The laws of binary black hole mechanics

Transcription

1 The laws of binary black hole mechanics Laboratoire Univers et Théories Observatoire de Paris / CNRS 2π/ω H Σ κ H k a γ u a k a A 1 A 2 A H κ ω H r + M,J A 3 A 1 +A 2 time i + H γ n a k a H s a k B a S k a Σ u v i I + u a i 0

2 Black hole uniqueness theorem in GR [Israel (1967); Carter (1971); Hawking (1973); Robinson (1975)] The only stationary vacuum black hole solution is the Kerr solution of mass M and angular momentum S Black holes have no hair. (J. A. Wheeler) Black hole event horizon H characterized by: Angular velocity ω H Surface gravity κ Surface area A A κ ω H r + M,S

3 The laws of black hole mechanics [Hawking (1972); Bardeen et al. (1973)] Zeroth law of mechanics: ω H κ const. (over H) First law of mechanics: A κ r + M,S δm ω H δs ` κ 8π δa A 1 Second law of mechanics: H A 3 A 1 +A 2 δa ě 0 A 2 time

4 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

5 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

6 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

7 Generalized zeroth law of mechanics [Friedman, Uryū & Shibata (2002)] Black hole spacetimes with helical Killing vector field k a On each component H i of the horizon, the expansion and shear of the null generators vanish Generalized rigidity theorem H Ť i H i is a Killing horizon è Constant horizon surface gravity κ 2 i 1 2 p a k b a k b q Hi Lack of asymptotic flatness overall scaling of κ i is free è k a k a k a κ 1 H 1 H 2 κ 2

8 Generalized first law of mechanics [Friedman, Uryū & Shibata (2002)] Spacetimes with black holes + perfect fluid matter sources One-parameter family of solutions tg ab pλq, u a pλq, ρpλq, spλqu Globally defined Killing field k a Ñ conseved Noether charge Q δq ÿ i ż κ i 8π δa i ` h δpdmb q ` T δpdsq ` v a δpdc a q Σ Q h,t u a v a k a κi A i M b,s,c a Σ

9 Issue of asymptotic flatness [Friedman, Uryū & Shibata (2002)] Binaries on circular orbits have a helical Killing symmetry k a Helically symmetric spacetimes are not asymptotically flat [Gibbons & Stewart (1983); Detweiler (1989); Klein (2004)] Asymptotic flatness can be recovered if radiation can be turned off : Conformal Flatness Condition Post-Newtonian approximation For asymptotically flat spacetimes: k a Ñ t a ` Ω φ a and δq δm ADM Ω δj

10 Application to black hole binaries [Friedman, Uryū & Shibata (2002)] Rigidity theorem Ñ black holes are in a state of corotation Conformal flatness condition Ñ asymptotic flatness recovered preferred normalization of κ i è For binary black holes the generalized first law reduces to δm ADM Ω δj`ÿ i κ i 8π δa i A 1 Ω κ 2 r + M ADM,J

11 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

12 First law for point-particle binaries [Le Tiec, Blanchet & Whiting (2012)] For balls of dust, the generalized first law reduces to ż δq z δpdm b q `, where z k a u a Σ Conservative PN dynamics Ñ asymptotic flatness recovered Two spinless compact objects modelled as point masses m i and moving along circular orbits obey the first law δm ADM Ω δj ` ÿ z i δm i i m 1 Ω z 2 r + MADM,J

13 Extension to spinning binaries [Blanchet, Buonanno & Le Tiec (2013)] ADM Hamiltonian Hpxi, pi, Si; m i q of two point particles with rest masses m i and canonical spins Si [Steinhoff et al. (2008)] Redshift observables and spin precession frequencies: BH Bm i z i and BH BSi Ω i First law for aligned spins (J L ` ři S i) and circular orbits: δm ADM Ω δl`ÿ pz i δm i ` Ω i δs i q i m 1,S 1 Ω r + z 2,Ω 2 M ADM,L

14 Corotating point particles [Blanchet, Buonanno & Le Tiec (2013)] A point particle with rest mass m i and spin S i is given an irreducible mass µ i and a proper rotation frequency ω i via δm i ω i δs i ` c i δµ i and m 2 i µ 2 i ` S 2 i {p4µ 2 i q The first law of binary point-particle mechanics becomes δm ADM Ω δj ` ÿ rz i c i δµ i ` pz i ω i ` Ω i Ωq δs i s i Comparing with the first law for corotating black holes, δm ADM Ω δj ` ři p4µ iκ i q δµ i, the corotation condition is z i ω i Ω Ω i ÝÑ ω i pωq ÝÑ S i pωq

