r t rt trs - PowerPoint PPT Presentation

❚❤❡ ♣r♦❜❧❡♠ ♦❢ ♠♦t✐♦♥ ✐♥ ❣r❛✈✐t② t❤❡♦r✐❡s history, and ♥❡✇ ♣❡rs♣❡❝t✐✈❡s ✿ The example of binary black-holes in Einstein-Maxwell-Dilaton (EMD) theory Nathalie Deruelle, with F´ elix-Louis Juli´ e and Marcela C´ ardenas CNRS, APC-Paris Diderot Kyoto, 23 February 2018 – Typeset by Foil T EX – 1

❚❤❡ ♥❡✇ ❡r❛ ✐♥ ❛str♦♥♦♠② GW150914 : first observation of a BBH coalescence by LIGO GW170817: first observation of a BNS coalescence by LIGO/Virgo with EM counterparts Will allow to probe modified theories of gravity , in the strong-field regime near merger, an “important and doable problem, which is still in infancy” (to paraphrase Takashi Nakamura). – Typeset by Foil T EX – 2

Needles in a haystack (from T. Damour conference, Hannover 2016) – Typeset by Foil T EX – 3

“Knowing the chirp to hear it”... – Typeset by Foil T EX – 4 from L. Blanchet conference, Hannover 2016

The “effective-one-body” (EOB) approach A. Buonanno and T. Damour, 1998 • maps the two-body general relativistic Post-Newtonian (PN) dynamics to the motion of a test particle in an effective SSS metric • defines a resummation of the PN dynamics to describe analytically the coalescence of 2 compact objects from inspiral to merger • is instrumental to build libraries of waveform templates for LIGO/Virgo z 2 / M h 15 0.2 10 0.1 5 t / M z 1 / M 2750 2800 2850 2900 2950 3000 - 15 - 10 - 5 5 10 15 - 5 - 0.1 - 10 - 0.2 - 15 Aim : extend the EOB approach to modified gravities – Typeset by Foil T EX – 5

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ 1. The Einstein-Maxwell-Dilaton (EMD) black hole as a simple example of a “hairy” black hole 2. The action for a binary EMD black hole system or, how to “skeletonize” hairy black holes 3. The (conservative) dynamics of an EMD black hole binary vs “state-of-the-art” in scalar-tensor theories and GR • Lagrangian and Hamiltonian for the relative motion • Mapping to an effective-one-body (EOB) hamiltonian • A first flavour of possible tests – Typeset by Foil T EX – 6

References (all in arXiv) Thermodynamics sheds light on black hole dynamics Marcela C´ ardenas, F´ elix-Louis Juli´ e, Nathalie Deruelle, arXiv:1712.02672 On the motion of hairy black holes in EMD theories F´ elix-Louis Juli´ e, JCAP 1801 (2018) Reducing the 2-body problem in ST theories to the motion of a test particle : a ST-EOB approach F´ elix-Louis Juli´ e Phys.Rev. D97 (2018) no.2, 024047 Two body pb in ST theories as a deformation of GR : an EOB approach F´ elix-Louis Juli´ e, Nathalie Deruelle Phys.Rev. D95 (2017) 12, 124054 On conserved charges and thermodynamics of AdS4 dyonic BHs Marcela C´ ardenas, Oscar Fuentealba, Javier Matulich, JHEP 1605 (2016) Einstein-Katz action, variational principle, Noether charges and the thermodynamics of AdS-BHs Andr´ es Anabal´ on, Nathalie Deruelle, F´ elix-Louis Juli´ e, JHEP 1608 (2016) – Typeset by Foil T EX – 7

