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

U = M r M R

Newtonian theory of Love numbers

U = M r − 1 2xaxbEab M R Eab = −∂a∂bUext(0)

Newtonian theory of Love numbers

U = M r − 1 2xaxbEab + 3 2 xaxbQab r5 M R Eab = −∂a∂bUext(0) Qab

Newtonian theory of Love numbers

U = M r − 1 2xaxbEab + 3 2 xaxbQab r5 M R Eab = −∂a∂bUext(0) Qab = λ2Eab

Newtonian theory of Love numbers

U = M r − 1 2xaxbEab + 3 2 xaxbQab r5 M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

U = M r − 1 2xaxbEab

• 1 + 2k2

R r 5 M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

U = M r −

• ℓ2

(ℓ − 2)! ℓ! xa1 · · · xaℓEa1···aℓ

• 1 + 2kℓ

R r 2ℓ+1 M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

U = M r −

• ℓ2
• |m|ℓ

(ℓ − 2)! ℓ! rℓEℓm

• 1 + 2kℓ

R r 2ℓ+1 Yℓm M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

ψ0 =

• ℓ2
• |m|ℓ
• (ℓ + 2)(ℓ + 1)

ℓ(ℓ − 1) rℓ−2 Eℓm

• 1 + 2kℓ

R r 2ℓ+1

2Yℓm

M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

ψ0 =

• ℓ2
• |m|ℓ
• (ℓ + 2)(ℓ + 1)

ℓ(ℓ − 1) rℓ−2 Eℓm

• 1 + 2kℓm

R r 2ℓ+1

2Yℓm

M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

ψ0 =

• ℓ2
• |m|ℓ
• (ℓ + 2)(ℓ + 1)

ℓ(ℓ − 1) rℓ−2 Eℓm

• 1 + 2kℓm

R r 2ℓ+1

2Yℓm

kℓm = k(0)

ℓ

+ imχ k(1)

ℓ

+ O(χ2) M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Newtonian theory of Love numbers

ψ0 =

• ℓ2
• |m|ℓ
• (ℓ + 2)(ℓ + 1)

ℓ(ℓ − 1) rℓ−2 Eℓm

• 1 + 2kℓm

R r 2ℓ+1

2Yℓm

Tidal Love numbers kℓm ← → body’s internal structure M R Eab = −∂a∂bUext(0) Qab = λ2Eab = − 2

3k2R5Eab

Internal structure of neutron stars

GW observations as probes of neutron star internal structure

Relativistic theory of Love numbers

• Electric-type and magnetic-type tidal moments:

EL ∝ ˆ C0a10a2;a3···aℓ and BL ∝ εa1bc ˆ Ca20bc;a3···aℓ

Relativistic theory of Love numbers

• Electric-type and magnetic-type tidal moments:

EL ∝ ˆ C0a10a2;a3···aℓ and BL ∝ εa1bc ˆ Ca20bc;a3···aℓ

• Metric and Geroch-Hansen multipole moments:

gαβ = ˚ gαβ + htidal

αβ

• ∼r ℓ

+ hresp

αβ

• ∼r −(ℓ+1)

− → ML = ˚ ML + δML SL = ˚ SL + δSL

Relativistic theory of Love numbers

• Electric-type and magnetic-type tidal moments:

EL ∝ ˆ C0a10a2;a3···aℓ and BL ∝ εa1bc ˆ Ca20bc;a3···aℓ

• Metric and Geroch-Hansen multipole moments:

gαβ = ˚ gαβ + htidal

αβ

• ∼r ℓ

+ hresp

αβ

• ∼r −(ℓ+1)

− → ML = ˚ ML + δML SL = ˚ SL + δSL

• Two families of tidal deformability parameters:

δML = λel

ℓ EL

and δSL = λmag

ℓ

BL

Relativistic theory of Love numbers

• Electric-type and magnetic-type tidal moments:

EL ∝ ˆ C0a10a2;a3···aℓ and BL ∝ εa1bc ˆ Ca20bc;a3···aℓ

• Metric and Geroch-Hansen multipole moments:

gαβ = ˚ gαβ + htidal

αβ

• ∼r ℓ

+ hresp

αβ

• ∼r −(ℓ+1)

− → ML = ˚ ML + δML SL = ˚ SL + δSL

• Two families of tidal deformability parameters:

δML = λel

ℓ EL

and δSL = λmag

ℓ

BL

• Dimensionless tidal Love numbers:

kel/mag

ℓ

≡ −(2ℓ − 1)!! 2(ℓ − 2)! λel/mag

ℓ

R2ℓ+1

Love numbers of spinning compact objects

• The spin breaks the spherical symmetry of the background
• No proportionality between (δML, δSL) and (EL, BL)
• 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 (δML, δSL) and (EL, BL)
• Degeneracy of the azimuthal number m lifted
• Parity mixing and mode couplings allowed
• Metric and Geroch-Hansen multipole moments:

gαβ = ˚ gαβ+htidal

αβ

• ∼rℓ

+ hresp

αβ

• ∼r−(ℓ+1)

− → Mℓm = ˚ Mℓm + δMℓm Sℓm = ˚ Sℓm + δSℓm

Love numbers of spinning compact objects

• The spin breaks the spherical symmetry of the background
• No proportionality between (δML, δSL) and (EL, BL)
• Degeneracy of the azimuthal number m lifted
• Parity mixing and mode couplings allowed
• Metric and Geroch-Hansen multipole moments:

gαβ = ˚ gαβ+htidal

αβ

• ∼rℓ

+ hresp

αβ

• ∼r−(ℓ+1)

− → Mℓm = ˚ Mℓm + δMℓm Sℓm = ˚ Sℓm + δSℓm

• Four families of tidal deformability parameters:

λME

ℓℓ′mm′ ≡ ∂δMℓm

∂Eℓ′m′ λSB

ℓℓ′mm′ ≡ ∂δSℓm

∂Bℓ′m′ λSE

ℓℓ′mm′ ≡ ∂δSℓm

∂Eℓ′m′ λMB

ℓℓ′mm′ ≡ ∂δMℓm

∂Bℓ′m′

Black holes have zero Love numbers

Reference Background Tidal field

[Binnington & Poisson 2009]

Schwarzschild weak, generic ℓ

[Damour & Nagar 2009]

Schwarzschild weak, generic ℓ

[Kol & Smolkin 2012]

Schwarzschild weak, electric-type

[Chakrabarti et al. 2013]

Schwarzschild weak, electric, ℓ = 2

[G¨ urlebeck 2015]

Schwarzschild strong, axisymmetric

[Landry & Poisson 2015]

Kerr to O(S) weak, quadrupolar

[Pani et al. 2015]

Kerr to O(S2) weak, (ℓ, m) = (2, 0) Problem of fine-tuning from an Effective-Field-Theory perspective

Investigating Kerr’s Love

(Eℓm, Bℓm) → ψ0 → Ψ → hαβ → (Mℓm, Sℓm) → λM/S,E/B

ℓm

Metric reconstruction through the Hertz potential Ψ M S = χM2 (Eℓm, Bℓm)

Perturbed Weyl scalar

• Recall that in the Newtonian limit we established

lim

c→∞ ψℓm

∝ Eℓm rℓ−2 1 + 2kℓm (R/r)2ℓ+1

2Yℓm(θ, φ)

• For a Kerr black hole the perturbed Weyl scalar reads

ψℓm ∝

• Eℓm +

3i ℓ+1 Bℓm

• Rℓm(r) 2Yℓm(θ, φ)
• Asymptotic behavior of general solution of static radial

Teukolsky equation: Rℓm(r) = rℓ−2 (1 + · · · )

• tidal field Rtidal

ℓm

+ κℓm r−ℓ−3 (1 + · · · )

• linear response Rresp

ℓm

Why analytic continuation?

