Horizon Surface Gravity in Black Hole Binaries

Transcription

1 Horizon Surface Gravity in Black Hole Binaries, Philippe Grandclément Laboratoire Univers et Théories Observatoire de Paris / CNRS gr-qc/ A 1 Ω κ 2 z 2 Ω Ω m 1 κ A z m

2 Black hole uniqueness theorem in GR [Israel 1967; Carter 1971; Hawking 1973; Robinson 1975] ˆ 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) ˆ 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, Carter & Hawking 1973] ˆ Zeroth law of mechanics: ω H κ = const. (on H) ˆ First law of mechanics: A κ r + M,S δm = ω H δs + κ 8π δa A 1 ˆ Second law of mechanics: H A 3 A 1 +A 2 δa 0 A 2 time

4 What is the horizon surface gravity?

5 What is the horizon surface gravity? ˆ For an event horizon H generated by a Killing field k α, κ ( α k β β k α ) H

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

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

8 First laws of binary mechanics κ A ω H δm = ω H δs + κ 8π δa test [Bardeen et al. 1973] A 1 Ω κ 2 δm = Ω δj + a κ a 8π δa a [Friedman et al. 2002] Ω z 2 δm = Ω δj + a z a δm a [Le Tiec et al. 2012] [Blanchet et al. 2013] m 1 A κ Ω z m δm = Ω δj + κ δa + z δm 8π [Gralla & Le Tiec 2013]

9 Surface gravity and redshift [Pound 2015 (unpublished)] (Credit: Zimmerman, Lewis & Pfeiffer 2016)

10 Source modelling for compact binaries log 10 (r /m) 4 1 (compactness) post Newtonian theory numerical relativity perturbation theory & self force mass ratio 4 log 10 (m 2 /m 1 )

11 Source modelling for compact binaries m 2 log 10 (r /m) r m 1 v 2 c 2 Gm c 2 r (compactness) post Newtonian theory numerical relativity perturbation theory & self force mass ratio 4 log 10 (m 2 /m 1 )

12 Source modelling for compact binaries log 10 (r /m) q m 1 m m 2 1 (compactness) post Newtonian theory numerical relativity perturbation theory & self force m mass ratio 4 log 10 (m 2 /m 1 )

13 Source modelling for compact binaries log 10 (r /m) 4 m 2 1 (compactness) post Newtonian theory numerical relativity perturbation theory & self force m mass ratio 4 log 10 (m 2 /m 1 )

14 ˆ 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 )

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

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

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

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

19 Quasi-equilibrium initial data ˆ 3+1 decomposition of the metric: ds 2 = N 2 dt 2 ( + γ ij dx i + N i dt ) ( dx j + N j dt ) ˆ Conformal flatness condition approximation: γ ij = Ψ 4 f ij + h ij ˆ Assume exact helical Killing symmetry: L k g αβ = 0 with k α = ( t ) α + Ω ( φ ) α ˆ Solve five elliptic equations for (N, N i, Ψ) ˆ Determine orbital frequency Ω by imposing M ADM = M Komar ˆ Impose vanishing linear momentum to find rotation axis

20 Surface gravity for mass ratio 2 : 1

21 Surface gravity for mass ratio 2 : 1

22 Variations in horizon surface gravity 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 ) :1 2:1 1: µω

23 Surface gravity vs orbital frequency 0.95 q µ = ICO 4µ a κ a N 1PN 2PN 3PN 4PN CFC µω

24 Surface gravity vs orbital frequency PN 4PN CFC 3PN 4PN CFC 4µ a κ a ICO q µ = 1/ µω

25 Surface gravity vs orbital frequency q µ = 1/ µ a κ a ICO CFC 4PN BHP CFC 4PN BHP µω

26 Surface gravity vs orbital frequency µ a κ a :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: µω

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

28 Summary ˆ The celebrated laws of black hole (BH) mechanics have been extended to binary BH systems ˆ In corotating binaries, the surface gravity κ a is constant ˆ We computed κ a (Ω) from quasi-equilibrium initial data for corotating BH binaries with comparable masses ˆ We compared those numerical results to the analytical predictions from the PN approximation and linear BH perturbation theory and found excellent agreement Prospects ˆ Perturbation theory may prove useful to build templates for IMRIs and even comparable-mass binaries

29 Additional Material

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

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

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 1 : 1 3PN PN EOB(3PN) Schw J/(mμ)

33 Periastron advance vs orbital frequency [Le Tiec, Mroué et al. 2011] K :1 ν = m 1 m 2 /m 2 q = m 1 /m 2 Schw EOB GSFν PN GSFq δk/k mω ϕ

34 Periastron advance vs mass ratio [Le Tiec, Mroué et al. 2011] 0 δk/k EOB GSFν PN Schw GSFq q

35 Waveform from head-on collision [Sperhake, Cardoso et al. 2011] ψ 20 / η : 100 PP limit L/M = 9.58 L/M = PP limit L/M = 9.58 L/M = ψ 30 / η t / M

36 Waveform from head-on collision [Sperhake, Cardoso et al. 2011] ψ 20 / η : 10 PP limit L/M = L/M = PP limit L/M = L/M = ψ 30 / η t / M

37 Recoil velocity vs symmetric mass ratio [Nagar 2013]

38 Self-force from numerical relativity [Le Tiec, Buonanno et al. 2013] 0.06 χ 1 = χ 2 = 0 q = 1 q = 1.5 W NR -W 0.04 q = 3 q = 5 q = mω ϕ

39 Self-force from numerical relativity [Le Tiec, Buonanno et al. 2013] (W NR -W) / ν (W NR -W) / ν χ 1 = χ 2 = mω ϕ

40 Self-force from numerical relativity [Le Tiec, Buonanno et al. 2013] (W NR -W) / ν GSF (W NR -W) / ν χ 1 = χ 2 = 0 GSF mω ϕ

41 Self-force from numerical relativity [Le Tiec, Buonanno et al. 2013] 0.04 q = 1.5 q = 3 q = W NR -W 0.02 q = χ 1 = 0.5, χ 2 = mω ϕ

42 Self-force from numerical relativity [Le Tiec, Buonanno et al. 2013] (W NR -W) / ν (W NR -W) / ν χ 1 = 0.5, χ 2 = mω ϕ

43 Prediction confirmed! [van de Meent 2017] Δ NR NR err Δ PN O(η) Periapsis shift ρ GSF ρ NR ρ PN (m 1 +m 2 )Ω ϕ