15 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

16 Rotating black hole + orbiting moon Kerr black hole of mass M and spin S perturbed by a moon of mass m! M: g ab pεq ḡ ab ` ε Dg ab ` Opε 2 q M,S ω H E,L m Perturbation Dg ab obeys the linearized Einstein equation with point-particle source ż DG ab 8π DT ab 8π m dτ δ 4 px, yq u a u b Particle has energy E m t a u a and ang. mom. L m φ a u a Physical Dg ab : retarded solution, no incoming radiation, perturbations DM B E and DJ L [Keidl et al. (2010)] γ

17 Rotating black hole + corotating moon We choose for the geodesic γ the unique equatorial, circular orbit with azimuthal frequency ω H, i.e., the corotating orbit Gravitational radiation-reaction is Opε 2 q and neglected ë the spacetime geometry has a helical symmetry In adapted coordinates, the helical Killing field reads χ a t a ` ω H φ a Conserved orbital quantity associated with symmetry: z χ a u a m 1 pe ω H Lq Σ 2π/ω H γ u a χ a

18 Zeroth law for a black hole with moon [Gralla & Le Tiec (2013)] Because of helical symmetry and corotation, the expansion and shear of the perturbed future event horizon H vanish Rigidity theorems then imply that H is a Killing horizon [Hawking (1972); Chruściel (1997); Friedrich et al. (1999); etc] The horizon-generating Killing field must be of the form k a pεq t a ` ` ω loooooomoooooon H ` ε Dω H φa ` Opε 2 q circular orbit frequency Ω The surface gravity κ is defined in the usual manner as κ p a k b a k b q H Since κ const. over any Killing horizon [Bardeen et al. (1973)], we have proven a zeroth law for the perturbed event horizon

19 Angular velocity vs black hole spin [Gralla & Le Tiec (2013)] last stable corotating orbit ω H M Dω H M 2 / m χ = S / M 2

20 Surface gravity vs black hole spin [Gralla & Le Tiec (2013)] κ M last stable corotating orbit Dκ M 2 / m χ = S / M 2

21 First law for a black hole with moon [Gralla & Le Tiec (2013)] Adapting [Iyer & Wald (1994)] to non-vacuum perturbations of non-stationary spacetimes we find (with Q ab ε abcd c k d ) ż ż ż pδq ab Θ abc k c q 2 δ ε abcd G de k e ε abcd k d G ef δg ef BΣ Σ Σ Applied to nearby BH with moon spacetimes, this gives the first law δm B Ω δj ` κ δa ` z δm 8π Features variations of the Bondi mass and angular momentum B H H k a i + I + k a Σ k a S γ u a i 0 i

22 Black holes and point particles κ A ω H δm ω H δs κ 8π δa M 2ω H S κa 4π A 1 Ω κ 2 δm Ω δj ÿ i κ i 8π δa i M 2ΩJ ÿ i κ i A i 4π Ω z 2 z i δm i z i m i m 1 δm Ω δj ÿ i M 2ΩJ ÿ i A κ Ω z m δm Ω δj κ δa`z δm 8π M 2ΩJ κa 4π ` zm

23 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

24 Particle Hamiltonian first law Geodesic motion of test mass m in Kerr geometry ḡ ab derives from canonical Hamiltonian Hpx µ, p µ q 1 2m ḡ ab pxqp a p b Hamilton-Jacobi equation completely separable [Carter (1968)] Canonical transformation px µ, p µ q Ñ pq α, J α q to generalized action-angle variables [Schmidt (2002); Hinderer & Flanagan (2008)] dj α dτ B H Bq α 0, dq α dτ B H BJ α ω α Varying HpJ α q yields a particle Hamiltonian first law valid for generic bound orbits [Le Tiec (2014)] δe Ω ϕ δl ` Ω r δj r ` Ω θ δj θ ` xzy δm

