Relationship between apparent, event and Cauchy horizons - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:37:26Z http://mathoverflow.net/feeds/question/46486 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46486/relationship-between-apparent-event-and-cauchy-horizons Relationship between apparent, event and Cauchy horizons Anirbit 2010-11-18T14:53:15Z 2010-11-19T15:13:41Z <p>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. </p> <p>I would like to know a reference for or a proof of the following two concepts,</p> <ol> <li><p>For static/stationary space-times, the event horizon must equal the apparent horizon. </p></li> <li><p>For static space-times, the event horizon is where the static Killing field becomes null.</p></li> </ol> <p>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?)</p> <p>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. </p> <ol> <li><p>I would like to know what is the most precise statement one can make about the relationship between Cauchy horizons and event horizons. </p></li> <li><p>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. </p></li> </ol> <p>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.</p> <p>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. </p> <p>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? </p> <p>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?</p> <p>What is the most precise connection known between existence of a black-hole and global hyperbolicity of its interior and exterior? </p> <p>(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!) </p> http://mathoverflow.net/questions/46486/relationship-between-apparent-event-and-cauchy-horizons/46499#46499 Answer by drbobmeister for Relationship between apparent, event and Cauchy horizons drbobmeister 2010-11-18T16:32:04Z 2010-11-18T17:13:37Z <p>I would begin by checking out <em>The Large Scale Structure of Space-Time</em> by Hawking and Ellis. Note: I would probably have submitted this brief remark as a comment, but on my system "the software" won't allow comments until after at least one answer has been posted.</p>