to cease to exist, or be somehow ill-defined.

This confuses me a bit. Wouldn't the "natural" topology be the one on the hyperbola itself as a parametrized curve? Or is that the same as the topology inherited from Minkowski space?

It is not the same as the topology inherited from Minkowski space. On the hyperbola (which is just the real line), as a point recedes to infinity, it does not approach any limit.

The fact is that in both cases (paths with endpoints on a given spacelike surface, or paths of constant total proper time), the correct choice of topology is the one that agrees with the ambient Minkowski space. But in the case of paths with endpoints on a given spacelike surface, these topologies agree.

There is nothing mysterious about 1+1-dimensional Minkowski space. Topologically, it is exactly the same thing as R^2, the Euclidean plane. So all you're asking is whether a sequence of line segments inside some triangle converges to the edge of the triangle.

 

[PeterDonis]

Still irrelevant. Replace "Minkowski space" and "Lorentz-invariant" with "Euclidean space" and "rotation-invariant". Surely you would agree that objects exist which are not rotationally-invariant?

The fact that you can look at something from a different angle (cf. a different inertial frame) does not cause that thing to cease to exist, or be somehow ill-defined.

Of course. I certainly wasn't trying to claim that any given surface of simultaneity of a particular inertial observer doesn't exist, or that it isn't well-defined. I had thought, though, that the Lorentz invariance of the hyperbola, vs. the non-invariance of the surface of simultaneity, might make a difference. In view of your further comments, I'm not sure that intuition of mine actually leads anywhere. See below.

It is not the same as the topology inherited from Minkowski space. On the hyperbola (which is just the real line), as a point recedes to infinity, it does not approach any limit.

Ah, I see. So in a sense, I was right in saying that a sequence of timelike geodesics, all with the same Minkowski length, doesn't approach any null geodesic as a limit--since that sequence doesn't approach any limit at all. But that pretty much says everything that can be said about that particular sequence, so it doesn't really help with understanding anything else. Fair enough.

The fact is that in both cases (paths with endpoints on a given spacelike surface, or paths of constant total proper time), the correct choice of topology is the one that agrees with the ambient Minkowski space.

"Correct" meaning "such that the sequence has a well-defined limit", yes? But that still leaves me a bit confused about the hyperbola case; see below.

There is nothing mysterious about 1+1-dimensional Minkowski space. Topologically, it is exactly the same thing as R^2, the Euclidean plane. So all you're asking is whether a sequence of line segments inside some triangle converges to the edge of the triangle.

In the case of the paths all ending on a surface of simultaneity, I agree. But in the case of the paths all having the same total proper time, i.e., all ending on the hyperbola, there is no triangle that encloses them all, is there?

 

[Ben Niehoff]

In the case of the paths all ending on a surface of simultaneity, I agree. But in the case of the paths all having the same total proper time, i.e., all ending on the hyperbola, there is no triangle that encloses them all, is there?

No, and this is part of why this sequence converges only pointwise, but not uniformly.

 

[TrickyDicky]

"Correct" meaning "such that the sequence has a well-defined limit", yes? But that still leaves me a bit confused about the hyperbola case; see below.



In the case of the paths all ending on a surface of simultaneity, I agree. But in the case of the paths all having the same total proper time, i.e., all ending on the hyperbola, there is no triangle that encloses them all, is there?

It is a matter of choosing what possible topology is the adequate to keep considering the spacetime a manifold, and what topology we must ignore, in this case the topology induced by the Lorentzian metric is formally ignored, so we discard the cases with pointwise convergence, since only the stronger uniform convergence assures properties like continuity or Riemann integrability that allow to comply with the smooth manifold topology.
Quoting from the mathpages link:
" To do this, we can simply take as our basis sets all the finite intersections of Minkowski neighborhoods. Since the contents of an e-neighborhood of a given point are invariant under Lorentz transformations, it follows that the contents of the intersection of the e-neighborhoods of two given points are also invariant. Thus we can define each basis set by specifying a finite collection of events with a specific value of e for each one, and the resulting set of points is invariant under Lorentz transformations. This is a more satisfactory approach than defining neighborhoods as the set of points whose coordinates (with respect to some arbitrary system of coordinates) differ only a little, but the fact remains that by adopting this approach we are still tacitly abandoning the Lorentz-invariant sense of nearness and connectedness, because we are segregating null-separated events into disjoint open sets. "

 

[TrickyDicky]

In the case of Minkoski spacetime the fact that it admits a Cauchy surface foliation and the relativity of simultaneity is what allows us to make compatible the Lorentzian metric induced topology and the smooth manifold one. But this cannot be done in general with curved spacetimes, we saw it is possible for certain compact manifolds but at the price of having closed timelike curves.
Most times we have to deal with singularities that make the concept of connected smooth manifold (that seemed so important to keep) lose its meaning (what's smooth in a space full of discontinuities?). Nobody seems to care about it so I guess there is no much point worrying about it anyway.