Rℓm(r) = rℓ−2 (1 + · · · )

• tidal field Rtidal

ℓm

+ κℓm r−ℓ−3 (1 + · · · )

• linear response Rresp

ℓm

Ambiguity in the linear response [Fang & Lovelace 2005; Gralla 2018]

The decaying solution Rresp

ℓm

is affected by a radial coord. transfo.

Ambiguity in the tidal field [Pani, Gualtieri, Maselli & Ferrari 2015]

The growing solution Rtidal

ℓm +αRresp ℓm

still qualifies as a tidal solution

Kerr black hole linear response

Rℓm(r) = Rtidal

ℓm (r)

• ∼rℓ−2

+ 2kℓm Rresp

ℓm (r) ∼r−(ℓ+3)

• The coefficients kℓm can be interpreted as the Newtonian

Love numbers of a Kerr black hole and read kℓm = −imχ (ℓ + 2)!(ℓ − 2)! 4(2ℓ + 1)!(2ℓ)!

ℓ

• n=1
• n2(1 − χ2) + m2χ2
• 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 hresp

αβ via Ψresp

Love numbers of a Kerr black hole

• We compute the Love numbers to linear order in χ ≡ S/M2

Love numbers of a Kerr black hole

• We compute the Love numbers to linear order in χ ≡ S/M2
• The modes of the mass/current quadrupole moments are

δM2m . = imχ 180 (2M)5 E2m and δS2m . = imχ 180 (2M)5 B2m

Love numbers of a Kerr black hole

• We compute the Love numbers to linear order in χ ≡ S/M2
• The modes of the mass/current quadrupole moments are

δM2m . = imχ 180 (2M)5 E2m and δS2m . = imχ 180 (2M)5 B2m

• The black hole tidal bulge is rotated by 45◦ with respect to

the quadrupolar tidal perturbation

Love numbers of a Kerr black hole

• We compute the Love numbers to linear order in χ ≡ S/M2
• The modes of the mass/current quadrupole moments are

δM2m . = imχ 180 (2M)5 E2m and δS2m . = imχ 180 (2M)5 B2m

• The black hole tidal bulge is rotated by 45◦ with respect to

the quadrupolar tidal perturbation

• The associated dimensionless tidal Love numbers are

kME

2m = kSB 2m

. = −imχ 120 and kMB

2m = kSE 2m

. = 0

Love numbers of a Kerr black hole

• We compute the Love numbers to linear order in χ ≡ S/M2
• The modes of the mass/current quadrupole moments are

δM2m . = imχ 180 (2M)5 E2m and δS2m . = imχ 180 (2M)5 B2m

• The black hole tidal bulge is rotated by 45◦ with respect to

the quadrupolar tidal perturbation

• The associated dimensionless tidal Love numbers are

kME

2m = kSB 2m

. = −imχ 120 and kMB

2m = kSE 2m

. = 0

• For a dimensionless black hole spin χ = 0.1 this gives

|k2,±2| ≃ 2 × 10−3 − → black holes are “rigid”

Love tensor of a Kerr black hole

• For a nonspinning compact body we have the proportionality

relations δMab = λel

2 Eab

and δSab = λmag

2

Bab

• For a spinning black hole we have the more general tensorial

relations δMab = λabcdEcd and δSab = λabcdBcd

• To linear order in the black hole spin vector Sa we find

δMab . = 16 45 M3 ScEd

(aεb)cd

δSab . = 16 45 M3 ScBd

(aεb)cd

Love tensor of a Kerr black hole

(λabcd) . = χ 180 (2M)5   I11 I12 I13 I12 −I11 I23 I13 I23   I11 ≡   1 1   I12 ≡   −1 +1   I13 ≡  

1 2 1 2

  I23 ≡   − 1

2

− 1

2

 

Newtonian static quadrupolar tide

Eab = µ r3   2 −1 −1   δMab . = 3Q   1 1   M S µ +q −q +q −q d d ↑

χ 180 (2M)5 µ r3 = qd2

Tidal torquing of a spinning black hole

[Thorne & Hartle 1980; Poisson 2004]

