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I am using these lectures by Rodnianksi and Dafermos as the reference for this question.

In third point in the list on the top of page 19 they emphasize the importance of completeness of the future null infinity in being able to define a black hole through a Penrose diagram. I guess by completeness they mean geodesic completeness.

  • What is the issue that is being hinted at?

In the statement just below the second diagram on page 21 they seem to be able to read off that the part of the future null infinity that is intersected by the maximal Cauchy development of their chosen Cauchy surface is incomplete.

  • I could not understand how this is obvious.

This observation is what they are using to motivate Christodoulou's thinking of incompleteness of future null infinity as a defining criteria of naked singularity.

  • How does this relate to the other viewpoint of Christodoulou that naked singularity is characterized by the non-compactness of the intersection of the past of the future null infinity with the Cauchy surface? (This paper for reference)
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  • $\begingroup$ Where is Willie Wong when you need him? $\endgroup$
    – drbobmeister
    Jan 9, 2011 at 21:24
  • $\begingroup$ Off on vacation. :) There is something strange about your third question: I don't think you are reading Christodoulou right. Take Minkowski space. The past of future null infinity is the whole space, so its intersection with any Cauchy surface is non-compact. In fact what you wrote has more to do with asymptotic flatness and less with naked singularity... $\endgroup$
    – Willie Wong
    Jan 13, 2011 at 15:27
  • $\begingroup$ @Willie Great to see you back from vacation! :) In my third question I was comparing the statements made just below the 2nd diagram on page 21 of the Dafermos-Rodnianski review and the italicized statement at the end of page 13 on Christodoulou's review. Both seem to be trying to characterize the weak censorship conjecture but the arguments are not obviously equivalent to me. Hence seeking help. $\endgroup$
    – Anirbit
    Jan 14, 2011 at 12:51
  • $\begingroup$ Dear Scholars, I have a question as follows : Let M be an asymptotically flat globally hyperbolic spacetime, and let M* denotes its conformal completion. Consider the set A which is the intersection of future null infinity and closure of causal future of a large sphere S, closure being taken with respect to M*.( assume that intersection is non-empty). By definition, we usually assume that future null infinity is closed and connected. Is it possible that under some condition(s), this set A is open also ? $\endgroup$
    – user24004
    May 27, 2012 at 12:44
  • $\begingroup$ I think it would be best if you created a separate question with what you wrote above. (Click on the "Ask Question" button at the top of the page.) $\endgroup$
    – Igor Khavkine
    May 27, 2012 at 22:48

2 Answers 2

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It will be helpful if you look up the definition of a singularity in spacetime in a standard textbook on GR, Hawking/Ellis "The Large Scale Structure of Spacetime" comes to mind.

About your first question: The important point is that the definition of a singularity is non-trivial, it cannot simply be defined as a point on a Lorentzian manifold where some tensor diverges (despite the name "singularity", the reason for this is explained in most GR textbooks, the Hawking/Ellis book contains a particularly lucid discussion of the involved ideas). Instead, the defining property is that in the presence of a singularity, there are observers aka reference frames that do not live as long as the universe exists, they either end before the universe itself comes to an end (these are the ones who fall into the singularity) or did not exist when the universe was created (these are the ones escaping from the singularity).

The basic idea is therefore to define a spacetime without singularities to be a spacetime that is geodesically complete. Given a spacetime with a singularity, we can remove all geodesics from the spacetime that end up in the singularity and the ones that emerge from the singularity, and get a spacetime without singularities. The minimum region that we have to remove is then defined as the region comprising the singularity. Therefore the "black hole region" or "singularity" of a spacetime is defined to be the minimum subset one has to remove to get a "complete" spacetime in the sense that there are no geodesics that end or begin life prematurely.

Page 19 of your reference is a first step in making these ideas a little bit more precise, it is not a mathematical definition, but a heuristic motivation.

Penrose diagrams are a tool in this context, but not a necessary tool.

I don't understand your second question, page 21 seems to be an explanation of the above ideas using generic Penrose diagrams, this is simply a graphical explanation of what the authors have done before, the diagrams are supposed to show a spacetime with a singularity.

As for your third question, I cannot access the Christodoulou paper.