 

[yenchin]

Not sure if this has been mentioned already earlier in the thread, but the issue of topology arise already in special relativity. Zeeman observed that the ordinary topology on R^4 has no "physical significance" from SR point of view. He thus introduced so-called "fine topology". Later on, Hawking, King and McCarthy introduced another useful topology called the "path topology", which had the same homeomorphism group as Zeeman's fine topology, which is essentially the Lorentz group.

Appendix A of the book "The Geometry of Minkowski Spacetime" by Naber has a good discussion about this. Also see http://en.wikipedia.org/wiki/Spacetime_topology.

 

[stevendaryl]

Isn't this what we have been talking about all along and finally clarified?
It seems as you are now trying to do what you said can't be done:using proper time as parameter for null geodesics.
Sure it is a limit case, try reading the last page of the thread again.

Nothing that I wrote assumes that the parameter s is proper time.

 

[stevendaryl]

You can always use the affine definition, which never has the extra term. This equation constrains (when you solve it) the parameter to be a linear function of proper time for time like geodesic, or some suitable affine parameter for null or spacelike geodesic. If, instead, you use a more general formula, you can get any admissable parameter - but then you can always do a simple parameter transform in which the geodesic equation takes the simplest form. This explained well on pages 70-71 of:

http://arxiv.org/pdf/gr-qc/9712019v1.pdf

I would say that it discusses these issues, but not that it explains them. It confirms what I thought, that the two equations for geodesics (extremizing proper time and parallel transport using the connection coefficients) are only equivalent for an affine parametrization. But it doesn't really explain why parallel transport implies a particular choice of parameters. I don't quite understand that.

Also, it seems to me that null geodesics should be understandable in terms of the calculus of variations. But it's not clear in what sense. Since every null path has the same proper time--zero, the geodesics are obviously not singled out by the calculus of variations. That's why I was thinking that there was some other more general criterion that would select the null geodesics. The answer--forget about the calculus of variations, and instead just use parallel transport--is a little unsatisfying, partly because of the restriction to affine parametrizations. Isn't there a way to characterize the non-affinely parametrized null geodesics?

 

[PAllen]

I would say that it discusses these issues, but not that it explains them. It confirms what I thought, that the two equations for geodesics (extremizing proper time and parallel transport using the connection coefficients) only hold for an affine parametrization. But it doesn't really explain why parallel transport implies a particular choice of parameters. I don't quite understand that.

Also, it seems to me that null geodesics should be understandable in terms of the calculus of variations. But it's not clear in what sense. Since every null path has the same proper time--zero, the geodesics are obviously not singled out by the calculus of variations. That's why I was thinking that there was some other more general criterion that would select the null geodesics. The answer--forget about the calculus of variations, and instead just use parallel transport--is a little unsatisfying, partly because of the restriction to affine parametrizations. Isn't there a way to characterize the non-affinely parametrized null geodesics?

To me, the issue is explained. For any actual curve in spacetime, changing parameter does not change the curve. The most general form of geodesic equation reduces to the simplest by some general parameter substitution. Therefore, there is no need to ever use the general form - use the simplest, then change parameter if you want to for some perverse reason.

The way the affine definition selects a simplest affine parameter (which turns out to be a linear function of proper time for timelike geodesics) is by imposing the simplest form of geodesic equation. All possible parameters chosen by the affine equation are related by a linear function.

To my mind, and this is a philosophic bias, the variational definition of geodesic is not meaningful to me for a Lorentzian metric except for timelike curves. For spacelike paths, it is not an even a local extremal of any sort (neither maximum nor minimum). Thus, I prefer to adopt the parallel transport definition, and then derive the fact that it is a local maximum for timelike geodesics.

 

[TrickyDicky]

Nothing that I wrote assumes that the parameter s is proper time.

You are right, I misunderstood something at the beginning of the post.

