Help me visualize this, boundary of the future

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SUMMARY

The discussion focuses on visualizing the boundary of the chronological future of a subset S within a spacetime (M, g_ab). It establishes that this boundary is an achronal, three-dimensional embedded manifold, as demonstrated through the relationship between points on the boundary and their future counterparts. The argument presented relies on the properties of null geodesics and the open nature of the past of points in spacetime. The reference to Proposition 6.3 and Definition 25 in the linked material further supports the claim that the boundary is not trivially defined.

PREREQUISITES
  • Understanding of spacetime concepts, specifically (M, g_ab)
  • Familiarity with null geodesics and their properties
  • Knowledge of manifold theory, particularly embedded manifolds
  • Basic grasp of chronological and achronal boundaries in relativity
NEXT STEPS
  • Study the properties of null geodesics in general relativity
  • Explore the definitions and implications of achronal sets in manifold theory
  • Review Proposition 6.3 and Definition 25 in the referenced material for deeper insights
  • Investigate examples of embedded manifolds in various spacetime scenarios
USEFUL FOR

This discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students of general relativity seeking to deepen their understanding of spacetime boundaries and manifold properties.

JasonJo
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Let S be any subset of M, where (M, g_ab) is a spacetime.

Can you guys help me kind of visualize why the boundary of the chronological future of S is an achronal, 3 dimensional embedded manifold?

I am just having a hard time seeing why this is so. I'm picturing a sphere, and then having all the null geodesics emanating from it. The null geodesics form the boundary of the chronological future. Why is this thing achronal? Why is it a submanifold?
 
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I can visualise it only for various flat space scenarios. I googled it and these guys give an argument which seems to make sense, and they also say it's not obvious (Proposition 6.3, and Definition 25): http://books.google.com/books?id=zJ5rPOpiKjYC&printsec=frontcover#PPA85,M1

I think it's roughly like this. Suppose x is on the boundary of the future of S, and suppose y is in the future of x. Then x is in the past of y. The past of y is open, so there must be some open region surrounding x that is in the past of y. Because x is the boundary of the future of S, every open region surrounding x contains a point z in the future of S. So y is the future of an open region surrounding x containing some point z in the future of S. So y, the future of x, is in the future of S. So we don't have to worry that future of the boundary is also on the boundary.
 

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