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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

Innerstructure of blackholes

Anna Borkowska

Faculty of Mathematics, Physics and Computer Science UMCS Lublin


Outline
Outline

  • Extremelyshortintroduction

  • Types of blackholes

  • Singularity... whatisthat?

  • Gravitationalcollapse

  • PhysicalfieldsinsideSchwarzschildblack hole

  • Interiors


Who to begin with
Who to beginwith?

Gab = Tab

Rab – ½Rgab= Tab

Rab = Tab – ½Tgab

Albert Einstein (1879 - 1955)


Solution is
Solutionis...

...themetricstructure of spacetime.


Carter penrose diagrams
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


No hair theorem
No hairtheorem...

black hole solutions of general relativityequationsarecompletelycharacterized by onlythreeexternallyobservableparameters:

  • mass M

  • electric charge Q

  • specificangularmomentum a

John Archibald Wheeler (ur. 1911)


Schwarzschild black hole
Schwarzschildblack hole

  • sphericallysymmetric, static, vacuum

  • characterized by mass M

  • twosingular regions: r = 0 → spacelikesingularityr = 2M → eventhorizon


Reissner nordstr m black hole
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


Kerr black hole
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


Kerr newman black hole
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


What exactly is singularity
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...


Types of singularities
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


Cosmic censor conjecture
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)


What about the interior
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


Physical fields inside a schwarzschild black hole
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:


masslessscalar field - effectivepotential:

massless field with non-zero spin - effectivepotential:

  • * s = 1 – electromagneticwaves * s = 2 – gravitationalwaves


masslessscalarfields:

masslessfieldswith non-zero spin:

(radiativemodes: l ≥s)

perturbationsaredamped out: t → ∞, fixedr

perturbationsgrowinfinitely: fixed t, r→ 0

theboundary of the region, whereperturbationsaresmall:


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


Interior of reissner nordstr m black hole
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


Cauchyhorizon:

slowlycontracting (withretarded time) lightlikemildlysingularthree-cylinder

shrinks to form a strongspacelikesingularityatlate-time region


Interior of kerr black hole interior of kerr newman black hole
Interior of Kerrblack hole Interior of Kerr - Newman black hole

  • probably... similar to theReissner - Nordströmblack hole interior


Bibliography
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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