The laws of binary black hole mechanics: An update

The laws of binary black hole mechanics: An update Laboratoire Univers et Théories Observatoire de Paris / CNRS A 1 Ω κ 2 z 2 Ω Ω m 1 κ A z m

The laws of black hole mechanics [Hawking 1972, Bardeen, Carter & Hawking 1973] Zeroth law of mechanics: ω H κ = const. (on 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 0 A 2 time

What is the horizon surface gravity?

What is the horizon surface gravity? For an event horizon H generated by a Killing field k α, i.e. for a Killing horizon: κ 2 1 2 ( α k β β k α ) H

What is the horizon surface gravity? For an event horizon H generated by a Killing field k α, i.e. for a Killing horizon: κ 2 1 2 ( α k β β k α ) H For a Schwarzschild black hole of mass M, this yields κ = 1 4M = GM R 2 S

Outline 1 Laws of binary black hole mechanics In general relativity In post-newtonian theory First laws for generic bound orbits 2 Applications Calibration of EOB models Horizon surface gravity

Generalized zeroth law of mechanics [Friedman, Uryū & Shibata 2002] Black hole spacetimes with helical Killing vector field k α On each component H i of the event horizon, the expansion and shear of the generators vanish Generalized rigidity theorem: H = i H i is a Killing horizon Constant horizon surface gravity κ 2 i = 1 2 ( α k β β k α ) Hi The binary black hole system is in a state of corotation

Generalized first law of mechanics [Friedman, Uryū & Shibata 2002] Spacetimes with black holes + perfect fluid matter sources One-parameter family of solutions {g αβ (λ), u α (λ), ρ(λ), s(λ)} Globally defined Killing field k α conseved Noether charge Q δq = i κ i 8π δa [ i + h δ(dmb ) + T δ(ds) + v α δ(dc α ) ] Σ

Issue of asymptotic flatness [Friedman, Uryū & Shibata 2002] Binaries on circular orbits have a helical Killing symmetry k α 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 α t α + Ω φ α and δq = δm ADM Ω δj

Application to black hole binaries [Friedman, Uryū & Shibata 2002] Rigidity theorem black holes are in a state of corotation Conformal flatness condition asymptotic flatness recovered preferred normalization of κ i [Le Tiec & Grandclément 2017] For binary black holes the generalized first law reduces to δm ADM = Ω δj+ i κ i 8π δa i A 1 Ω κ 2 r + M ADM,J

Outline 1 Laws of binary black hole mechanics In general relativity In post-newtonian theory First laws for generic bound orbits 2 Applications Calibration of EOB models Horizon surface gravity

First law for point-particle binaries [Le Tiec, Blanchet & Whiting 2012] For balls of dust, the generalized first law reduces to δq = z δ(dm b ) +, where z = k α u α Σ Conservative PN dynamics 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 + i z i δm i Ω z 2 m 1 r + MADM,J

Extension to spinning binaries [Blanchet, Buonanno & Le Tiec 2013] Canonical ADM Hamiltonian H(x i, p i, S i ; m i ) of two point particles with masses m i and spins S i [Steinhoff et al. 2008] Redshift observables and spin precession frequencies: H m i = z i and H S i = Ω i First law for aligned spins (J = L + i S i) and circular orbits: δm = Ω δl+ i (z i δm i + Ω i δs i ) Ω m 1,S 1 r + z 2,Ω 2 M ADM,L

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 /(4µ 2 i ) The first law of binary point-particle mechanics becomes δm = Ω δj + i [z i c i δµ i + (z i ω i + Ω i Ω) δs i ] Comparing with the first law for corotating black holes, δm = Ω δj + i (4µ iκ i ) δµ i, the corotation condition is z i ω i = Ω Ω i ω i (Ω) S i (Ω)

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 = i κ i 8π δa i M 2ΩJ = i κ i A i 4π Ω z 2 z i δm i z i m i m 1 δm Ω δj = i M 2ΩJ = i A κ Ω z m δm Ω δj = κ δa+z δm 8π M 2ΩJ = κa 4π + zm

Outline 1 Laws of binary black hole mechanics In general relativity In post-newtonian theory First laws for generic bound orbits 2 Applications Calibration of EOB models Horizon surface gravity