25 Particle Hamiltonian first law Geodesic motion of test mass m in Kerr geometry ḡ ab derives from canonical Hamiltonian Hpx µ, p µ q 1 2m ḡ ab pxqp a p b Hamilton-Jacobi equation completely separable [Carter (1968)] Canonical transformation px µ, p µ q Ñ pq α, J α q to generalized action-angle variables [Schmidt (2002); Hinderer & Flanagan (2008)] dj α dτ B H Bq α 0, dq α dτ B H BJ α ω α Varying HpJ α q yields a particle Hamiltonian first law valid for generic bound orbits [Le Tiec (2014)] δe Ω δl ` z δm

26 Inclusion of conservative GSF effects [Isoyama et al. (in preparation)] Geodesic motion of self-gravitating mass m in time-symmetric regular metric g ab ` hab R derives from canonical Hamiltonian Hrx µ, p µ ; γs Hpx µ, p µ q ` H int rx µ, p µ ; γs It is still possible to perform a canonical transformation px µ, p µ q Ñ pq α, J α q to generalized action-angle variables Varying HpJ α q yields a first law valid for generic bound orbits δe Ω ϕ δl ` Ω r δj r ` Ω θ δj θ ` xzy δm The actions J α, fundamental frequencies Ω α, and averaged redshift xzy include conservative GSF corrections from H int

27 Inclusion of conservative GSF effects Discussed by Tanaka [Isoyama et al. (in preparation)] Geodesic motion of self-gravitating mass m in time-symmetric regular metric g ab ` hab R derives from canonical Hamiltonian Hrx µ, p µ ; γs Hpx µ, p µ q ` H int rx µ, p µ ; γs It is still possible to perform a canonical transformation px µ, p µ q Ñ pq α, J α q to generalized action-angle variables Varying HpJ α q yields a first law valid for generic bound orbits δe Ω ϕ δl ` Ω r δj r ` Ω θ δj θ ` xzy δm The actions J α, fundamental frequencies Ω α, and averaged redshift xzy include conservative GSF corrections from H int

28 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

29 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

30 ADM mass, Bondi mass, binding energy Conservation of mass-energy i + M B (u 1 ) M ADM M B puq ` ż u 8 Fpu 1 q du 1 u = u 1 M B (u 0 ) Bondi-Sachs mass loss formula dm B du Fpuq E(t 1 ) m,m t = t 1 u = u 0 t = t 0 i 0 Binding energy of the binary source M ADM Eptq M B puq pm ` mq E(t 0 ) i

31 Binding energy beyond the test-mass limit [Le Tiec, Barausse & Buonanno (2012)] The binding energy E is a function of x rpm ` mqωs 2{3 In the extreme mass ratio limit q m{m! 1, z? 1 3x ` q z GSF pxq ` Opq 2 q ˆ E 1 2x µ? 1 ` q E GSF pxq ` Opq 2 q 1 3x The exact conservative self-force effect z GSF pxq is known [Detweiler (2008); Shah et al. (2011); Akcay et al. (2012); etc] The first law provides a relationship E Ø z, which implies E GSF pxq 1 2 z GSFpxq x 3 z1 GSFpxq ` f pxq A similar result holds for the total angular momentum J

32 Binding energy vs angular momentum [Le Tiec, Barausse & Buonanno (2012)] δe/μ E/μ GSFq GSFν NR GSFν EOB(3PN) 3PN GSFq GSFν NR Schw 3PN PN EOB(3PN) Schw J/(mμ)

33 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

34 Innermost stable circular orbit (ISCO) 0.96 L 2 = V 0.9 9r 2 E V pr; Lq r / M

35 Innermost stable circular orbit (ISCO) 0.96 L 2 = 15 L 2 = V 0.9 9r 2 E V pr; Lq r / M

36 Innermost stable circular orbit (ISCO) 0.96 L 2 = 15 L 2 = 14 L 2 = V 0.9 9r 2 E V pr; Lq r / M

37 Innermost stable circular orbit (ISCO) L 2 = 15 L 2 = 14 L 2 = 13 L 2 = 12 V 0.9 9r 2 E V pr; Lq r / M

38 Innermost stable circular orbit (ISCO) The innermost stable circular orbit is identified by a vanishing restoring radial force under small-e perturbations: B 2 H Br 2 0 ÝÑ Ω ISCO The minimum energy circular orbit is the most bound orbit along a sequence of circular orbits: BE BΩ 0 ÝÑ Ω MECO ISCO m For Hamiltonian systems [Buonanno et al. (2003)] M,S Ω ISCO Ω MECO no stable circular orbit

