The Topology of Spacetimes: Exploring the Global Structure of Curved Manifolds

[kevinferreira]

Mod note: This thread contains an off-topic discussion from the thread https://www.physicsforums.com/showthread.php?p=4216768

The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically...

So a notion of distance is used... I wonder how.

 

[dextercioby]

But that's troubling, as in the wikipedia article of the exponential map it is said explicitly

<The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically...>

So a notion of distance is used... I wonder how.

The manifold is riemannian, it has a metric tensor which creates a metric from the topological point of view. See http://en.wikipedia.org/wiki/Riemannian_manifold particularly this part <The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product.> The inner product (which creates a norm, hence a metric topology) is induced by the geometrical metric (tensor).

 

[WannabeNewton]

So a notion of distance is used... I wonder how.

The riemannian metric / metric tensor defines an inner product on Tp(M). An inner product induces a norm.
EDIT: dextercioby beat me to it while I was typing =D

 

[kevinferreira]

Of course, thank you both.
But then, aha, the tangent space is indeed a topological vector space with a topology induced by the notion of distance induced by the norm induced by the inner product, itself defined by the metric of the Riemannian manifold!

 

[Naty1]

why I do not regret leaving mathematical interpretations to others! {LOL}

But then, aha, the tangent space is indeed a topological vector space with a topology induced by the notion of distance induced by the norm induced by the inner product, itself defined by the metric of the Riemannian manifold!

 

[pervect]

Of course, thank you both.
But then, aha, the tangent space is indeed a topological vector space

I'd agree - at least off the top of my head.

with a topology induced by the notion of distance induced by the norm induced by the inner product, itself defined by the metric of the Riemannian manifold!

No, you need to define the topological "neighborhood" (open balls) by something that's always greater than zero, like x^2 + y^2 + z^2 + t^2, not x^2 + y^2 + x^2 - t^2, which would be the inner product.

But I don't think there's anything that prevents you from doing that if you want.

 

[kevinferreira]

No, you need to define the topological "neighborhood" (open balls) by something that's always greater than zero, like x^2 + y^2 + z^2 + t^2, not x^2 + y^2 + x^2 - t^2, which would be the inner product.

But I don't think there's anything that prevents you from doing that if you want.

Hmm, indeed. It is true that for two vectors $X,Y\in T_p(M)$ we may have $g(X,Y)≤0$, so that distance is not positively defined, and therefore it isn't really a metric space. This is very problematic. Maybe we can just take the absolute value of
$g(X,Y)$, but then we have to check that the triangular inequality is still true.
Hmm, $g$ doesn't even define a distance in M, as M is pseudo-Riemannian, the distance is usually defined as
$$\int_a^b ds=\int_a^b\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$$
and this is always positive.

 

[Dale]

Hmm, indeed. It is true that for two vectors $X,Y\in T_p(M)$ we may have $g(X,Y)≤0$, so that distance is not positively defined, and therefore it isn't really a metric space. This is very problematic.

I think that a pseudo-Reimannian manifold is probably not a metric space. Because of path dependency issues there is probably not a unique measure of distance between two points in every manifold. E.g. some manifolds may have two events where there are two geodesics with different lengths that connect them.

In any case, even if that can be overcome the topology of a pseudo-Riemannian manifold is inherited from the underlying manifold, not from the metric nor from the inner product of vectors in the tangent spaces.

 

[George Jones]

Hmm, indeed. It is true that for two vectors $X,Y\in T_p(M)$ we may have $g(X,Y)≤0$, so that distance is not positively defined, and therefore it isn't really a metric space.

There isn't any problem.

But I don't think there's anything that prevents you from doing that if you want.

Take any basis for $T_p(M)$. This basis can contain any combination of timelight, lightlike, and spacelike vectors. Use this basis to set up a bijection between $T_p(M)$ and $\mathbb{R}^4$ in the standard way. Call a subset of $T_p(M)$ open if the corresponding subset of $\mathbb{R}^4$ is open in its standard topology. This turns $T_p(M)$ into a topological vector space that is homeomorphic to $\mathbb{R}^4$. The toploogy (class of open sets) of $T_p(M)$ arrived at in this way is independent of the original basis used.