Averaged redshift for eccentric orbits Generic eccentric orbit parameterized by the two requencies t = 0 = 0 t = P = T m 1 Ω r = 2π P, Ω φ = Φ P Time average of redshift z = dτ/dt over one radial period m 2 P z 1 P 0 z(t) dt = T P

First law of mechanics for eccentric orbits [Le Tiec 2015, Blanchet & Le Tiec 2017] Canonical ADM Hamiltonian H(x i, p i ; m i ) of two point particles with constant masses m i

First law of mechanics for eccentric orbits [Le Tiec 2015, Blanchet & Le Tiec 2017] Canonical ADM Hamiltonian H(x i, p i ; m i ) of two point particles with constant masses m i Variation δh + Hamilton s equation + orbital averaging: δm = Ω φ δl + Ω r δr + i z i δm i First integral associated with the variational first law: M = 2 (Ω φ L + Ω r R) + i z i m i

Particle Hamiltonian first law Geodesic motion of test mass m in Kerr geometry ḡ αβ derives from canonical Hamiltonian H(x µ, p µ ) = 1 2m ḡ αβ (x)p α p β

Particle Hamiltonian first law Geodesic motion of test mass m in Kerr geometry ḡ αβ derives from canonical Hamiltonian H(x µ, p µ ) = 1 2m ḡ αβ (x)p α p β Hamilton-Jacobi equation completely separable [Carter 1968] Canonical transformation (x µ, p µ ) (q α, J α ) to generalized action-angle variables [Schmidt 2002, Hinderer & Flanagan 2008] dj α dτ = H q α = 0, dq α dτ = H J α ω α

Inclusion of conservative GSF effects [Fujita et al. 2017] Geodesic motion of self-gravitating mass m in time-symmetric regular metric ḡ αβ + hαβ R derives from canonical Hamiltonian H(x µ, p µ ; γ) = H(x µ, p µ ) + H int (x µ, p µ ; γ) In class of canonical gauges, one can define a unique effective Hamiltonian H(J) = H(J) + 1 2 H int (J) yielding a first law valid for generic bound orbits: δe = Ω φ δl + Ω r δj r + Ω θ δj θ + z δm

Outline 1 Laws of binary black hole mechanics In general relativity In post-newtonian theory First laws for generic bound orbits 2 Applications Calibration of EOB models Horizon surface gravity

Applications of the first law Fix ambiguity parameters in 4PN 2-body equations of motion [Jaranowski & Schäfer 2012, Damour et al. 2014, Bernard et al. 2016]

Applications of the first law Fix ambiguity parameters in 4PN 2-body equations of motion [Jaranowski & Schäfer 2012, Damour et al. 2014, Bernard et al. 2016] Compute GSF contributions to energy and angular momentum [Le Tiec, Barausse & Buonanno 2012] Calculate Schwarzschild and Kerr ISCO frequency shifts [Le Tiec et al. 2012, Akcay et al. 2012, Isoyama et al. 2014] Test cosmic censorship conjecture including GSF effects [Colleoni & Barack 2015, Colleoni et al. 2015] Calibrate Effective One-Body potentials [Barausse et al. 2012, Akcay & van de Meent 2016, Bini et al. 2016]

Applications of the first law Calibrate Effective One-Body potentials [Barausse et al. 2012, Akcay & van de Meent 2016, Bini et al. 2016] Compare particle redshift to black hole surface gravity [Zimmerman, Lewis & Pfeiffer 2016, Le Tiec & Grandclément 2017]

Outline 1 Laws of binary black hole mechanics In general relativity In post-newtonian theory First laws for generic bound orbits 2 Applications Calibration of EOB models Horizon surface gravity

EOB dynamics beyond circular motion [Le Tiec 2015] H real m 2 m 1 EOB m 1 + m 2 H (A,D,Q) eff Conservative EOB dynamics determined by potentials A = 1 2M/r + ν a(r) + D = 1 + ν d(r) + Q = ν q(r) p 4 r + Functions a(r), d(r) and q(r) controlled by z GSF (Ω r, Ω φ )

