quasinormal modes of a blackhole with quadrupole moment · a.allahyari, h.firouzjahi and...
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Alireza AllahyariSchool of Astronomy
Quasinormal modes ofa blackhole with quadrupole
moment
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AstronomyBiological SciencesCognitive SciencesComputer ScienceMathematicsNano ScienceParticles and AcceleratorsPhilosophyPhysics
Iranian National Observatory Brain Engineering CenterCondensed Matter NationalGrid Computing GroupIranian Light Source Facility
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Outline
• Blackholes
• Axisymmetric metrics
𝛿 −metric
Hartle-Thorne metric
• Light ring method
• Quasinormal modes
SQ-metric
Hartle-Thorne metric
• Future work
• Conclusion
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Blackholes
• 1905 special relativity was postulated.
• 1907 Minkowski wrote the theory using a spacetime metric.
• 1907-1915
• Equivalence principle: locally gravitational field can be removed
• Locally special relativity applies.
• Singularities appear in the solutions.
Hawking- Penrose singularity theorem showed singularities are unavoidable if some form of energy condition holds
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Blackholes
Shadow of a blackhole
www.uanews.arizona.edu
• Observational signature of black holes• detection of gravitational waves from mergingblack holes
• motions of stars orbiting a blackhole
• emission of vast amounts of electromagnetic radiation (X-rays)
• lensing
• shadow of a blackhole
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Blackholes
physics.aps.org www.ligo.org/
Merging blackholes undergo different phases of evolution
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Blackholes
• No hair theorem: mass, charge, angular momentum.
• Cosmic censorship: singularities should be covered by event horizons.
• It is believed that other moments are radiated away (Price).
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Blackholes
• Our motivation is why the mass estimation by LIGO-VIRGO observations confirm large mass blackholes.
• It could be that the observed state is not a blackhole in the usual sense.
• We consider generalized blackholes.
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Axisymmetric metrics
• Axisymmetric metrics can be written in Weyl canonical line element as
• Axis is regular
• Prolate spheroidal coordinates
• The solutions are not unique.
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Axisymmetric metrics\ 𝛿 -metric
• The solution can be given by
• In spherical coordinates we have
• 𝛿 -metric is (Bach and Weyl (1922) )
D. Papadopoulos, B. Stewart and L. Witten, “Some Properties of a Particular Static, Axially Symmetric Space-time,”
• Schwarzschild
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Axisymmetric metrics\ 𝛿 -metric
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Axisymmetric metrics\ 𝛿 -metric
• Consider r=2m hypersurface, its normal N satisfies
• We find that this surface is null for
• Also r=2m is an infinite redshift surface
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Axisymmetric metrics\ 𝛿 -metric
• To interpret 𝛿 -metric, we expand the metric in the weak field limit and bring it to the Newtonian form
• where we have defined
• This metric describes a source with mass and quadrupole
• 𝛿 metric expanded to first order in q for the physical mass M is the SQ metric.
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Axisymmetric \Hartle -Thorne metric
• Erez_Rosen metric is given by
• The Hartle-Thorne metric is
(J. B. Hartle and K. S. Thorne, “Slowly Rotating Relativistic Stars”)
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Light ring method
• Quasinormal modes are damped oscillations of a BH which are ingoing at the horizon
and out going at infinity
• After merger, we may consider a blackhole with slight perturbations.
• The QNMs are related to the decay of the photon congruence in the unstable orbit.
• we consider the bundle of null rays in the unstable circular equatorial orbit at r = r0
• For a congruence of null rays, the law of conservation of rays is given by
E. Berti et.al V. Ferrari et.al
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Light ring method
• Null geodesics
• Slight perturbations of unstable photon orbits results in
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Quasinormal modes
• SQ metric QNMs
A. Allahyari, H. Firouzjahi and B. Mashhoon, ``Quasinormal Modes of a Black Hole with Quadrupole Moment,''
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Quasinormal modes
• HT metric QNMs
A.Allahyari, H.Firouzjahi and B.Mashhoon, ``Quasinormal Modes of a Black Hole with Quadrupole Moment,''
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Ongoing
• Rotating 𝛿-metric
• Rotation contribute to the quadrupole
• The mere fact that the source rotates increases the corresponding range of quadrupole.
A. Allahyari, H. Firouzjahi and B. Mashhoon, “Quasinormal Modes of Generalized Black Holes: delta-Kerr Spacetime”, Classical and Quantum Gravity, 10.1088/1361-6382/ab6860
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Ongoing
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Ongoing
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Ongoing
• The quasinormal modes are
• QNM for H-T metric
• Numerically equal to the quasinormal modes of rotating H-T metric.
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Conclusion
• Axisymmetric configuration can undergo quasinormal oscillations.
• We could determine mass and quadrupole of two solutions.
• Essentially the QNMs for both SQ and static HT spacetimes to linear order in the quadrupole moment are the same.
• The quadrupole increases the mass estimate in the real part of QNM spectrum.
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Thanks…