Tidal Love numbers of Kerr black holes Alexandre Le Tiec - PowerPoint PPT Presentation

Tidal Love numbers of Kerr black holes Alexandre Le Tiec Laboratoire Univers et Th´ eories Observatoire de Paris / CNRS Collaborators: M. Casals & E. Franzin Submitted to PRL, gr-qc/2007.00214

Newtonian theory of Love numbers R M U = M r

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) R M U = M r − 1 2 x a x b E ab

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab R M x a x b Q ab U = M r − 1 2 x a x b E ab + 3 r 5 2

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R M x a x b Q ab U = M r − 1 2 x a x b E ab + 3 r 5 2

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M x a x b Q ab U = M r − 1 2 x a x b E ab + 3 r 5 2

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � 5 � � R U = M r − 1 2 x a x b E ab 1 + 2 k 2 r

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � 2 ℓ +1 � � R ( ℓ − 2)! U = M � x a 1 · · · x a ℓ E a 1 ··· a ℓ r − 1 + 2 k ℓ ℓ ! r ℓ � 2

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � 2 ℓ +1 � � R U = M ( ℓ − 2)! � � r ℓ E ℓ m r − 1 + 2 k ℓ Y ℓ m ℓ ! r ℓ � 2 | m | � ℓ

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � � 2 ℓ +1 � � R ( ℓ + 2)( ℓ + 1) � � r ℓ − 2 E ℓ m ψ 0 = 1 + 2 k ℓ 2 Y ℓ m ℓ ( ℓ − 1) r ℓ � 2 | m | � ℓ

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � � 2 ℓ +1 � � R ( ℓ + 2)( ℓ + 1) � � r ℓ − 2 E ℓ m ψ 0 = 1 + 2 k ℓ m 2 Y ℓ m ℓ ( ℓ − 1) r ℓ � 2 | m | � ℓ

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � � 2 ℓ +1 � � R ( ℓ + 2)( ℓ + 1) � � r ℓ − 2 E ℓ m ψ 0 = 1 + 2 k ℓ m 2 Y ℓ m ℓ ( ℓ − 1) r ℓ � 2 | m | � ℓ k ℓ m = k (0) + im χ k (1) + O ( χ 2 ) ℓ ℓ

Newtonian theory of Love numbers E ab = − ∂ a ∂ b U ext ( 0 ) Q ab = λ 2 E ab R = − 2 3 k 2 R 5 E ab M � � � 2 ℓ +1 � � R ( ℓ + 2)( ℓ + 1) � � r ℓ − 2 E ℓ m ψ 0 = 1 + 2 k ℓ m 2 Y ℓ m ℓ ( ℓ − 1) r ℓ � 2 | m | � ℓ Tidal Love numbers k ℓ m ← → body’s internal structure

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Relativistic theory of Love numbers • Electric-type and magnetic-type tidal moments: E L ∝ ˆ B L ∝ ε a 1 bc ˆ C 0 a 1 0 a 2 ; a 3 ··· a ℓ and C a 2 0 bc ; a 3 ··· a ℓ

Relativistic theory of Love numbers • Electric-type and magnetic-type tidal moments: E L ∝ ˆ B L ∝ ε a 1 bc ˆ C 0 a 1 0 a 2 ; a 3 ··· a ℓ and C a 2 0 bc ; a 3 ··· a ℓ • Metric and Geroch-Hansen multipole moments: � M L = ˚ M L + δ M L + h resp g αβ + h tidal g αβ = ˚ − → αβ S L = ˚ αβ S L + δ S L ���� ���� ∼ r ℓ ∼ r − ( ℓ +1)

Relativistic theory of Love numbers • Electric-type and magnetic-type tidal moments: E L ∝ ˆ B L ∝ ε a 1 bc ˆ C 0 a 1 0 a 2 ; a 3 ··· a ℓ and C a 2 0 bc ; a 3 ··· a ℓ • Metric and Geroch-Hansen multipole moments: � M L = ˚ M L + δ M L + h resp g αβ + h tidal g αβ = ˚ − → αβ S L = ˚ αβ S L + δ S L ���� ���� ∼ r ℓ ∼ r − ( ℓ +1) • Two families of tidal deformability parameters: δ M L = λ el δ S L = λ mag ℓ E L and B L ℓ

Relativistic theory of Love numbers • Electric-type and magnetic-type tidal moments: E L ∝ ˆ B L ∝ ε a 1 bc ˆ C 0 a 1 0 a 2 ; a 3 ··· a ℓ and C a 2 0 bc ; a 3 ··· a ℓ • Metric and Geroch-Hansen multipole moments: � M L = ˚ M L + δ M L + h resp g αβ + h tidal g αβ = ˚ − → αβ S L = ˚ αβ S L + δ S L ���� ���� ∼ r ℓ ∼ r − ( ℓ +1) • Two families of tidal deformability parameters: δ M L = λ el δ S L = λ mag ℓ E L and B L ℓ • Dimensionless tidal Love numbers: λ el/mag ≡ − (2 ℓ − 1)!! k el/mag ℓ ℓ 2( ℓ − 2)! R 2 ℓ +1

Love numbers of spinning compact objects • The spin breaks the spherical symmetry of the background ◦ No proportionality between ( δ M L , δ S L ) and ( E L , B L ) ◦ Degeneracy of the azimuthal number m lifted ◦ Parity mixing and mode couplings allowed

