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SPIN AND HIGHER MULTIPOLE CORRECTIONS TO EMRIS Jiang LONG Asia Pacific Center for Theoretical Physics Jan 10, 2019 TSIMF Sanya CONTENTS Introduction & motivation Assumptions MPD formalism Orbits Gravitational

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SPIN AND HIGHER MULTIPOLE CORRECTIONS TO EMRIS

Jiang LONG Asia Pacific Center for Theoretical Physics Jan 10, 2019 TSIMF Sanya

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CONTENTS

  • Introduction & motivation
  • Assumptions
  • MPD formalism
  • Orbits
  • Gravitational waves
  • Conclusion & discussion
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INTRODUCTION & MOTIVATION

  • GR has passed varies tests, including deflection of light, precession of Mercury…
  • Recently, gravitational waves, one of its prediction, has been detected by LIGO.
  • Binary black hole (BBH) systems are perfect laboratory to test GR.

1)Newton gravity: i) two-body problem, exactly solvable ii) three-body problem? 2)Einstein gravity: i) one-body problem, Schwarzchild, Kerr ii) two-body problem, not easy to find an analytic solution 3)Gravitational waves are radiated from BBHs.

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INTRODUCTION & MOTIVATION

  • Discovery from LIGO…
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INTRODUCTION & MOTIVATION

  • Two facts

1) the mass ratio of two BHs (NS): 𝑁1

𝑁2 β‰ˆ 1

2) The mass of the BH in this picture: M~10π‘π‘‘π‘π‘šπ‘π‘ , stellar black holes

  • BHs:

1) Stellar black hole: gravitational collapse of a star , 1~102π‘π‘‘π‘π‘šπ‘π‘  2) Intermediate mass black hole (IMBH): no strong evidence, 102~105π‘π‘‘π‘π‘šπ‘π‘  3) Supermassive black hole (SMBH): center of galaxies, 105~109π‘π‘‘π‘π‘šπ‘π‘ 

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INTRODUCTION & MOTIVATION

  • The parameter space of BBH: 𝑛, 𝑇, 𝑁, 𝐾, π‘šπ‘π‘ π‘, …
  • LIGO just tests the region: 𝑛

𝑁 β‰ˆ 1, 𝑁 β‰ˆ 10π‘π‘‘π‘π‘šπ‘π‘ 

  • Intermediate and extreme mass ratio: q ≑ 𝑛

𝑁 β‰ͺ 1

  • The small BH 𝑛 β‰ˆ π‘π‘‘π‘π‘šπ‘π‘ 
  • Intermediate mass ratio π‘Ÿ β‰ˆ (10βˆ’2~10βˆ’5)
  • Extreme mass ratio π‘Ÿ β‰ˆ (10βˆ’6~10βˆ’9)
  • We will discuss the perturbation theory to compute gravitational wave in the

region π‘Ÿ β‰ͺ 1

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ASSUMPTIONS

  • Large BH

1) Near-extreme Kerr black hole (High spin Kerr black hole) 2) Near horizon region: emergence of conformal symmetry the last stage of black hole merger

  • Ξ» =

1 βˆ’ 𝐾2

𝑁4 β‰ͺ 1

J.Bardeen &G.Horowitz (1999) M.Guica, T.Hartman, W.Song &A.Strominger (2009)

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ASSUMPTIONS

  • Coalescence of a binary black hole
  • Three steps(phases):

1) Inspiral 2) Merger 3) Ringdown

  • Large high spin Kerr black hole as a background
  • Three patches of Kerr black hole

1) far region 2) NHEK region 3) near-NHEK region

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KERR BLACK HOLE

  • Three patches of a high spin

Kerr black hole Last stage of a small black hole Falls into a large high spin Kerr black hole is in NHEK and near-NHEK region

  • J.Bardeen, W.Press, S.Teukolsky (1972)
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FAR REGION

  • Far region
  • ො

𝑦 β‰ͺ 1, π‘œπ‘“π‘π‘  β„Žπ‘π‘ π‘—π‘¨π‘π‘œ π‘ π‘“π‘•π‘—π‘π‘œ

  • ො

𝑦 β†’ ∞, observer

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NHEK REGION

  • NHEK region
  • It can be obtained by coordinate transformation

and take the limit Ξ» β†’ 0

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NEAR-NHEK REGION

  • Near-NHEK region

It can be obtained by coordinate transformation and take the limit Ξ» β†’ 0 𝑠 β†’ 0, β„Žπ‘π‘ π‘—π‘¨π‘π‘œ 𝑠 β†’ ∞, π‘π‘’π‘’π‘π‘‘β„Ž 𝑒𝑝 𝑔𝑏𝑠 π‘ π‘“π‘•π‘—π‘π‘œ

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ASSUMPTIONS

  • Small black hole
  • 1) mass m (π‘žπœˆ)
  • 2) spin S (π‘‡πœπœ)
  • 3) black hole is not a point particle, it has a size!
  • 4) As a first step, we ignore any backreaction from gravitational waves
  • How to describe the movement of an extended object in curved

spacetime?