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  • $\begingroup$ @Tim Thanks for your answer. I am aware of the motivations for defining the singularity through geodesic completeness as you explained. But in the lectures above they seem to emphasize the need to have the future null infinity geodesically complete, which is a different thing altogether. Thats the point I am not clear about. $\endgroup$
    – Anirbit
    Jan 10, 2011 at 4:14
  • $\begingroup$ @Tim The point in page 21 that I can't see is how they are able to read off automatically from the diagram that the part of the future null infinity intersected by the Cauchy development of their chosen Cauchy surface is geodesically incomplete (in contrast to what they wanted to be satisfied for defining a black hole) Is there something trivial that I am missing here? $\endgroup$
    – Anirbit
    Jan 10, 2011 at 4:18
  • $\begingroup$ @Tim Can you point me to the exact reference in the Hawking and Ellis book that you are referring to? $\endgroup$
    – Anirbit
    Jan 10, 2011 at 4:18
  • $\begingroup$ Hi Anirbit, I was not thinking of any specific page of Hawking/Ellis, but about the discussion "how to define a singularity", which is in chapter 5. Maybe it would be helpful if you explain how the definition of your reference is different from the definition in Hawking/Ellis (I don't see an important difference). Dafermos/Rodnianski don't say that the future null infinity has to be geodesically complete on p.19. Actually they don't define what they mean with "complete" on the page, but mention one possibily: When the past is the complete spacetime (that is when there is no singularity). $\endgroup$
    – Tim van Beek
    Jan 10, 2011 at 6:09
  • $\begingroup$ To your second remark: It is not the future null infinity that is supposed to be geodesically complete, the very notion of this statement does not make any sense. The intersection of the Cauchy surface and future null infinity is empty in the first diagram on page 21, as it should be, the Cauchy surface exists in finite time, for a properly chosen reference frame. $\endgroup$
    – Tim van Beek
    Jan 10, 2011 at 6:14
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At least about the definition of "complete future null infinity" I found some answers on the 8th page of these lectures by Klainerman.

I would be glad to hear of some explanations about how the two definitions given on that page relate to each other and how are they related to the notion explained by Tim in the comments to his answer. (Tim is calling the future null infinity to be complete if the whole manifold is geodesically complete)

Also these seem closely related to the idea of calling a hypersurface as being "generated by complete null geodesics". I would like to know what this means and why this is often used as a condition for the event horizon to satisfy.

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  • $\begingroup$ Future null infinity is not "physically present" in the space-time: it is a formal boundary and you cannot really consider it a null hypersurface of your Lorentzian manifold. In the case of strongly asymptotically flat space-times, you can conformally compactify and take a portion of the conformal boundary to be future null infinity, and you can then look at its completeness. The second definition given by Klainerman in that survey article applies to a larger class of space-times. When your space-time is sufficiently nice that both can be applied, the definitions agree. $\endgroup$
    – Willie Wong
    Jan 13, 2011 at 15:31
  • $\begingroup$ The first definition on pg 8 of your linked article is essentially the definition of "generated by future-complete null geodesics". If you switch time orientation you get the definition for past-complete null geodesics. In general it is hard for a hypersurface to be generated by both future and past complete null geodesics, as it often will imply a null-splitting theorem arxiv.org/abs/math/9909158 (and so the space-time will be geometrically special). $\endgroup$
    – Willie Wong
    Jan 13, 2011 at 15:34
  • $\begingroup$ @Willie Thanks! So you are saying that a hypersurface being "generated by future complete null geodesics" is the same as it being "geodesically complete with respect to future directed null geodesics". (I wonder then why these two different terminologies exist!) So can you shed light on why a demand is made for the event horizon to be "generated by complete null geodesics" ? Does it follow from say the 3 conditions and definition of a black hole as given on the top of page 19 of the Dafermos-Rodnianski lecture? $\endgroup$
    – Anirbit
    Jan 14, 2011 at 13:26
  • $\begingroup$ @Willie Thought of adding that I have often seen lines of argument based on the fact that the boundary of causal/chronal future/past of a point is "generated by future/past directed null geodesics". In globally hyperbolic spaces an extension of this that I see getting used often is that the boundary of the chronal future/past of compact orientable spacelike submanifolds are also "generated by null geodesics" orthogonal to them (and some more about not containing any conjugate points) $\endgroup$
    – Anirbit
    Jan 14, 2011 at 14:35
  • $\begingroup$ @Willie I wonder how much of the above holds if one replaces points by compact or closed in the general case and just closed sets in the globally hyperbolic case. At least in the black hole area theorem of Hawking one seems to using the generation by conjugate point free null geodesic property for boundaries of causal future of closed sets though the initial theorem seemed to be proven only for some special kind of compact sets. The black hole area theorem hinges on the future directed null geodesic completeness of the event horizon. $\endgroup$
    – Anirbit
    Jan 14, 2011 at 14:43

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