Deductions about extending a pseudo-riemannian manifold

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In summary: Thanks for the help.In summary, the conversation discusses the possibility of a pseudo-riemannian 4 manifold S, without boundary and not simply connected, being a regular submanifold of another pseudo-riemannian 4 manifold M that preserves its geometry and topology. The question is whether if a pair of spacelike geodesics in S do not intersect, but their extension in M does intersect at the boundary of S in M, then this is true for any M containing S as a regular submanifold. This is an issue in General Relativity and has been raised in a previous thread. The paper referenced in the conversation shows that this particular example can be extended in surprising ways with different topology. The answer to
  • #1
PAllen
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Suppose we have a pseudo-riemannian 4 manifold S (sometimes also called a Minkowskian manifold) that is without boundary, and not simply connected. Suppose there is at least one pseudo-riemannian 4 manifold M (also without boundary) that has S as a regular submanifold, preserving all geometry (pseudo-riemannian metric) and topology of S.

Suppose in S, there are a pair of spacelike geodesics that do not intersect in S, but their extension in M does intersect. Further, the intersection occurs at a boundary of S in M. I would like to conclude that if this is true for one M as described, then it is true for any M (any pseudo-riemannian 4-manifold, without boundary, containing S as a regular submanifold and preserving all of its geometry and topology).

A difficulty for pseudo-riemannian manifolds is that a geodesic of zero interval can still be topologically a line (a null geodesic). There can also be topological lines that are not geodesics that have invariant interval of zero (null paths). Thus, arguments about interval being zero or tending to zero don't tell you anything about intersection without additional reasoning.

Questions:

1) Is my hoped for conclusion true? If so, what is a strategy for establishing it?

The specific question I have concerns geometries of interest in General Relativity, where I posted the specific version of this question and got no response:

https://www.physicsforums.com/showthread.php?t=732569

And these wonderments of mine were triggered by the following paper which shows that my particular example of interest (SC exterior geometry) can be extended in some surprising ways, with very different topology than Kruskal (by which I mean the maximal analytic extension; the following paper uses it only as a style of coordinates):

http://arxiv.org/abs/0910.5194
 
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  • #2
In the other thread, I believe I have answered this.

The answer is yes, the hoped for property is true.

The method is simply that for an argument that would use balls in Riemannian geometry, for pseudo-Rieamannian geometry you need an argument in terms of causal diamonds: region bounded by a past light cone from P1 with a future light cone from P2 (P1 in causal future of P2).
 
  • #3
Analytic extensions are not unique. For example, consider the metric on some open region of a round sphere:

[tex]ds^2 = d\theta^2 + \sin \theta \; d\phi^2[/tex]
We can consider ##\theta \in (0,\pi/4)##, ##\phi \in [0,2\pi)## so that we're talking about a spherical cap. There are at least two distinct maximal extensions of this. We could extend to the usual 2-sphere, or we could extend to ##RP^2##. The metric tensor is the same.

However, in ##S^2##, geodesics through (0,0) intersect in two points, whereas in ##RP^2##, geodesics through (0,0) intersect in one point.

I'm not sure how this relates to your question, because it's unclear to me what you're trying to achieve.
 
  • #4
Ben Niehoff said:
Analytic extensions are not unique. For example, consider the metric on some open region of a round sphere:

[tex]ds^2 = d\theta^2 + \sin \theta \; d\phi^2[/tex]
We can consider ##\theta \in (0,\pi/4)##, ##\phi \in [0,2\pi)## so that we're talking about a spherical cap. There are at least two distinct maximal extensions of this. We could extend to the usual 2-sphere, or we could extend to ##RP^2##. The metric tensor is the same.

However, in ##S^2##, geodesics through (0,0) intersect in two points, whereas in ##RP^2##, geodesics through (0,0) intersect in one point.

I'm not sure how this relates to your question, because it's unclear to me what you're trying to achieve.

Unless I misunderstand you, this does not contradict what I'm proposing (and now think is true except for possible technical wording details) in the first two paragraphs of post #1 of this thread. Focus just on those (the rest is background and related questions).
 
  • #5


I cannot definitively say whether your hoped for conclusion is true without further analysis and evidence. However, based on the information provided, it seems plausible that it could be true.

One possible strategy for establishing this conclusion would be to use mathematical proofs and rigorous reasoning to show that the conditions described in the scenario are sufficient for the conclusion to hold true. This could involve examining the properties of pseudo-riemannian manifolds and their regular submanifolds, as well as the behavior of geodesics and their extensions.

Additionally, it may be helpful to consider other examples and counterexamples, and to explore the implications of the scenario in different contexts. Collaborating with other experts in the field and seeking feedback and input from peers could also be beneficial in further investigating and potentially proving this conclusion.

It is also important to keep in mind that there may be limitations and exceptions to this conclusion, and further research and analysis would be needed to fully understand and establish its validity. Overall, the key to establishing this conclusion would be to approach it with a scientific mindset and utilize critical thinking, evidence-based reasoning, and rigorous analysis.
 

1. What is a pseudo-Riemannian manifold?

A pseudo-Riemannian manifold is a mathematical object that describes the geometric properties of a space that has a combination of both positive and negative curvature. It is a generalization of a Riemannian manifold, which only has positive curvature.

2. How are deductions made about extending a pseudo-Riemannian manifold?

Deductions about extending a pseudo-Riemannian manifold are made by studying the geometric and topological properties of the manifold. This involves analyzing its curvature, connections, and other geometric invariants.

3. What is the significance of extending a pseudo-Riemannian manifold?

Extending a pseudo-Riemannian manifold allows us to better understand the geometry and topology of the space in question. It also has important applications in physics, particularly in general relativity and the study of spacetime.

4. Can all pseudo-Riemannian manifolds be extended?

No, not all pseudo-Riemannian manifolds can be extended. The extendability of a manifold depends on its geometric and topological properties, such as its curvature and dimension. Some manifolds may have singularities or other obstructions that prevent them from being extended.

5. Are there any open problems or unsolved questions related to extending pseudo-Riemannian manifolds?

Yes, there are still many open problems and unsolved questions related to extending pseudo-Riemannian manifolds. One such problem is the extendability of Lorentzian manifolds, which have a special type of pseudo-Riemannian metric and are used to model spacetime in general relativity. Another open question is whether all compact pseudo-Riemannian manifolds can be extended.

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