# Relationship between apparent, event and Cauchy horizons

I would use the definition of an event horizon as being the boundary of the past of the future null infinity of a space-time, future/past Cauchy horizon of a closed achronal surface as the boundary of its future/past domain of dependence and apparent horizon as the outermost trapped surfaces.

I would like to know a reference for or a proof of the following two concepts,

1. For static/stationary space-times, the event horizon must equal the apparent horizon.

2. For static space-times, the event horizon is where the static Killing field becomes null.

In the maximally extended Reissner-Nordstrom black-hole space-time the inner horizon is a Cauchy horizon for the "t=0" space-like surfaces. ( Was there a way to see the above without doing the extension?)

I can't prove it but I think the outer horizon of a Reissner-Nordstrom black-hole space-time is not a Cauchy horizon for any closed achronal surface.

1. I would like to know what is the most precise statement one can make about the relationship between Cauchy horizons and event horizons.

2. Definition of a black hole as in Yvonne's book is the complement of the past of the set covered by the null geodesics which have an infinite future canonical parameter.

This definition doesn't seem to guarantee global hyperbolicity for either the outside or inside of a black hole and neither does it even demand time-orientability of the space-time nor does this want the space-time to have a "regular" Penrose compactification.

She needs to put in an extra definition of calling a space-time to be asymptotically strongly predictable" if the complement of the closure of the black hole region is globally hyperbolic.

Does the above criteria get automatically implied if one uses the definition of black hole as the complement of the past of the future null infinity for those space-times which have a regular Penrose compactifiaction?

Hence my question as to in how general a situation can one guarantee that the space-time in the complement of the black-hole region is globally hyperbolic?

What is the most precise connection known between existence of a black-hole and global hyperbolicity of its interior and exterior?

(Somehow I can't number the questions as 1,2,3,4 and the software insists on calling it 1,2 and again 1,2!)

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As drbobmeister indicated, the first two questions are in Large Scale Structure of Space-Time, section 9.3. Question 1 is answered by Proposition 9.3.1, Question 2 by 9.3.4. Question 4 is non-sensical. To even define a black-hole (event horizon), one generally needs asymptotic predictability, which requires existence of a partial Cauchy surface, so the DOC is globally hyperbolic by assumption. The region inside does not have to be not globally hyperbolic, as you can see in Schwarzschild, and many scalar-field collapse scenarios. – Willie Wong Nov 18 '10 at 19:03
See the meta discussion tea.mathoverflow.net/discussion/760/preview-of-numberperiod/… about ways of getting the questions numbered the way you want them. – José Figueroa-O'Farrill Nov 18 '10 at 19:51
@Willie Thanks for your reply. Can you expand "DOC" ? How does one prove that in the Schwarszchild case the region inside the event horizon is globally hyperbolic? Is there a characterization of such scenarios? (essentially my 4th question) – Anirbit Nov 18 '10 at 21:33
DOC = domain of outer communications = complement of black hole region. And what's there to prove about Schwarzschild? You draw the Penrose diagram, and it's trivial. For spherically symmetric space-times existence of Cauchy Horizon is equivalent to existence of boundaries of space-time where the area radius extends continuously to some finite value. For general space-times, there are no clear characterizations besides the definition. – Willie Wong Nov 18 '10 at 23:22
@Willie thanks for the clarification. Can you kindly give reference or explicitly state what is the definition of a black-hole that you are using? Now I have made my 4th question referenced and backed it up with motivations. I guess now it makes sense. – Anirbit Nov 19 '10 at 15:19