This is equivalent to introducing a (positive-definite) norm on $T_p(M)$.

 

[kevinferreira]

I think that a pseudo-Reimannian manifold is probably not a metric space. Because of path dependency issues there is probably not a unique measure of distance between two points in every manifold. E.g. some manifolds may have two events where there are two geodesics with different lengths that connect them.

In any case, even if that can be overcome the topology of a pseudo-Riemannian manifold is inherited from the underlying manifold, not from the metric nor from the inner product of vectors in the tangent spaces.

Even if there isn't a unique way of measuring path lengths, it's enough to have one with distance properties in order to have a metric space, I think.
Yes, that is true, a manifold is a topological space before any other thing you may define on it. Can that be used on the tangent bundle?

 

[micromass]

I think that a pseudo-Reimannian manifold is probably not a metric space.

Every manifold is a metric space by Whitney's embedding theorem.

 

[dextercioby]

It's high time I threw in the reference for the one interested

Naber G.L. - The geometry of Minkowski spacetime (Springer, 1992)(271p)

 

[pervect]

Hmm, indeed. It is true that for two vectors $X,Y\in T_p(M)$ we may have $g(X,Y)≤0$, so that distance is not positively defined, and therefore it isn't really a metric space.

Maybe my point was too simple. We know that a manifold is a topological space. But we also know that the Minkowskii notion of distance isn't always positive. So it's not suitiable for defining "open balls".

So we ask - do we actually use the Minkowskii notion of distance to define the topology of our 4-d space time? (Forget about the tangent space, for the moment, I'm talking about how space-time is a 4-d manifold).

For instance, if we are now observing a distant event in the andromeda galaxy, and it's Lorentz interval is zero, does that mean it's close to us, in our neighborhood?

THe answer is no, and no.

 

[TrickyDicky]

There isn't any problem.


Take any basis for $T_p(M)$. This basis can contain any combination of timelight, lightlike, and spacelike vectors. Use this basis to set up a bijection between $T_p(M)$ and $\mathbb{R}^4$ in the standard way. Call a subset of $T_p(M)$ open if the corresponding subset of $\mathbb{R}^4$ is open in its standard topology. This turns $T_p(M)$ into a topological vector space that is homeomorphic to $\mathbb{R}^4$. The toploogy (class of open sets) of $T_p(M)$ arrived at in this way is independent of the original basis used.

This is equivalent to introducing a (positive-definite) norm on $T_p(M)$.

Yes, what I always have a hard time understanding is that if both the topology and the distance function is the same in a pseudo-Riemannian manifold as in any Riemannian manifold as pervect and micromass also point out(metric space per Whitney theorem, R^4 topology, etc), then what is the deal with the different kind of vectors (timelike, lightlike, spacelike) of the Lorentzian metric tensor, how can they give rise to so much physics if mathematically it is all equivalent to using a positive-definite inner product wrt the integrated distance function and the topology?
IOW, if the Lorentzian metric tensor only has a local significance, why are its pseudometric features extended to determine the global features of the manifold?

 

[Dale]

Every manifold is a metric space by Whitney's embedding theorem.

OK, I guess they must just use the length of the shortest path, regardless of whether or not there are multiple geodesics.

 

[WannabeNewton]

OK, I guess they must just use the length of the shortest path, regardless of whether or not there are multiple geodesics.

Any topological manifold is metrizable. As the requirement is a topological manifold, this is done before a riemannian or pseudo riemannian metric is even equipped to the manifold.

