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New results for electromagneticquasinormal modes of black holesDenitsa Staicova, Plamen FizievSofia UniversityValencia, 22.03.2013


“Small, dark, and heavy: But is it a blackhole?”*●●●Theoretical BHs: Vacuum solutions of the Einstein equations with infinite curvatureand regions, causally disconnected from the rest of the Universe.Observational BHs: compact massive objects with mass more than the TOV limit(M>2-3M ⊙)What do we observe?– Accretion of matter– X-ray emission – Quiescence period– Gravitational lensing– Astrophysical jetsGRB 080916C', E iso=8.8x10 54 ergCredit: NASA/Swift/Stefan ImmlerWe cannot observe the BHs directly! We can only observe the movement of matterin vicinity of the compact mass.●Observational BHs seem everywhere but are they equivalent to the theoreticalBHs? For example, do GRBs need a BH?/*M.Visser, C. Barcelo , S. Liberati, S. Sonego PoS BHs,GRandStrings 2008:010,2008, arXiv:0902.0346v2/


How to see a BH?●●●●●Trough its spectrum!Gravitational waves – perfectly suited to expose a BH:– bigger luminosity– no interaction with the surrounding medium– no absorption– characteristic spectrumThe biggest problem – they have not yet been observed.Multi-messenger approachThe perturbation theory approach:a way to glimpse into thefundamental physics● Quasi-normal modes – observed also in full GR *Jaramillo et al. PRD 85, 084030 (2012), Rezzolla at al. PRL 104.221101 (2010)


Confluent Heun Functionsd 2dt H (z)+ [β+1α+ + γ+1] 2 z z−1dH ( z)dz+( μ z + νz−1 ) H ( z)=0●●●●●●Second Order ODE, 2 regular singularities: z=0,1 and oneirregular: infinityIn the domain z є [0,1], well-defined, converging series solutionFor z>1, no such solution (only asymptotic one), the only workingcode is implemented in MAPLEUnlike the case of the confluent hypergeometric function – noknown integral representationWorkarounds: continued fraction method, phase integral method,matched expansions etc.Problem: Branch cuts, Stokes and Anti-stokes lines?


The Teukolsky equations●The Teukolsky Angular Equation:Ψ=e i(ωt+m ϕ) S (θ) R(r)¿●where u=cos(θ). Singularities: ±π,∞●Angular regularity on the sphereThe Teukolsky Radial Equation:Unknowns: {ωn,m,l, En,m,l}●Singularities: r-,r+,∞ Both ODEs of the CHE type!!!


QNM s=-2, a=0The Regge-Wheeler vs. TREBoth equations of the type:●RWE:●TRE:


QNM s=-2, a=0The Regge-Wheeler vs. TREBoth equations of the type:●RWE:●TRE:The QNMs from both equations coincide with precision of at least 8 digits!Also coincide with well-known values.


P. Fiziev,D.S. PRD.84, 127502 (2011)What about the“mysterious” line?Important for the socalledAS mode!Maassen van denBrink (2000)Leung et al.(2003)


The EM QNMs of the KBH (s=-1)●In this case, one needs to solve a transcendental system oftwo equations featuring the CHE with unknowns: ω and A●●TRE (imposing BHBC):TAE (requiring regularity on the sphere):D. S. and Fiziev P., arXiv:1112.0310v1 [astro-ph.HE]


The results, a=0:We find 2 sets of modes: one of which coinciding with the QNMs.i((3+ϵ)/2−arg ω)What is the physics of the other? r=∣r∣e


a>0m=1, l=1,n=0..10In physical units:ω=0.8kHz-1.5kHz,t=0.53-4.2msFor a=0..0.999m=0,1, l=1,2


m=1, l=1,n=0..9Again the same“mysterious line”ε=0ε=0.15m=0, l=1,n=3m=0, l=1,n=4


So what about the second set?Could it be a branch cut?For z>1 there is one major BC – along the positive real axis(BC1) defined by the following equation:i((3+ϵ)/2−arg ω)r=∣r∣e– It can also be reached for ω – real and negative– It can be reached for ω-imaginary and ω – even, forexample for the AS modesThe additional line (BC2):Works by :●Maassen van den Brink (2000)●Casals, Ottewill (2012)●Yang et al. (2012)[0.099 + 4.596 i, 0.157 + 4.556 i]


What about the asymptotics?The TRE asymptotics:A ← =r −1+2i ω M e i ω r A → =r 1−2i ω M e −i ω rThe asymptotics is usedessentially to fixthe solution going into infinity.R=C ← R ← +C → R →New spectrum should correspondto different boundary conditions.But what?BH∞Credit: R.Fitzpatrick


The boundary conditions do matterQNM/??? spectraJets spectrumD.S. P. Fziev Astrophys Space Sci (2011) 332


Thank you!Supported by the Foundation "Theoretical and ComputationalPhysics and Astrophysics", by the Bulgarian National ScientificFund under contracts DO-1-872, DO-1-895, DO-02-136, and SofiaUniversity Scientific Fund, contract 185/26.04.2010.The Heun Project:http://tcpa.uni-sofia.bg/heun/home.html


Values of the parameters for TAE:


Values of the parameters for TRE:


Berti, E et al. Class.Quant.Grav. 26 (2009)


The Heun functions● Karl L. W. Heun 1889:d 2dt 2 H + [ γ z +δz−1 +ϵ] dwz−a dz +αβ z−qz( z−1)(z−a) H =0ε=α+β-γ-δ+1, 4 regular singularities, z=0,1,a,∞●Confluence: CHE, BHE, DHE, THE●●●Generalize: the hypergeometric function, the Lame function,Mathieu function, the spheroidal wave functionsNumerous applications!Because of regularity of the singular points, seriesexpansion everywhere.● Group of symmetries of order 192

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