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Inner structure of black holes

Inner structure of black holes. Anna Borkowska Faculty of Mathematics, Physics and Computer Science UMCS Lublin . Outline. Extremely short introduction Types of black holes Singularity ... what is that ? Gravitational collapse Physical fields inside Schwarzschild black hole

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Inner structure of black holes

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  1. Innerstructure of blackholes Anna Borkowska Faculty of Mathematics, Physics and Computer Science UMCS Lublin

  2. Outline • Extremelyshortintroduction • Types of blackholes • Singularity... whatisthat? • Gravitationalcollapse • PhysicalfieldsinsideSchwarzschildblack hole • Interiors

  3. Who to beginwith? Gab = Tab Rab – ½Rgab= Tab Rab = Tab – ½Tgab Albert Einstein (1879 - 1955)

  4. Solutionis... ...themetricstructure of spacetime.

  5. Carter - Penrosediagrams types of infinity: I+futuretimelikeinfinity: t → +∞ atfinite radius rI- past timelikeinfinity: t → –∞ atfinite radius rI0spacelikeinfinity: r → +∞ atfinite time t I +futurenullinfinity: t + r → +∞ atfinite time t –rI - past nullinfinity: t – r→ –∞atfinite time t +r two-dimensional diagram, thatallow to depictcausalrelationsbetweenpointsinspacetime, themetric of a diagram isconformallyequivalentto themetric of spacetime

  6. No hairtheorem... black hole solutions of general relativityequationsarecompletelycharacterized by onlythreeexternallyobservableparameters: • mass M • electric charge Q • specificangularmomentum a John Archibald Wheeler (ur. 1911)

  7. Schwarzschildblack hole • sphericallysymmetric, static, vacuum • characterized by mass M • twosingular regions: r = 0 → spacelikesingularityr = 2M → eventhorizon

  8. Reissner - Nordströmblack hole • sphericallysymmetric, static • characterized by mass M and electric charge Q • threesingular regions: r = 0 → timelikesingularity r+ = M + (M2 – Q2)½ → eventhorizon r– = M – (M2 – Q2)½ → inner (Cauchy) horizon

  9. Kerrblack hole • stationary, rotating, vacuum • characterized by mass M, specificangularmomentum a • threesingular regions: r = 0 → timelike ring singularity r+ = M + (M2 – a2)½ → eventhorizon r– = M – (M2 – a2)½ → inner (Cauchy) horizon

  10. Kerr - Newman black hole • stationary, rotating • characterized by mass M, specificangularmomentum a and charge Q • threesingular regions: r = 0 → timelike ring singularity r+ = M + (M2 – a2 – Q2)½ → eventhorizon r– = M – (M2 – a2 – Q2)½ → inner (Cauchy) horizon

  11. Whatexactlyissingularity? • ‘place’, wheresomepathologicalbehavior of thespacetimemetricoccurs • incompletness of particleorphotonworldlinesinspacetime thenotion of a ‘place’ is not definedwherethesingularityoccurs– undefinedmetricexcludesthe point fromthespacetimemanifold theBig Bang singularity of Robertson - Walker cosmologicalsolutionτ = 0 orSchwarzschildsingularityr = 0 are not incorporatedinspacetime...

  12. Types of singularities • spacelike – attimelikeinfinity, unavoidable (Schwarzschild) • timelike (null) – atspacelikeinfinity, avoidable (Reissner - Nordström, Kerr) • point – occursat a point of model coordinates • (Schwarzschild) • ring – occurs on a circularlinein model coordinates • (Kerr, Kerr - Newman) • strong – unboundeddeformationdue to tidalforces • (Schwarzschild, Kerr) • weak – finitedeformationdue to tidalforces • (Cauchyhorizon of Reissner - Nordström, Kerr) • static – homogeneouscollapsemodels • (Friedmann, Robertson, Walker) • oscillatory – inhomogeneouscollapsemodels • (Belinskii, Khalatnikov, Lifshitz ) • notnaked – hiddenwithineventhorizon, impossible to see • naked – visible for distantobservers