Love numbers of spinning compact objects • The spin breaks the spherical symmetry of the background ◦ No proportionality between ( δ M L , δ S L ) and ( E L , B L ) ◦ Degeneracy of the azimuthal number m lifted ◦ Parity mixing and mode couplings allowed • Metric and Geroch-Hansen multipole moments: � M ℓ m = ˚ M ℓ m + δ M ℓ m + h resp g αβ + h tidal g αβ = ˚ − → αβ S ℓ m = ˚ αβ S ℓ m + δ S ℓ m ���� ���� ∼ r ℓ ∼ r − ( ℓ +1)

Love numbers of spinning compact objects • The spin breaks the spherical symmetry of the background ◦ No proportionality between ( δ M L , δ S L ) and ( E L , B L ) ◦ Degeneracy of the azimuthal number m lifted ◦ Parity mixing and mode couplings allowed • Metric and Geroch-Hansen multipole moments: � M ℓ m = ˚ M ℓ m + δ M ℓ m + h resp g αβ + h tidal g αβ = ˚ − → αβ S ℓ m = ˚ αβ S ℓ m + δ S ℓ m ���� ���� ∼ r ℓ ∼ r − ( ℓ +1) • Four families of tidal deformability parameters: ℓℓ ′ mm ′ ≡ ∂δ M ℓ m ℓℓ ′ mm ′ ≡ ∂δ S ℓ m λ M E λ S B ∂ E ℓ ′ m ′ ∂ B ℓ ′ m ′ ℓℓ ′ mm ′ ≡ ∂δ S ℓ m ℓℓ ′ mm ′ ≡ ∂δ M ℓ m λ S E λ M B ∂ E ℓ ′ m ′ ∂ B ℓ ′ m ′

Black holes have zero Love numbers Reference Background Tidal field Schwarzschild weak, generic ℓ [Binnington & Poisson 2009] Schwarzschild weak, generic ℓ [Damour & Nagar 2009] Schwarzschild weak, electric-type [Kol & Smolkin 2012] Schwarzschild weak, electric, ℓ = 2 [Chakrabarti et al. 2013] Schwarzschild strong, axisymmetric [G¨ urlebeck 2015] [Landry & Poisson 2015] Kerr to O ( S ) weak, quadrupolar Kerr to O ( S 2 ) weak, ( ℓ, m ) = (2 , 0) [Pani et al. 2015] Problem of fine-tuning from an Effective-Field-Theory perspective

Investigating Kerr’s Love S = χ M 2 ( E ℓ m , B ℓ m ) M ( E ℓ m , B ℓ m ) → ψ 0 → Ψ → h αβ → ( M ℓ m , S ℓ m ) → λ M / S , E / B ℓ m Metric reconstruction through the Hertz potential Ψ

Perturbed Weyl scalar • Recall that in the Newtonian limit we established ∝ E ℓ m r ℓ − 2 � 1 + 2 k ℓ m ( R / r ) 2 ℓ +1 � c →∞ ψ ℓ m lim 2 Y ℓ m ( θ, φ ) 0 • For a Kerr black hole the perturbed Weyl scalar reads � � ψ ℓ m 3 i ∝ E ℓ m + ℓ +1 B ℓ m R ℓ m ( r ) 2 Y ℓ m ( θ, φ ) 0 • Asymptotic behavior of general solution of static radial Teukolsky equation: R ℓ m ( r ) = r ℓ − 2 (1 + · · · ) + κ ℓ m r − ℓ − 3 (1 + · · · ) � �� � � �� � linear response R resp tidal field R tidal ℓ m ℓ m

Why analytic continuation? R ℓ m ( r ) = r ℓ − 2 (1 + · · · ) + κ ℓ m r − ℓ − 3 (1 + · · · ) � �� � � �� � linear response R resp tidal field R tidal ℓ m ℓ m Ambiguity in the linear response [Fang & Lovelace 2005; Gralla 2018] The decaying solution R resp is affected by a radial coord. transfo. ℓ m Ambiguity in the tidal field [Pani, Gualtieri, Maselli & Ferrari 2015] ℓ m + α R resp The growing solution R tidal still qualifies as a tidal solution ℓ m

Kerr black hole linear response R ℓ m ( r ) = R tidal + 2 k ℓ m R resp ℓ m ( r ) ℓ m ( r ) � �� � � �� � ∼ r ℓ − 2 ∼ r − ( ℓ +3) • The coefficients k ℓ m can be interpreted as the Newtonian Love numbers of a Kerr black hole and read ℓ k ℓ m = − im χ ( ℓ + 2)!( ℓ − 2)! � � n 2 (1 − χ 2 ) + m 2 χ 2 � 4(2 ℓ + 1)!(2 ℓ )! n =1 • The linear response vanishes identically when: ◦ the black hole spin vanishes ( χ = 0) ◦ the tidal field is axisymmetric ( m = 0) • Reconstruct the Kerr black hole response h resp αβ via Ψ resp

Love numbers of a Kerr black hole • We compute the Love numbers to linear order in χ ≡ S / M 2

Love numbers of a Kerr black hole • We compute the Love numbers to linear order in χ ≡ S / M 2 • The modes of the mass/current quadrupole moments are = im χ = im χ δ M 2 m . δ S 2 m . 180 (2 M ) 5 E 2 m 180 (2 M ) 5 B 2 m and

Love numbers of a Kerr black hole • We compute the Love numbers to linear order in χ ≡ S / M 2 • The modes of the mass/current quadrupole moments are = im χ = im χ δ M 2 m . δ S 2 m . 180 (2 M ) 5 E 2 m 180 (2 M ) 5 B 2 m and • The black hole tidal bulge is rotated by 45 ◦ with respect to the quadrupolar tidal perturbation