❚❤❡ ❊✐♥st❡✐♥✲▼❛①✇❡❧❧✲❉✐❧❛t♦♥ ✭❊▼❉✮ ❜❧❛❝❦ ❤♦❧❡ – Typeset by Foil T EX – 8

Isolated EMD black holes G. W. Gibbons 1982, GWG and K. i. Maeda 1988, GWG 1996 D. Garfinkle, G. T. Horowitz and A. Strominger 1991 Vacuum Einstein-Maxwell-dilaton action of gravity d 4 x √− g � R − 2 g µν ∂ µ ϕ ∂ ν ϕ − e − 2 aϕ F 2 � ´ 16 π I vac [ g µν , A µ , ϕ ] = Field equations : µ F νλ − 1 R µν = 2 ∂ µ ϕ ∂ ν ϕ + 2 e − 2 aϕ � 4 g µν F 2 � F λ e − 2 aϕ F µν � � ϕ = − 1 2 e − 2 aϕ F 2 � D µ = 0 , Static, spherically symmetric, solutions depend a priori on 5 integration constants. “Electric” black hole solutions depend on only 3. For a = 1 : � − 1 dr 2 + r 2 � ds 2 = − dt 2 + 1 − r + 1 − r + 1 − r − d Ω 2 � � � � r r r � e ϕ ∞ r + r − 1 − r − ϕ = ϕ ∞ + 1 � � A t = − , A i = 0 , 2 ln 2 r r – Typeset by Foil T EX – 9

EMD black hole thermodynamics (case a = 1 ) 1 Temperature : T = (or surface gravity κ = 2 πT ) 4 πr + � e ϕ ∞ r + r − Electric potential : Φ = A t ( r → ∞ ) − A t ( r + ) = 2 r + � � � 1 − r − Entropy : S = πr 2 A (or area : A = 4 S ; or M irr = 4 π ) + r + Associated global charges : � r + r − M = 1 2 r + − 1 e − ϕ ∞ ´ Q = , r − dϕ ∞ 2 2 (see M. Henneaux et al 2002,..., C´ ardenas et al 2016, Juli´ e et al 2016) The variations of S , Q , and M wrt r + , r − and ϕ ∞ , are such that TδS = δM − Φ δQ – Typeset by Foil T EX – 10

❚❤❡ ❛❝t✐♦♥ ❢♦r ❛ ❜✐♥❛r② ❊▼❉ ❜❧❛❝❦ ❤♦❧❡ s②st❡♠ – Typeset by Foil T EX – 11

“Skeletonizing” an EMD black hole in GR : Mathisson 1931, Infeld 1950,... d 4 x √− g 1 R − 2 g µν ∂ µ ϕ ∂ ν ϕ − e − 2 aϕ F 2 � + I bh [Ψ , g µν , ϕ, A µ ] ´ � I = 16 π A µ dx µ ´ ´ I bh = − m ( ϕ ) ds + q Linear coupling to A µ , and q constant, to preserve U (1) symmetry ; m A ( ϕ ) : m � = const because ϕ cannot be “gauged away” (Eardley 1975, Damour Esposito-Farese 1992) Question : how are q and m ( ϕ ) related to the parameters characterizing the black hole, that is, r + , r − and ϕ ∞ ? Answer : by identifying the EMD black hole solution to that of the field equations for the skeletonized body above. F´ elix-Louis Juli´ e, 2017 – Typeset by Foil T EX – 12

The “sensitivity” of an EMD black hole ds m ( ϕ ) δ (4) ( x − z ) (with T µν = u µ u ν ) ´ • Field equations √− g R µν = 2 ∂ µ ϕ∂ ν ϕ + e − 2 aϕ � − 1 2 g µν F 2 � T µν − 1 2 F µα F α � � +8 π 2 g µν T ν ds δ (4) ( x − z ) e − 2 aϕ F µν � u µ � ´ D ν = 4 πq √− g ds δ (4) ( x − z ) � ϕ = − a 2 e − 2 aϕ F 2 +4 π dm ´ √− g dϕ • Lowest order asymptotic solution in the body rest-frame : = − q e 2 ϕ ∞ , ϕ asym = ϕ ∞ − 1 , A asym � 2 m ∞ g asym dm � = η µν + δ µν dϕ | ∞ µν t r r r to be identified with the EMD black hole solution (case a = 1 ) : � e ϕ ∞ r , ϕ asym = ϕ ∞ − r − � r + , A asym r + r − g asym � = η µν + δ µν = − µν t r 2 2 r Hence a differential equation, with a unique solution � 2 e − ϕ ∞ | ∞ so that q 2 = 2 m dm r + r − r + = 2 m ∞ , r − = 2 dm dϕ e 2 ϕ | ∞ dϕ , q = � µ 2 + q 2 e 2 ϕ m ( ϕ ) = F´ elix-Louis Juli´ e, 2017 2 – Typeset by Foil T EX – 13