  13. CosmicCensorConjecture • theonlynakedsingularityintheUniverseisthe Big Bang singularity WEAK:A nakedsingulatitycannotevolvefrom a regularinitial state of the system under anyphysicallyreasonableassumptionsconcerningtheproperties of matter. STRONG: In the general casethesingularitiesproduced by gravitationalcollapsearespacelike so that no observercanseethemuntilhefallsintothem. Roger Penrose (ur. 1931)

  14. Whataboutthe interior? • evolutionary problem → exchange of temporal and spatialcoordinates what to do? • conditions on thesurface of a black hole→integrationin time of Einstein equations→structure of spacetimeinsidetheblack hole... what’sthe problem? • internalstructure of a black hole stronglydepends on theconditions on an eventhorizonintheinfinitefuture of an externalobserver • inapplicability of general relativity tospacetimefragments, wherethecurvatureapproaches Planck scales – existence of singularity

  15. Physicalfieldsinside a Schwarzschildblack hole • perturbationcreated by a test objectfallingonto a black hole (scalar, electromagnetic, gravitational) Whathappens to fieldslong time aftertheobjecthasfalleninto a black hole? evolvesaccording to Klein - Gordon equation: because of sphericalsymmetry of themetric, themodemay be introduced: harmonic time dependence: Regge - Wheeler equation:

  16. masslessscalar field - effectivepotential: massless field with non-zero spin - effectivepotential: • * s = 1 – electromagneticwaves * s = 2 – gravitationalwaves

  17. masslessscalarfields: masslessfieldswith non-zero spin: (radiativemodes: l ≥s) perturbationsaredamped out: t → ∞, fixedr perturbationsgrowinfinitely: fixed t, r→ 0 theboundary of the region, whereperturbationsaresmall:

  18. Whataboutnon-radiativeperturbationmultipoles (l < s)? electromagneticperturbations l = 0 → electric charge gravitationalperturbations l = 0 → mass l = 1 → angularmomentum perturbations do not damp out: t → ∞, fixedr perturbationsgrowinfinitely: fixed t, r→ 0 ...metricchangesintoKerrorReissner - Nordström! • Whataboutperturbationsproducedinsideeventhorizon? • → propagationin a small region, ram intothesingularity

  19. Interior of Reissner - Nordströmblack hole • externalperturbationsgrowinfinitely near r-,1 • hypersurface r-,1 – infiniteblueshift • enormousconcentration of energy →changeinspacetimestructure → scalarmild (weak) singularity • stargatemay not be totallyclosed • mass inflation m(v,r) ~ v-peκv • horizon r-,2 – stablewithrespect to smallperturbationsoutsidetheblack hole

  20. Cauchyhorizon: slowlycontracting (withretarded time) lightlikemildlysingularthree-cylinder shrinks to form a strongspacelikesingularityatlate-time region

  21. Interior of Kerrblack hole Interior of Kerr - Newman black hole • probably... similar to theReissner - Nordströmblack hole interior

  22. Bibliography • R. M. Wald „General relativity”. TheUniversity of Chicago Press, Chicago 1984. • V. P. Frolov, I. D. Novikov „Black Hole Physics: Basic Concepts and New Developments”. KluwerAcademicPublishers, Dordrecht 1998. • C. Misner, K. Thorne, J. Wheeler „Gravitation”. W. H. Freeman & Company, San Francisco 1973. • A. Ori; Gen. Rel. Grav.7, 881-929 (1997). • R. A. Matzner, N. Zamorano; Phys. Rev. D 19, 2821-2826 (1979). • E. Poisson, W. Israel; Phys. Rev. Lett.63, 1663-1666 (1989). • E. Poisson, W. Israel; Phys. Rev. D41, 1796-1810 (1990). • A. Bonnano, S. Droz, W. Israel, S. M. Morsink; Phys. Rev. D50, 7372-7375 (1994). • S. Hod, T. Piran; Gen. Rel. Grav.30, 1555-1562 (1998). Thankyou for yourattention

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