  • Generalization of geodesics
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MATHISSON-PAPAPETROU-DIXON FORMALISM

  • Geodesics of a point particle without spin π‘£πœˆπ›Ό

πœˆπ‘£πœ = 0

  • A particle with momentum p and spin S (MP equation)
  • π‘žπœˆ ≑ Χ¬ π‘ˆπœˆπœ π‘’Ξ£πœ, 𝑇𝛽𝛾 ≑ Χ¬ 𝑦𝛽 βˆ’ 𝑨𝛽 π‘ˆπ›Ύπ›Ώπ‘’Ξ£Ξ³ βˆ’ (𝛽 ↔ 𝛾)
  • Conservation of stress tensor
  • Spin Supplementary Condition (SSC)
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MATHISSON-PAPAPETROU-DIXON FORMALISM

  • Evolution equations of an extended body
  • Force and torque: presence of higher multipoles
  • 2𝑂 βˆ’ π‘žπ‘π‘šπ‘“: described by a tensor with N+2 indices
  • 𝐾𝜈1πœˆπ‘‚π›½π›Ύπ›Ώπœ€ with symmetry structure
  • 𝑕𝛽𝛾,𝜈1β‹―πœˆπ‘‚: a extension of metric in the sense
  • f Veblen and Thomas
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MATHISSON-PAPAPETROU-DIXON FORMALISM

  • Mass:

1) 𝑛2 = βˆ’π‘ž2 2) 𝑛 = βˆ’π‘ž βˆ™ 𝑣

  • In general, 𝑛 β‰  𝑛
  • Spin:

π‘‡πœˆ = 1 2𝑛 πœ—πœˆπ›½π›Ύπ›Ώπ‘žπ›½π‘‡π›Ύπ›Ώ

  • Spin length:

𝑇2 = 1 2 𝑇𝛽𝛾𝑇𝛽𝛾 = π‘‡πœˆπ‘‡πœˆ

  • 𝑛, 𝑛, 𝑇 𝑏𝑠𝑓 π‘œπ‘π‘’ π‘‘π‘π‘œπ‘‘π‘“π‘ π‘€π‘“π‘’ π‘—π‘œ π‘’β„Žπ‘“ π‘žπ‘ π‘“π‘‘π‘“π‘œπ‘‘π‘“ 𝑝𝑔 β„Žπ‘—π‘•β„Žπ‘“π‘  π‘›π‘£π‘šπ‘’π‘—π‘žπ‘π‘šπ‘“π‘‘
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MATHISSON-PAPAPETROU-DIXON FORMALISM

  • Conserved quantities: Given a Killing vector ΞΎΞ±

𝑅ξ = ΞΎΞ±π‘žΞ± + 1

2 𝑇αβ𝛼 𝛽ξ𝛾 is conserved even in the presence of multipoles

  • Stress tensor: MPD equations are equivalent to 𝛼

πœˆπ‘ˆπœˆπœ = 0

  • Up to quadrupole,
  • The stress tensor is
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QUADRUPOLE MODEL

  • To solve MPD equations, one should construct explicit higher multipole model
  • Some effects that could contribute to quadrupole

1) spin-induced quadrupole 2)gravito-electric tidal field induced quadrupole 3)gravito-magnetic tidal field induced quadrupole

  • The quadrupole is a linear combination of these terms
  • J.Steinhoff & D.Puetzfeld (2012)
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QUADRUPOLE MODEL

  • Dimensional analysis
  • κ𝑇2 = 1 for black hole, κ𝑇2 β‰ˆ 5 for neutron stars

W.Laarakkers &E.Poisson (1999)

  • 𝑛, 𝑛 𝑏𝑠𝑓 π‘œπ‘π‘œ βˆ’ π‘‘π‘π‘œπ‘‘π‘“π‘ π‘€π‘“π‘’, π‘’β„Žπ‘π‘£π‘•β„Ž π‘’β„Žπ‘“π‘§ 𝑏𝑠𝑓 π‘“π‘Ÿπ‘£π‘π‘š π‘£π‘ž 𝑒𝑝 𝑃(𝑇3)
  • 𝜈 is conserved up to 𝑃(𝑇3), it is the mass term in perturbation theory
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CIRCULAR ORBIT