 

[TrickyDicky]

The way the affine definition selects a simplest affine parameter (which turns out to be a linear function of proper time for timelike geodesics) is by imposing the simplest form of geodesic equation. All possible parameters chosen by the affine equation are related by a linear function.

The simplest form of geodesic equation happens to be the one imposed by a torsionless (pseudo) Riemannian connection.

 

[PeterDonis]

No, and this is part of why this sequence converges only pointwise, but not uniformly.

Ok, I'm still cogitating about that part.

 

[PeterDonis]

Ok, I'm still cogitating about that part.

Ok, let me see if I've got this straight. Apologies if I'm belaboring the obvious.

First, the case where we look at convergence along a line of simultaneity. We start at a point (x, t) = (0, T) and move along a line of constant t to the point (T, T), and look at the slopes of the lines from the origin to each point. So we are looking at the convergence of x/t, where x/t goes from 0 to 1. Obviously this converges to 1, and the rate of convergence is independent of T; i.e., it's independent of which line of simultaneity we choose. (And it's also independent, of course, of which frame we choose; we can run this same analysis in any frame we like, and while the specific lines of constant t will be different, the rate of convergence is still independent of T.)

Second, the case where we look at convergence along the hyperbola, t^2 - x^2 = T^2. So we start at the point (0, T) and move along the hyperbola to ever larger values of x, and again we look at the slopes of the lines from the origin to each point. Here again we are looking at the convergence of x/t. It still converges to 1, as before, but now the rate of convergence depends on T, i.e., it depends on which hyperbola we are on. (Here again we can do this in any frame we like, since the hyperbolas are left invariant by a change of frame.)

 

[TrickyDicky]

Not sure if this has been mentioned already earlier in the thread, but the issue of topology arise already in special relativity. Zeeman observed that the ordinary topology on R^4 has no "physical significance" from SR point of view. He thus introduced so-called "fine topology". Later on, Hawking, King and McCarthy introduced another useful topology called the "path topology", which had the same homeomorphism group as Zeeman's fine topology, which is essentially the Lorentz group.

Appendix A of the book "The Geometry of Minkowski Spacetime" by Naber has a good discussion about this. Also see http://en.wikipedia.org/wiki/Spacetime_topology.

Yes we have been discussing Minkowski spacetime too, I mentioned that the Alexandrov topology plus a strong causal condition fulfilled by Minkowski spacetime coincides with the smooth manifold topology.

Do you know the difference between Zeeman and path topologies, (the other two spacetime topologies mentioned in the wiki link)?

 

[Ben Niehoff]

Second, the case where we look at convergence along the hyperbola, t^2 - x^2 = T^2. So we start at the point (0, T) and move along the hyperbola to ever larger values of x, and again we look at the slopes of the lines from the origin to each point. Here again we are looking at the convergence of x/t. It still converges to 1, as before, but now the rate of convergence depends on T, i.e., it depends on which hyperbola we are on. (Here again we can do this in any frame we like, since the hyperbolas are left invariant by a change of frame.)

Uniform convergence can only be defined when you have a notion of "closeness", as is explained on the Wiki page. A sufficient condition is a metric (mathematics definition; i.e., positive distance function).

The underlying Minkowski space has a topology that agrees with the topology induced by the standard Euclidean metric on R^2; hence that is the metric we should use. Thus we have a sequence of lines from the origin to a hyperbola, which asymptotically approach the asymptote of that hyperbola. However, no matter how "close" these two lines get near the origin, sufficiently far away from the origin, they are "far" from each other. Therefore this sequence converges only pointwise; not uniformly.

 

[Samshorn]

There is nothing mysterious about 1+1-dimensional Minkowski space. Topologically, it is exactly the same thing as R^2, the Euclidean plane.
--
Most confusion in GR arises by imbuing coordinates with excessive meaning.

Is the irony intentional? Surely to claim that THE topology of Minkowski spacetime is R^2 is to conflate the coordinates with spacetime itself. This is to imbue coordinates with excessive meaning (ontological status). It's certainly true that the topology of the usual coordinate systems that we apply to spacetime is the same as the topology of R^2, but this doesn't imply that the topology (or, better, every physically meaningful topology) of spacetime is necessarily the same as the topology of R^2. See, for example, the topology underlying Penrose's twistor view of spacetime, in which null rays are points, etc.