 

[kevinferreira]

Yes, what I always have a hard time understanding is that if both the topology and the distance function is the same in a pseudo-Riemannian manifold as in any Riemannian manifold as pervect and micromass also point out(metric space per Whitney theorem, R^4 topology, etc), then what is the deal with the different kind of vectors (timelike, lightlike, spacelike) of the Lorentzian metric tensor, how can they give rise to so much physics

Topologies and notions of neighbourhood do not determine uniquely all the properties we want spacetime to have. The notion of manifold is introduced in order to have local coordinates, and we make furthermore the assumption that it is a Riemannian manifold, i.e. that a line element on the manifold is given by
$$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$
It is the form of $g_{\mu\nu}$ that will therefore determine all the odd properties of the manifold that we're used to see as nice in euclidean space.

if mathematically it is all equivalent to using a positive-definite inner product wrt the integrated distance function and the topology?
IOW, if the Lorentzian metric tensor only has a local significance, why are its pseudometric features extended to determine the global features of the manifold?

I don't think it is 'equivalent', the metric tensor defines a inner product on each tangent space, while the integrated distance function is defined on the manifold itself. You may define a line on your manifold with only one vector, for example by
$$\frac{d\gamma}{ds}(0) = X$$
and therefore the path $\gamma(s)$ will give you a line 'along' X, on the manifold.
The metric tensor only has local significance, but then you can imagine that its local features compose the global properties as seen locally, so that in the end the whole ensemble of local features will 'add up' to form the manifold. You can find a lot of these things in mathematics, e.g. the Cantor set, which is just a set defined by very simple rules on euclidean space with the usual distance, but then in the end you get an ensemble which has completely different properties and very strange ones indeed.
You may also think about Minkowski space (flat spacetime), and about how you may define this light cone lat an event, and this is not just a drawing on paper, space itself gets different properties on different regions. You may want to view this as only locally defined, but there's nothing that contradicts the fact that you can extend this (relational) properties to the whole space.

 

[Dale]

Any topological manifold is metrizable. As the requirement is a topological manifold, this is done before a riemannian or pseudo riemannian metric is even equipped to the manifold.

I don't see how that works if you have a manifold without a metric. A metric space needs to have a unique notion of distance defined; how can you do that without defining a metric and making it a (pseudo-) Riemannian manifold?

Distances on a coffee cup and a donut are different even though they are the same topologically. Do you have an explanation how that works?

 

[micromass]

I don't see how that works if you have a manifold without a metric. A metric space needs to have a unique notion of distance defined; how can you do that without defining a metric and making it a (pseudo-) Riemannian manifold?

Distances on a coffee cup and a donut are different even though they are the same topologically. Do you have an explanation how that works?

Metrizable means that there exists a metric that makes it into a metric space. It doesn't mean that this metric is unique. Your example of coffee cups and donuts indeed gives you two different possible metrics.

A topological manifold is certainly metrizable, but this doesn't mean a unique metric. Nor does this imply that your metric agrees with the metric tensor you might define later.

 

[WannabeNewton]

I don't see how that works if you have a manifold without a metric. A metric space needs to have a unique notion of distance defined; how can you do that without defining a metric and making it a (pseudo-) Riemannian manifold?

Distances on a coffee cup and a donut are different even though they are the same topologically. Do you have an explanation how that works?

Indeed even though the two topological spaces mentioned are homeomorphic, they need not have same distance functions. Metrizable implies there exists some metric for the set but it doesn't state there is a single, unique metric. By the way, I think there is some confusion arising here in the terminology. Metric here is not the same thing as the metric tensor. The metric tensor is what gives rise to notions like riemannian or pseudo riemannian. Of course if you give your topological manifold a smooth structure then you can always endow it with some riemannian metric.

 

[zonde]

I think that a pseudo-Reimannian manifold is probably not a metric space.

Every manifold is a metric space by Whitney's embedding theorem.

This does not seem right.

Whitney embedding theorem:
"The strong Whitney embedding theorem states that any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean 2m-space, if m>0."
Hausdorff space:
"Pseudometric spaces typically are not Hausdorff"

So I suppose that pseudo-Reimannian manifold is not necessarily Hausdorff.

 

[WannabeNewton]

So I suppose that pseudo-Reimannian manifold is not necessarily Hausdorff.

pseudo - Riemannian manifold has a pseudo - riemannian metric. This is not the same thing as a pseudometric.