  • Solve MPD equations in near-NHEK region to find the trajectory of the small BH
  • Spinless case:

Equatorial plane

  • Spin and size effect: small mass ratio expansion
  • In small q expansion, one can prove

π‘žπœˆ = 𝑃 π‘Ÿ1 , 𝑇𝛽𝛾 = 𝑃 π‘Ÿ2 , 𝜈2 = 𝑃 π‘Ÿ5 , 𝜏2 = 𝑃(π‘Ÿ5)

  • Gravito-electric and magnetic tidal deformations are higher order
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CIRCULAR ORBIT

  • Assumptions:
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CIRCULAR ORBIT

  • Solution

NHEK: ΞΊ0 β†’ 0

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CIRCULAR ORBIT

  • π‘šβˆ— is the orbital angular momentum of NHEK circular orbit, critical angular momentum in near-NHEK
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GENERAL EQUATORIAL ORBITS

  • Conformal transformation: 𝑇𝑀(2, 𝑆) Γ— 𝑉(1) Γ— π‘„π‘ˆ

1) preserve NHEK 2) preserve near-NHEK 3) NHEK↔near-NHEK

  • Near-NHEK: Circular(π‘šβˆ—)

NHEK: π·π‘—π‘ π‘‘π‘£π‘šπ‘π‘ 

βˆ—

  • Spinless case: all plunging or osculating equatorial orbits

entering into near-NHEK or NHEK are conformally related to a circular orbit. G.Compere, K.Fransen, T.Hertog, J.Long (2017)

  • MPD equations are covariant. We expect any equatorial orbit can

be obtained by applying conformal maps.

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GRAVITATIONAL WAVES

  • Teukolsky equation, Linearized perturbation equation of Kerr black hole

𝐻μν = 8πœŒπ»π‘‚π‘ˆ

ΞΌΞ½

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GRAVITATIONAL WAVES

  • Spin coefficicent & Weyl scalar
  • Ξ΄Οˆβˆ’2 = πœβˆ’4δψ4 encodes complete information of gravitational waves

δψ4(r β†’ ∞)=1

2 ( ሷ

β„Ž+ βˆ’ 𝑗 ሷ β„ŽΓ—)(𝑠 β†’ ∞)

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GRAVITATIONAL WAVES

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GRAVITATIONAL WAVES

  • Teukolsky equation
  • 1) far region: source free, outgoing at infinity
  • 2) NHEK or near-NHEK region: source stress tensor, ingoing at horizon
  • Stress tensor with quadrupole correction
  • For 2𝑂-pole,
  • Matching at intermediate region
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GRAVITATIONAL WAVES

  • Circular(π‘šβˆ—)
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GRAVITATIONAL WAVES

  • Circular(π‘šβˆ—)
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GRAVITATIONAL WAVES

  • Radial source term of Teukolsky equation
  • are fixed by circular orbit
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GRAVITATIONAL WAVES

  • is independent of M
  • β„Ž+ βˆ’ π‘—β„ŽΓ— ∝ 𝜈

ΖΈ 𝑠 typical fall off behavior

  • For extreme Kerr black holes, the frequency of the emitted GWs is locked by

kinematics to be extremal value ෝ πœ•π‘“π‘¦π‘’ = 𝑛

2𝑁

  • For near-extreme Kerr black holes, the frequency is relatively shifted
  • Near-NHEK approximation requires
  • π‘š π‘‘π‘π‘œ 𝑐𝑓 𝑀𝑓𝑠𝑧 π‘‘π‘šπ‘π‘‘π‘“ 𝑒𝑝 π‘šβˆ— 𝑐𝑣𝑒 π‘‘π‘π‘œ π‘œπ‘“π‘€π‘“π‘  𝑐𝑓 π‘ π‘“π‘π‘‘β„Žπ‘“π‘’ π‘—π‘œ π‘œπ‘“π‘π‘  βˆ’ 𝑂𝐼𝐹𝐿.
  • Maximal: π‘š β†’ π‘šβˆ—, minimal: π‘š β†’ ∞
  • Vanishes at first order of Ο‡π‘Ÿ =

𝑇 πœˆπ‘

  • Vanishes at second order of S for black holes ( κ𝑇2 = 1) , non-zero for neutron stars
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GRAVITATIONAL WAVES

  • Amplitude is independent of 𝑠
  • The leading contribution is from the modes with β„Ž = 1

2 βˆ’ π‘—πœ€π‘šπ‘›

  • Scaling behavior in the limit π‘š β†’ π‘šβˆ—
  • Generalization of the scaling behavior with spin and higher multipole

corrections. G.Compere, K.Fransen, T.Hertog, J.Long (2017)

