1 / 26

# Inner structure of black holes - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Inner structure of black holes

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

### Who to beginwith?

Gab = Tab

Rab – ½Rgab= Tab

Rab = Tab – ½Tgab

Albert Einstein (1879 - 1955)

### 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 casethesingularitiesproducedby gravitationalcollapsearespacelike so thatno observercanseethemuntilhefallsintothem.

Roger Penrose (ur. 1931)

• 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:

perturbationsaredamped out: t → ∞, fixedr

perturbationsgrowinfinitely: fixed t, r→ 0

theboundary of the region, whereperturbationsaresmall:

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!

• → 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

Cauchyhorizon:

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