Some doubts concerning the mathematical bases of GR

[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.