  • No divergent in the limit π‘š β†’ π‘šβˆ—
  • The orbit is completely fixed given energy and orbital angular momentum, using

Boyer-Linquist coordinates ො 𝑦 =

ΖΈ π‘ βˆ’ ΖΈ 𝑠+ ΖΈ 𝑠+ , ො

𝑦0 = Ξ»

ΞΊ0

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GRAVITATIONAL WAVES

  • Energy flux (working in progress)
  • Since we already obtained the waveform at infinity and horizon, the energy flux can

be found to be (NHEK)

  • ሢ

𝐹∞ = π‘Ÿ2 ො 𝑦0[π‘βˆž

0 + π‘βˆž 1 Ο‡π‘Ÿ + π‘βˆž 2 + κ𝑇2 ΰ·€

π‘βˆž

2

(Ο‡π‘Ÿ)2+ β‹― ]

  • ሢ

𝐹𝐼 = π‘Ÿ2 ො 𝑦0[𝑏𝐼

0 + 𝑏𝐼 1 Ο‡π‘Ÿ + 𝑏𝐼 2 + κ𝑇2 ΰ·€

𝑏𝐼

2

(Ο‡π‘Ÿ)2+ β‹― ]

  • π‘βˆž

(𝑗), 𝑏𝐼 (𝑗) are constants which should be evaluated numerically.

  • π‘βˆž

(0) = 0.987, 𝑏𝐼 (0) = βˆ’0.133

S.Gralla, S.Hughes & N.Warburton (2016)

  • π‘βˆž

(1) =? , 𝑏𝐼 (1) =?

First order correction from spin effect

  • π‘βˆž

(2) =? , 𝑏𝐼 (2) =?

Second order correction from spin effect

  • ΰ·€

π‘βˆž

(2) =? , ΰ·€

𝑏𝐼

(2) =?

First order correction from size (quadrupole) effect

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DISCUSSION & CONCLUSION

  • Detectability
  • Extremely small πœ‡, rapidly spinning Kerr black hole
  • Existence?
  • K.S.Thorne bound (1974): 𝐾 ≲ 0.998𝑁2
  • X-ray observing campaigns for AGNs
  • L.Brenneman, β€œMeasuring Supermassive

Black Hole Spins in Active Galactic Nuclei” 2013

  • Maybe we can assume the existence of

high spin Kerr black hole

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DISCUSSION & CONCLUSION

  • Detectability
  • Leading order frequency is locked by extreme Kerr black hole
  • 𝑔

𝐼 = 1 4πœŒπ‘ = 1.6 Γ— 10βˆ’2(106π‘π‘‘π‘π‘šπ‘π‘  𝑁

)

  • SMBH: space-based detectors, LISA
  • IMBH: ground-based detectors, Advanced LIGO, VIRGO
  • Precise observation?
  • Black holes and Neutron stars are different at second order of spin
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DISCUSSION & CONCLUSION

  • Detectability
  • Extreme mass ratio coalescence, π‘Ÿ~10βˆ’6, spin and size effect are too small
  • Intermediate mass ratio coalescence (IMRAC), π‘Ÿ~10βˆ’2, maybe it is more

closely related to experiments.

  • Two types of IMRACs

1) stellar mass BH falls into IMBH, LIGO 2) IMBH falls into SMBH, LISA

  • The method is reliable for IMRACs?
  • Existence of IMBH?
  • The self force effect should be comparable to spin effect
  • Convergence problem for higher multipole?
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DISCUSSION & CONCLUSION

  • Future direction
  • 1) Self force correction
  • 2) Circular orbits out of equatorial plane (spin effect is necessary)
  • 3) Plunging orbits from conformal transformation
  • 4) MPD equation at the full level by including all higher multipoles for BHs?
  • 5) Exact critical orbital angular momentum with all higher multipole

corrections?

  • 6) Numerical simulation and confirm our results
  • 7) β‹―
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THANKS FOR YOUR ATTENTION!

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TECHNICAL DETAILS

CONFORMAL TRANSFORMATION

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TECHNICAL DETAILS

CONFORMAL TRANSFORMATION

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TECHNICAL DETAILS

CONFORMAL TRANSFORMATION

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TECHNICAL DETAILS

NHEK, NEAR-NHEK & KERR

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TECHNICAL DETAILS

TEUKOLSKY EQUATION

  • Angular part
  • Radial part
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TECHNICAL DETAILS

TEUKOLSKY EQUATION

  • Radial equation with Delta function
  • Assume 𝑆1,2(𝑠) are two independent solutions of homogeneous equation
  • Define Wronskian
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