EOB dynamics beyond circular motion [Akcay & van de Meent 2016] d(v) 0.6 0.6 0.5 0.4 0.2 0.4 0.15 0.16 ISCO Numerical Δd ISCO (v) d ISCO (v) 6.5 d pn BDG (v) d Pade BDG (v) 0.01 0.05 0.1 0.15 ISCO v

EOB dynamics beyond circular motion [Akcay & van de Meent 2016] q(v) 0.45 0.4 0.3 0.3 0.2 0.15 0.125 0.15 0.16 ISCO Numerical Δq ISCO (v) 0.1 q ISCO (v) 4 pn q DJS(v) 0.01 0.05 0.1 0.15 ISCO v

EOB dynamics for spinning bodies [Bini, Damour & Geralico 2016] First law for spinning bodies GSF contribution to redshift } = SO coupling function δg S (u, â)

Outline 1 Laws of binary black hole mechanics In general relativity In post-newtonian theory First laws for generic bound orbits 2 Applications Calibration of EOB models Horizon surface gravity

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

Surface gravity and redshift [Pound 2015 (unpublished)] (Credit: Zimmerman, Lewis & Pfeiffer 2016)

Redshift vs orbital frequency [Zimmerman, Lewis & Pfeiffer 2016] / = / / = / / = / / = / / = / Ω

Redshift vs orbital frequency [Zimmerman, Lewis & Pfeiffer 2016] Δ z 10 3 Δ z 10 3 Δ z 10 3 2 0-2 - 4-6 2 0-2 - 4-6 2 0-2 - 4-6 m 1 /m =1/2 m 1 /m =5/7 m 2 /m =2/7 m 1 /m =7/9 m 2 /m =2/9 0.02 0.04 0.06 0.08 0.10 mω

3+1 decomposition of the metric: Quasi-equilibrium initial data ds 2 = N 2 dt 2 + γ ij ( dx i + N i dt ) ( dx j + N j dt )

Quasi-equilibrium initial data 3+1 decomposition of the metric: ds 2 = N 2 dt 2 ( + γ ij dx i + N i dt ) ( dx j + N j dt ) Conformal flatness condition approximation: γ ij = Ψ 4 f ij + h ij

Quasi-equilibrium initial data 3+1 decomposition of the metric: ds 2 = N 2 dt 2 ( + γ ij dx i + N i dt ) ( dx j + N j dt ) Conformal flatness condition approximation: γ ij = Ψ 4 f ij + h ij Assume exact helical Killing symmetry: L k g αβ = 0 with k α = ( t ) α + Ω ( φ ) α Solve five elliptic equations for (N, N i, Ψ)

Surface gravity for mass ratio 2 : 1 Surface gravity (arbitrary units)

Surface gravity for mass ratio 2 : 1 Relative variations in surface gravity

Variations in horizon surface gravity 0.048 0.04 0.032 µ 1 /µ = 1/11 µ 1 /µ = 1/3 µ 1 /µ = 1/2 µ 2 /µ = 2/3 µ 2 /µ = 10/11 σ (κa ) 0.024 0.016 10:1 2:1 1:1 0.008 0 0.035 0.07 0.105 0.14 0.175 0.21 µω

Surface gravity vs orbital frequency 0.95 q µ = 1 0.9 ICO 4µ a κ a 0.85 0.8 0.75 N 1PN 2PN 3PN 4PN CFC 0.02 0.04 0.06 0.08 0.1 0.12 µω

Surface gravity vs orbital frequency 1 0.9 0.8 3PN 4PN CFC 3PN 4PN CFC 4µ a κ a 0.7 0.6 0.5 ICO q µ = 1/2 0.4 0.06 0.09 0.12 0.15 0.18 0.21 0.24 µω

Surface gravity vs orbital frequency 1 0.9 q µ = 1/10 0.8 4µ a κ a 0.7 0.6 0.5 0.4 0.3 CFC 4PN BHP ICO CFC 4PN BHP 0.03 0.06 0.09 0.12 0.15 0.18 0.21 µω