 

[zonde]

pseudo - Riemannian manifold has a pseudo - riemannian metric. This is not the same thing as a pseudometric.

To me it seems the same.

Still using wikipedia:
"pseudometric space is a generalized metric space in which the distance between two distinct points can be zero"
and
"The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite."
i.e. it can give zero distance between two distinct points.

 

[WannabeNewton]

To me it seems the same.

Still using wikipedia:
"pseudometric space is a generalized metric space in which the distance between two distinct points can be zero"
and
"The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite."
i.e. it can give zero distance between two distinct points.

Yes but a metric and a riemannian metric are not the same thing; just because a specific property is shared amongst them doesn't make them the same thing. They are different constructions, pseudo or not. By your logic, a riemannian metric would determine the topological property of being Hausdorff when such a property is determined before a riemannian structure is even endowed -> a metric space is not the same thing as a riemannian manifold.

 

[micromass]

To me it seems the same.

Still using wikipedia:
"pseudometric space is a generalized metric space in which the distance between two distinct points can be zero"
and
"The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite."
i.e. it can give zero distance between two distinct points.

A pseudo-Riemannian metric is a function

$$g:T_pM\times T_pM\rightarrow \mathbb{R}$$

for each p.

A pseudometric is a function

$$d:M\times M\rightarrow \mathbb{R}$$

So how can they be the same thing??

 

[zonde]

Yes but a metric and a riemannian metric are not the same thing

Please explain.

 

[micromass]

Please explain.

Like I said in my post: the domains of the two functions don't even agree.

 

[micromass]

Check http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf
Page 6:

Spacetime is a connected, Hausdorff, differentiable pseudo-Riemannian manifold of dimension 4 whose points are called events.

So all spaces are taken to be Hausdorff.

The reason why they specify Hausdorff in this case is because they don't define manifold to be Hausdorff (which is not standard practice). But in any case, this shows that all spacetimes are taken to be Hausdorff.

 

[WannabeNewton]

Please explain.

For example consider the set of all bounded sequences of real (or complex) numbers together with the metric $d(x,y) = sup_{i\in \mathbb{N}}|x_{i} - y_{i}|$; this is the metric space $l^{\infty }$. d is a map $d:l^{\infty } \times l^{\infty } \rightarrow \mathbb{R}$. This space isn't even 2nd countable so how do you expect to somehow equate this metric with a riemannian metric which requires a topological manifold endowed with a smooth structure?

 

[zonde]

Like I said in my post: the domains of the two functions don't even agree.

But result for both functions is distance, right?

 

[micromass]

But result for both functions is distance, right?

The pseudo-Riemannian metric induces a distance on the tangent space $T_p(M)$, not on M. I don't think it induces a well-defined distance on the manifold. And even if it does, nobody says that the topology of this distance should with the topology of the manifold.

 

[zonde]

Check http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf
Page 6:

So all spaces are taken to be Hausdorff.

The reason why they specify Hausdorff in this case is because they don't define manifold to be Hausdorff (which is not standard practice). But in any case, this shows that all spacetimes are taken to be Hausdorff.

Fine there is someone else who thinks like you.
Now can you provide arguments? In that link there is only definition (belief) and no arguments.

Why do you believe that all spacetimes are Hausdorff?

 

[micromass]

Fine there is someone else who thinks like you.
Now can you provide arguments? In that link there is only definition (belief) and no arguments.

Why do you believe that all spacetimes are Hausdorff?

Because it is the definition...
How do you define a spacetime??

Since you like wikipedia, here is another reference: http://en.wikipedia.org/wiki/Spacetime_topology

 

[WannabeNewton]

Why do you believe that all spacetimes are Hausdorff?

It is a definition. For example see pg 12 of Wald.

 

[zonde]

Because it is the definition...
How do you define a spacetime??

Since you like wikipedia, here is another reference: http://en.wikipedia.org/wiki/Spacetime_topology

Let me try such question:
How do you define neighborhood if you have distance function that can give zero for two distinct points?