• An arbitrary spinning body interacting with a tidal

environment suffers a tidal torquing: ˙ Sa = −εabcMbdEd

c + SbdBd c

• Applied to a spinning black hole this yields

˙ S . = − 8 45 M5χ [2E1 + B1 − 3E2 + B2] M Sa (Eab, Bab)

Observing black hole tidal deformability

[Pani & Maselli 2019]

Accumulated GW phase in LISA band during quasi-circular inspiral down to Schwarzschild ISCO: Φtidal ≃ −2 × 103 10−7 µ/M

• k2

0.002

• like 1st order dissipative self-force

↑ k2 µ M

Summary

• Love numbers of Kerr black holes do not vanish in general
• We computed in closed-form the leading (quadrupolar) Love

numbers to linear order in the black hole spin

• Kerr black holes deform like any other self-gravitating body,

despite being particularly “rigid” compact objects

• This is closely related to the phenomenon of tidal torquing
• The black hole tidal deformation contribution to the GW

phase of EMRIs could be detectable by LISA

• New black hole test of the Kerr-like nature of the massive

compact objects at the center of galaxies Spinning black holes fall in Love!

Two basis of independent solutions

• Dimensionless radial coordinate and spin parameter:

x ≡ r − r+ r+ − r− and γ = a r+ − r−

• Smooth and unsmooth solutions:

Rsmooth

ℓm

= x−2(1 + x)−2 F(−ℓ − 2, ℓ − 1, −1 + 2imγ; −x) Runsmooth

ℓm

= (1 + 1/x)2imγ F(−ℓ + 2, ℓ + 3, 3 − 2imγ; −x)

• Tidal and response solutions:

Rtidal

ℓm

= xℓ (1 + x)2 F(−ℓ − 2, −ℓ − 2imγ, −2ℓ; −1/x) ∼ xℓ−2 Rresp

ℓm =

x−ℓ−1 (1 + x)2 F(ℓ − 1, ℓ + 1 − 2imγ, 2ℓ + 2; −1/x) ∼ x−ℓ−3

To Love or not to Love?

Rℓm(r) = rℓ−2 (1 + · · · )

• tidal field Rtidal

ℓm

+ κℓm r−ℓ−3 (1 + · · · )

• linear response Rresp

ℓm

To Love or not to Love?

Rℓm(r) = rℓ−2 (1 + · · · )

• tidal field Rtidal

ℓm

+ κℓm r−ℓ−3 (1 + · · · )

• linear response Rresp

ℓm

Eric Poisson is not in Love

• The growing solution Rtidal

ℓm

is not unique

• Specify it uniquely by requiring its smoothness
• Since Rℓm is smooth, he concludes that κℓm = 0

To Love or not to Love?

Rℓm(r) = rℓ−2 (1 + · · · )

• tidal field Rtidal

ℓm

+ κℓm r−ℓ−3 (1 + · · · )

• linear response Rresp

ℓm

Eric Poisson is not in Love

• The growing solution Rtidal

ℓm

is not unique

• Specify it uniquely by requiring its smoothness
• Since Rℓm is smooth, he concludes that κℓm = 0

Marc Casals and I are in Love

• Analytic continuation of ℓ ∈ R
• The growing/decaying solutions are specified uniquely
• Smoothness of Rℓm yields a response coefficient κℓm = 0

To Love or not to Love?

Rℓm(r) = rℓ−2 (1 + · · · )

• tidal field Rtidal

ℓm

+ κℓm r−ℓ−3 (1 + · · · )

• linear response Rresp

ℓm

Eric Poisson is not in Love

• The growing solution Rtidal

ℓm

is not unique

• Specify it uniquely by requiring its smoothness
• Since Rℓm is smooth, he concludes that κℓm = 0

Marc Casals and I are in Love

• Analytic continuation of ℓ ∈ R
• The growing/decaying solutions are specified uniquely
• Smoothness of Rℓm yields a response coefficient κℓm = 0

Edgardo Franzin has mixed feelings