The parameters of a skeletonized ( a = 1 ) EMD black hole � � µ 2 + q 2 e 2 ϕ r + r − 2 e − ϕ ∞ , r + = 2 m ∞ , r − = 2 dm q = dϕ | ∞ , and m ( ϕ ) = 2 Recall : the global charges and entropy of an EMD black hole are � � � r + r − e − ϕ ∞ , M = 1 1 − r − 2 r + − 1 r − dϕ ∞ , and S = πr 2 ´ Q = + 2 2 r + Hence Q = q is a constant : δQ = 0 . Also : δM = δm ∞ − dm dϕ δϕ | ∞ = 0 Our skeletonized BHs exchange no charge nor energy with their environment. Now, since TδS = δM − Φ δQ , the black hole entropy is also a constant. Therefore µ can be identified to a function of the BH entropy. Indeed : � � 4 π + e 2 ϕ S S 2 Q 2 µ = = ⇒ m ( ϕ ) = 4 π with (for an Einstein-Hilbert action) S = A 4 and M 2 irr = S 4 π C´ ardenas, Juli´ e, ND, 2018 – Typeset by Foil T EX – 14

Hence, all in all, Skeletonized action for a binary EMD black hole system : d 4 x √− g 1 R − 2 g µν ∂ µ ϕ ∂ ν ϕ − e − 2 aϕ F 2 � � + I bbh [ g µν , ϕ, A µ ] ´ I = 16 π A µ dx µ I bbh = − � ´ m A ( ϕ ) ds A + � ´ A q A A A � 4 π + e 2 ϕ S A 2 Q 2 with q A = Q A and m A ( ϕ ) = (for a = 1 ) A where the charges Q A remain constant (true until coalescence) where the entropies S A also remain constant ( not true at coalescence). * The action I is the starting point to study the relative motion of the two black holes. – Typeset by Foil T EX – 15

❚❤❡ ✭❝♦♥s❡r✈❛t✐✈❡✮ ❞②♥❛♠✐❝s ♦❢ ❛♥ ❊▼❉ ❜❧❛❝❦ ❤♦❧❡ ❜✐♥❛r② Lagrangian and Hamiltonian for the relative motion – Typeset by Foil T EX – 16

The 1st Post-Newtonian (1PN) Lagrangian of an EMD BH binary ds A m A ( ϕ ) δ (4) ( x − z A ) (with T µν u µ A u ν ´ • Field equations A = A ) √− g R µν = 2 ∂ µ ϕ∂ ν ϕ + e − 2 aϕ � − 1 2 g µν F 2 � µν − 1 2 F µα F α � T A 2 g µν T A � + 8 π � ν A δ (4) ( x − z A ) u µ e − 2 aϕ F µν � � ´ D ν = 4 πq A � ds A √− g A A δ (4) ( x − z A ) 2 e − 2 aϕ F 2 + 4 π � dm A � ϕ = − a ´ ds A √− g A dϕ • Work in harmonic and Lorenz gauges g 00 = − e − 2 U , g 0 i = − 4 g i , g ij = δ ij e 2 V Write : A t = δA t , A i = δA i , ϕ = ϕ ∞ + δϕ Weak field O ( v 2 ) ∼ O ( m/r ) iteration. • Solve and obtain ′∞ m ∞ A v i m � v 6 � � v 5 � V = U + O , g i = � + O , ϕ = ϕ ∞ + � r A + · · · , etc A A A A r A – Typeset by Foil T EX – 17