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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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