 

[TrickyDicky]

Is the irony intentional? Surely to claim that THE topology of Minkowski spacetime is R^2 is to conflate the coordinates with spacetime itself. This is to imbue coordinates with excessive meaning (ontological status). It's certainly true that the topology of the usual coordinate systems that we apply to spacetime is the same as the topology of R^2, but this doesn't imply that the topology (or, better, every physically meaningful topology) of spacetime is necessarily the same as the topology of R^2.

The discussion was restricted to topologies that allow Lorentzian spacetimes to be considered smooth manifolds.
Of course there are many physically interesting spacetime topologies, problem is in most you can't even do calculus on them in a reliable way, and that is not a good thing.

 

[PeterDonis]

Thus we have a sequence of lines from the origin to a hyperbola, which asymptotically approach the asymptote of that hyperbola. However, no matter how "close" these two lines get near the origin, sufficiently far away from the origin, they are "far" from each other. Therefore this sequence converges only pointwise; not uniformly.

Is the bolded phrase another way of saying that the rate of convergence depends on T, i.e., on which hyperbola we are on? That seems to be the critical difference between pointwise and uniform convergence, at least from the Wiki pages.

 

[Ben Niehoff]

Is the bolded phrase another way of saying that the rate of convergence depends on T, i.e., on which hyperbola we are on?

No, we're staying on the same hyperbola. We're watching a sequence of lines converge to another line.

That seems to be the critical difference between pointwise and uniform convergence, at least from the Wiki pages.

That is an overly simplistic view.

 

[lavinia]

On the other hand the defining property of GR was explaining gravity thru curvature as an invariant, but Lorentzian manifolds, precisely due to their not being metric spaces, may be both flat and curved depending on what patch is chosen, in other words curvature is not a property of the manifold alone.

You are right that curvature is not a property of the manifold alone. But to say that the a Lorenzian manifold precisely because it is not a metric space can be flat or curved is wrong. It is easy for a metric space to be flat in some areas and curved in others.

Finally the assumption that the GR manifold is smooth seems to be contradicted by the existence of singularities, the condition usually imposed that one must only look at the space and time intervals that are singularity free doesn't seem a very rigorous mathematical prescription.

Here I don't know the Physics but why can't the metric be singular and not the manifold? I can imagine a Lorenzian space time where inside of a bounded region, the light cones converge to a point. This is a point on the smooth underlying manifold but Lorenzian metric is singular.

- Lorentz transformations are smooth and I think - you tell me - that they are supposed to described coordinate transformation between intertial frames of reference in Special Relativity and to closely - in fact as closely as you like in a small enough region- describe coordinate transformations between free float frames in General Relativity. This seems to imply that the manifold is smooth.

- It also seems to me naively that the topology of a space time can not be arbitrary. I say this only because the existence of light cones seems to imply that the manifold must have zero Euler characteristic.

 

[TrickyDicky]

You are right that curvature is not a property of the manifold alone. But to say that the a Lorenzian manifold precisely because it is not a metric space can be flat or curved is wrong. It is easy for a metric space to be flat in some areas and curved in others.

Yeah, this was clarified later in the thread.

Here I don't know the Physics but why can't the metric be singular and not the manifold? I can imagine a Lorenzian space time where inside of a bounded region, the light cones converge to a point. This is a point on the smooth underlying manifold but Lorenzian metric is singular.

My understanding is that the topology induced by the metric in a smooth manifold must coincide or be subordinate to the natural topology of the manifold, in that
sense you can't separate a metric singularity from a manifold singularity.

- Lorentz transformations are smooth and I think - you tell me - that they are supposed to described coordinate transformation between intertial frames of reference in Special Relativity and to closely - in fact as closely as you like in a small enough region- describe coordinate transformations between free float frames in General Relativity. This seems to imply that the manifold is smooth.

I agree, but this only assures local smoothness, and from the definition of differential manifold I deduce that global smoothness is required:
" In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space."
The existence of singularities seema naively to go against the globally defined smoothness, which would leave GR spactimes with singularities not qualifying as smooth manifolds, just as manifolds. But I haven't gotten any mathematician to either refute or confirm this.

- It also seems to me naively that the topology of a space time can not be arbitrary. I say this only because the existence of light cones seems to imply that the manifold must have zero Euler characteristic.

In a way, this naive idea is what induced me to start this thread, I subsequently learned that as I commented above the topology is imposed by the smooth manifold structure. I still have problems understanding why many of the physical features of GR seem to be taken from the topology of its pseudosemimetric, even though that should be overridden by the manifold topology.