Surface gravity vs orbital frequency 1 0.9 0.8 4µ a κ a 0.7 0.6 0.5 0.4 0.3 10:1 µ 1 /µ = 1/11 µ 1 /µ = 1/3 µ 1 /µ = 1/2 µ 2 /µ = 2/3 µ 2 /µ = 10/11 2:1 1:1 0.03 0.06 0.09 0.12 0.15 0.18 0.21 µω

Surface gravity vs orbital frequency 1 [Zimmerman, Lewis & Pfeiffer 2016] 0.9 0.8 4µ a κ a 0.7 0.6 0.5 0.4 0.3 10:1 µ 1 /µ = 1/11 µ 1 /µ = 1/3 µ 1 /µ = 1/2 µ 2 /µ = 2/3 µ 2 /µ = 10/11 2:1 1:1 0.03 0.06 0.09 0.12 0.15 0.18 0.21 µω

Perturbation theory for comparable masses log 10 (r /m) m 2 4 m 1 1 (compactness) 3 2 1 post Newtonian theory numerical relativity perturbation theory & self force 0 0 1 2 3 mass ratio 4 log 10 (m 2 /m 1 )

Summary The classical laws of BH 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 z(ω) provides crucial information about the binary dynamics: Binding energy E and total angular momentum J Innermost stable circular orbit frequency Ω ISCO EOB effective potentials A, D, Q, GS Horizon surface gravity κ

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 Redshift at second order O(q 2 ) corrections in E(Ω), J(Ω) Perturbation theory may prove useful to build templates for IMRIs and even comparable-mass binaries

Additional Material

Why does BHPT perform so well? In perturbation theory, one traditionally expands as k max f (Ω; m i ) = a k (m 2 Ω) q k where q m 1 /m 2 [0, 1] k=0 However, most physically interesting relationships f (Ω; m i ) are symmetric under exchange m 1 m 2 Hence, a better-motivated expansion is k max f (Ω; m i ) = b k (mω) ν k where ν m 1 m 2 /m 2 [0, 1/4] k=0 In a PN expansion, we have b n = O ( 1/c 2n) = npn +

Why does BHPT perform so well? In perturbation theory, each surface gravity is expanded as 4µ 1 κ 1 = a(µ 2 Ω) + q b(µ 2 Ω) + O(q 2 ) 4µ 2 κ 2 = c(µ 2 Ω) + q d(µ 2 Ω) + O(q 2 ) From the first law we know that the general form is 4µ i κ i = ν k f k (µω) ± 1 4ν ν k g k (µω) k 0 k 0 Each surface gravity can thus be rewritten as 4µ i κ i = A(µΩ) ± B(µΩ) 1 4ν + C(µΩ) ν ± D(µΩ) ν 1 4ν + O(ν 2 ) Expand to linear order in q and match A, B, C, D

Rotating black hole + orbiting moon Kerr black hole of mass M and spin S perturbed by a moon of mass m M: g ab (ε) = ḡ ab + ε Dg ab + O(ε 2 ) M,S ω H E,L m Perturbation Dg ab obeys the linearized Einstein equation with point-particle source DG ab = 8π DT ab = 8π m dτ δ 4 (x, y) 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

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 O(ε 2 ) and neglected 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 (E ω H L) Σ 2π/ω H γ u a χ a

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 (ε) = t a + ( ω ) H + ε Dω }{{ H φ } a + O(ε 2 ) circular orbit frequency Ω The surface gravity κ is defined in the usual manner as κ 2 = 1 2 ( a k b a k b ) H Since κ = const. over any Killing horizon [Bardeen et al. 1973], we have proven a zeroth law for the perturbed event horizon

Angular velocity vs black hole spin [Gralla & Le Tiec 2013] 0.5 0.4 0.3 0.2 0.1 last stable corotating orbit ω H M Dω H M 2 / m 0 0 0.2 0.4 0.6 0.8 1 χ = S / M 2

Surface gravity vs black hole spin [Gralla & Le Tiec 2013] 0.4 0.2 κ M 0-0.2-0.4-0.6-0.8 last stable corotating orbit Dκ M 2 / m 0 0.2 0.4 0.6 0.8 1 χ = S / M 2

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 ) (δq ab Θ abc k c ) = 2 δ ε abcd G de k e ε abcd k d G ef δg ef Σ 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