Topology of closed timelike curves (CTC)

[zankaon]

For less than BH_h, deep in gravitational potential well, with very extreme curvature, might one have a future light cone tipping over sufficiently to become spacelike and then wrap around to join up (glued) to past light cone? This is like a closed timelike curve, which can not be shrunk to a point. "[URL So it would have a torus like topology; very different from our future light cone, which is finite and bounded in timelike sense, and hence not closed. So also topologically, any CTC would seem quite different from topology of C_R for greater than and less than BH_h.

also: 'The Kerr vacuum is unobjectionable and realistic (for black hole models) in the exterior regions, and unobjectionable but perhaps unrealistic (for black hole models) in the "shallow interior" regions, but as several commentators have mentioned, it is objectionable in the "deep interior" regions, since it there admits closed timelike curves (CTCs), as does the Goedel lambdadust. These CTCs are problematical.' C. Hillman 1-05-2007 'tipping'

 

[Chris Hillman]

For less than BH_h, deep in gravitational potential well, with very extreme curvature,

Attempted translation: "Deep inside the event horizon of a black hole.."

might one have a future light cone tipping over sufficiently to become spacelike and then wrap around to join up (glued) to past light cone?

No, the whole point of the timelike/null/spacelike classification of vectors is that it applies everywhere everywhen; no timelike vector can ever be "boosted/rotated" by a Lorentz motion (applied at the level of a tangent space) into a null or spacelike vector. It follows that you have misunderstood what Hawking and Ellis mean by "tipping over". Look more closely at their figures! For example, in their discussion of the Goedel lambdadust, the "light cones in the large" (the null surface generated by all the future null geodesics issuing from an event) have surprising global structure, but near the event of origin, they always look like forward light cones in str. The "tipping" refers to appearance in a particular coordinate chart only. (Think of how Mercator versus polyconic maps of the Earth distort the continents, etc.)

And in future, if you wish to attribute some quotation to me, please provide a link so that readers can not only verify the alleged attribution (I don't recall signing my name "C. Hillman" anywhere) but can also see the original context, which seems crucial if there is to be any chance of avoiding misunderstanding. If you don't know how to link to another website at PF, look for "URL Hyperlinking" in https://www.physicsforums.com/misc.php?do=bbcode . Similarly for "internal linking" to other PF posts.

Context: this is a subtle subject, and there is almost always context, and discussions of subtle issues like global structure can only be understood by those who have mastered elementary gtr, which requires mastering perhaps the most important idea in elementary str, the nature of light cones.

Also, Wikipedia articles can be created (and edited) by absolutely anyone at any time, and the one you linked to illustrates, to my mind, some of the many reasons why this is Not A Good Idea if you want to provide to the world a free on-line high quality encyclopedia.

 

[JesseM]

And in future, if you wish to attribute some quotation to me, please provide a link so that readers can not only verify the alleged attribution (I don't recall signing my name "C. Hillman" anywhere) but can also see the original context, which seems crucial if there is to be any chance of avoiding misunderstanding.

A quick search shows zankaon was quoting post #40 from the thread "Light cones tipping over"--C. Hillman was not part of the quote, so I assume it was just zankaon's attempt to attribute the quote rather than a signature that you yourself had written. But I agree that instead of just providing a name and date it's always better to provide a link if you're quoting a statement from a discussion forum.

 

[Chris Hillman]

Thanks, Jesse, I couldn't find it just now, but the quotation seems to be from my Post#40 from that thread, so indeed the context was complicated.

This has been an amazingly confusing thread, but I sense that at least some readers with less experience working with gtr might be clearing up some misconceptions from reading some of the comments by those with more experience, so forging ahead, I have some comments on points I haven't yet addressed ...

...a local coordinate chart on some region (homeomorphic to ordinary R^4) in a Lorentzian manifold is associated with four almost arbitrary monotonic functions; hence, in general, coordinates are arbitrary labels lacking any physical interpretation... geometric or coordinate-free interpretation...asymptotically flat regions...Golden Age of Relativity...Bondi radiation theory ... positive energy theorem... coordinate-free thinking...

...exact solutions with clear physical interpretations (including a clear understanding of the limits on their applications to realistic physical scenarios) include plane wave solutions, some null dust solutions such as the Vaidya null dust, many cosmological models such as the FRW models and various generalizations, colliding plane wave (CPW) models, etc. Then there are solutions which have clear interpretations in that it is clear what one is trying to describe, but which on closer inspection have physically objectionable features; these include Weyl vacuum solutions with "struts", the Van Stockum "rotating" cylindrically symmetric dust, Robinson-Trautman vacuums with "pipes", and so on...

The Kerr vacuum is unobjectionable and realistic (for black hole models) in the exterior regions, and unobjectionable but perhaps unrealistic (for black hole models) in the "shallow interior" regions, but as several commentators have mentioned, it is objectionable in the "deep interior" regions, since it there admits closed timelike curves (CTCs), as does the Goedel lambdadust. These CTCs are problematical...

...in terms of the intrinsic geometry of a spacetime model, infinitesimal light cones do not really becoming "sheared" (although they appear that way when we draw them in the Painleve chart), or "stretched temporally and squeezed radially" (although in the exterior region, they appear that way when we draw them in the exterior Schwarzschild chart), or "rescaled without changing shape" (although they appear that way when we draw them in the Kruskal chart, or other "conformal" charts).

So zankaon failed to quote the stuff I said right after the paragraph he quoted, the stuff which is actually relevant here!

But enough of that! zankaon, can you reformulate your question using the above?