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### Innerstructure of blackholes

Anna Borkowska

Faculty of Mathematics, Physics and Computer Science UMCS Lublin

Outline

- Extremelyshortintroduction
- Types of blackholes
- Singularity... whatisthat?
- Gravitationalcollapse
- PhysicalfieldsinsideSchwarzschildblack hole
- Interiors

Solutionis...

...themetricstructure of spacetime.

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 hairtheorem...

black hole solutions of general relativityequationsarecompletelycharacterized by onlythreeexternallyobservableparameters:

- mass M
- electric charge Q
- specificangularmomentum a

John Archibald Wheeler (ur. 1911)

Schwarzschildblack hole

- sphericallysymmetric, static, vacuum
- characterized by mass M
- twosingular regions: r = 0 → spacelikesingularityr = 2M → eventhorizon

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

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

- 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

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

- 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

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)

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

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 - 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

slowlycontracting (withretarded time) lightlikemildlysingularthree-cylinder

shrinks to form a strongspacelikesingularityatlate-time region

Interior of Kerrblack hole Interior of Kerr - Newman black hole

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

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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