39 Kerr ISCO frequency vs black hole spin [Bardeen, Press & Teukolsky (1972)] MΩ Kerr ISCO χ = S / M 2

40 Kerr ISCO frequency vs black hole spin [Bardeen, Press & Teukolsky (1972)] 0.5 MΩ Kerr ISCO Astrophysical relevance: measure of a black hole spin from the inner edge of its accretion disk χ = S / M 2

41 Frequency shift of the Kerr ISCO test mass result [Isoyama et al. (2014)] The orbital frequency of the Kerr ISCO is shifted under the effect of the conservative self-force: " * pm ` mqω ISCO looooomooooon MΩ Kerr ISCO pχq 1 ` looomooon q C Ω pχq ` Opq 2 q conservative GSF effect The frequency shift can be computed from a stability analysis of slightly eccentric orbits near the Kerr ISCO Combining the Hamiltonian first law with the MECO conditio BE{BΩ 0 yields the same result: C Ω 1 2 z 2 GSF pωkerr ISCO q E 2 pω Kerr ISCO q

42 ISCO frequency shift vs black hole spin [Isoyama et al. (2014)] method (i) method (ii) 1.32 C Ω χ = S / M 2

43 ISCO frequency shift vs black hole spin [Isoyama et al. (2014)] [Barack & Sago (2009)] [Le Tiec et al. (2012)]

44 ISCO frequency shift vs black hole spin [Isoyama et al. (2014)] PN (ISCO) PN (MECO) EOB (ISCO) δc Ω /C Ω χ = S / M 2

45 ISCO frequency shift vs black hole spin [Isoyama et al. (2014)] PN (ISCO) PN (MECO) EOB (ISCO) δc Ω /C Ω Strong-field benchmark χ = S / M 2

46 Outline 1 The laws of binary black hole mechanics In general relativity In post-newtonian theory In black hole perturbation theory First laws for generic bound orbits 2 Applications Energy and angular momentum Kerr ISCO frequency shift Horizon surface gravity

47 Surface gravity and redshift observable [Blanchet, Buonanno & Le Tiec (2013)] First law for corotating black holes δm ADM Ω δj ` ÿ p4µ i κ i q δµ i i First law for corotating point particles δm ADM Ω δj ` ÿ z i c i δµ i Analogy between BH surface gravity and particle redshift 4µ i κ i ÐÑ z i c i κ i i z i μ i New invariant relations for NR/BHP/PN comparison: κ i pωq

48 Surface gravity vs orbital frequency [Grandclément & Le Tiec (work in progress)] PN BHP CFC 4PN BHP CFC 0.2 µ i κ i q = 1/ (µ 1 + µ 2 )Ω

49 Surface gravity vs orbital frequency [Grandclément & Le Tiec (work in progress)] PN CFC 4PN CFC µ i κ i q = 1/ (µ 1 + µ 2 )Ω

50 Surface gravity vs orbital frequency [Grandclément & Le Tiec (work in progress)] 0.22 N 1PN 2PN 3PN 4PN CFC 0.21 µ i κ i 0.2 q = (µ 1 + µ 2 )Ω

51 Summary and prospects The classical laws of black hole mechanics can be extended to binary systems of compact objects First laws of mechanics come in a variety of different forms: Context: exact GR, perturbation theory, PN theory Objects: black holes, point particles Orbits: circular, generic bound Derivation: geometric, Hamiltonian Combined with the first law, the redshift zpωq provides crucial information about the binary dynamics: Binding energy E, total angular momentum J Innermost stable circular orbit frequency Ω ISCO Horizon surface gravity κ

52 Summary and prospects Exploit the Hamiltonian first law for a particle in Kerr: Innermost spherical orbit Marginally bound orbits Extend PN Hamiltonian first law for two spinning particles: Non-aligned spins and generic precessing orbits Contribution from quadrupole moments Explore the surface gravity in corotating black holes: Compute Dκ from a direct analysis of h ret ab Perturbative prediction for κ with q Ñ ν Redshift at second order Ñ Opq 2 q corrections in EpΩq, JpΩq