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

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In summary: This is equivalent to introducing a (positive-definite) norm on...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.
  • #106
micromass said:
Third, Singular points on a manifold are not a concept depending on the topology.
Well, since there seems to be no commonly accepted definition of the singularity concept (only of singular spacetime), it is at the very least hard to say. It might not depend on the topology but it might be incompatible with it, just the same way a manifold won't admit certain metrics incompatible with its manifold topology.
 
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  • #107
George Jones said:
Because the idea is so attractive, over the decades, there have been a number of attempts to define spacetime singularities as missing points or adjoined boundaries, but many (all?) have had various problems.

The work of Susan Scott and collaborators has possibly shown the most promise, but I know very little about this stuff. An interesting recent paper:

http://iopscience.iop.org/0264-9381/28/16/165003/

Unfortunately, at this link, the paper is behind a paywall, and I can't find it on the arXiv. I have access to it, but I haven't a chance to look at it yet.

Thanks, I'm trying to clarify things with a book called "The Analysis of Space-Time Singularities" by C. J. S. Clarke. So far the impression I get is that this is a bit of a mined field.
 
  • #108
TrickyDicky said:
micromass, this is just a technical note that I really don't think adds much to the main discussion and I have no interest whatsoever in arguing about it but just for the record and since you have insisted on it several times...
You have said repeatedly that the Hausdorff condition is already required for topological manifolds...
I didn't look back to see if he actually did (I doubt he did though) but yes it depends on the author. Some authors take Hausdorff as one of the conditions for a topological space to be a manifold and others don't. In the context of space - times if you want physically relevant ones you would probably include the condition.
 
  • #109
TrickyDicky said:
micromass, this is just a technical note that I really don't think adds much to the main discussion and I have no interest whatsoever in arguing about it but just for the record and since you have insisted on it several times...
You have said repeatedly that the Hausdorff condition is already required for topological manifolds, well I'm not the only one here that has pointed out that the reference text for this stuff, the 1973 book by Ellis and Hawking cites a few examples of topological manifolds that are not Hausdorff.
Besides, I've consulted several standard texts on differential geometry and they also name the Hausdorff condition only when defining smooth/differentiable manifolds.

It does depend on the author. I'm sure there are people who do things differently (although I would like to know which differential geometry texts you are talking about). But I think the standard definition is to require topological manifold to be Hausdorff. My posts try to reflect the standard position as much as possible. But yes, there are probably some authors who do things differently.
 
  • #110
TrickyDicky said:
My confusion comes from not seeing how an structure that is supposed to act only locally can have global effects.
This is my original cause of confusion that began the discussion, still in the other thread. This happens for the metric tensor, the energy-momentum tensor, the Riemann tensor, etc.
 
  • #111
kevinferreira said:
This is my original cause of confusion that began the discussion, still in the other thread. This happens for the metric tensor, the energy-momentum tensor, the Riemann tensor, etc.

I think this the "global" in the other thread is different from the "global" in this thread. There you were wondering about the locality in the EP, and the non-locality in things like the Riemann tensor for which the EP fails. Both of those are "local" relative to the issues in this thread. For example, one can allow non-Hausdorff manifolds on which to place solutions of the Einstein equations. Or decide before you solve the equations that the manifold has the topology of a torus. So there are now at least 3 "levels" of "locality" - local for the EP (roughly less than first derivatives), non-local for the EP (roughly spacetime curvature or second derivatives or higher), and manifold topology (global). You cannot choose the manifold completely arbitrarily - for example - Ben Niehoff some time ago commented on PF that pseudo-Riemannian metrics cannot be placed on even-dimensional spheres (I think, I can't quite remember what the result was).
 
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<h2>1. What is topology?</h2><p>Topology is a branch of mathematics that studies the properties of geometric shapes and spaces that are preserved under continuous deformations. It is concerned with the study of the properties of a space that remain unchanged when the space is stretched, twisted, or bent, but not torn or glued together.</p><h2>2. How does topology relate to spacetimes?</h2><p>In the context of spacetimes, topology is used to describe the global structure of curved manifolds. It helps us understand the overall shape and connectivity of the universe and how it is affected by the presence of matter and energy.</p><h2>3. What is a curved manifold?</h2><p>A curved manifold is a mathematical space that is curved in a non-Euclidean way. In other words, it is a space that does not follow the rules of traditional Euclidean geometry, where parallel lines never meet and the angles of a triangle always add up to 180 degrees. Instead, a curved manifold can have non-parallel lines that intersect and the angles of a triangle can add up to more or less than 180 degrees.</p><h2>4. How do scientists explore the global structure of curved manifolds?</h2><p>Scientists use various mathematical tools and techniques to study the global structure of curved manifolds. This includes differential geometry, which deals with the study of curved spaces, and topology, which helps us understand the overall shape and connectivity of the universe. They also use computer simulations and observations from astronomical data to test and refine their theories.</p><h2>5. What are the implications of understanding the topology of spacetimes?</h2><p>Understanding the topology of spacetimes has significant implications for our understanding of the universe and its evolution. It can help us make predictions about the behavior of matter and energy in the universe, and it can also provide insights into the fundamental laws of physics. Additionally, it can aid in the development of new technologies, such as space travel and communication systems.</p>

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric shapes and spaces that are preserved under continuous deformations. It is concerned with the study of the properties of a space that remain unchanged when the space is stretched, twisted, or bent, but not torn or glued together.

2. How does topology relate to spacetimes?

In the context of spacetimes, topology is used to describe the global structure of curved manifolds. It helps us understand the overall shape and connectivity of the universe and how it is affected by the presence of matter and energy.

3. What is a curved manifold?

A curved manifold is a mathematical space that is curved in a non-Euclidean way. In other words, it is a space that does not follow the rules of traditional Euclidean geometry, where parallel lines never meet and the angles of a triangle always add up to 180 degrees. Instead, a curved manifold can have non-parallel lines that intersect and the angles of a triangle can add up to more or less than 180 degrees.

4. How do scientists explore the global structure of curved manifolds?

Scientists use various mathematical tools and techniques to study the global structure of curved manifolds. This includes differential geometry, which deals with the study of curved spaces, and topology, which helps us understand the overall shape and connectivity of the universe. They also use computer simulations and observations from astronomical data to test and refine their theories.

5. What are the implications of understanding the topology of spacetimes?

Understanding the topology of spacetimes has significant implications for our understanding of the universe and its evolution. It can help us make predictions about the behavior of matter and energy in the universe, and it can also provide insights into the fundamental laws of physics. Additionally, it can aid in the development of new technologies, such as space travel and communication systems.

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