FIRST LAW OF BINARY BLACK HOLE DYNAMICS dedicated to the 60 th - PowerPoint PPT Presentation

FIRST LAW OF BINARY BLACK HOLE DYNAMICS dedicated to the 60 th birthday of Toshi Futamase, Hideo Kodama, Misao Sasaki Luc Blanchet Gravitation et Cosmologie ( G R ε C O ) Institut d’Astrophysique de Paris 15 novembre 2012 Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 1 / 34

Four Laws of Black Hole Dynamics ZEROTH LAW Surface gravity κ is constant over the horizon of a stationary black hole FIRST LAW Mass M and angular momentum J of BH rotation frequency change according to [Bardeen, Carter & Hawking 1973] ω H δM − ω H δJ = κ 8 π δ A A horizon SECOND LAW area In any physical process involving one or several κ surface gravity BHs with or without an environment [Hawking 1971] δA � 0 THIRD LAW It is impossible to achieve κ = 0 in any process Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 2 / 34

Four Laws of Black Hole Dynamics ZEROTH LAW Surface gravity κ is constant over the horizon of a stationary black hole FIRST LAW Mass M and angular momentum J of BH rotation frequency change according to [Christodoulou 1970, Smarr 1973] ω H M − 2 ω H J = κ 4 π A A horizon SECOND LAW area In any physical process involving one or several κ surface gravity BHs with or without an environment [Hawking 1971] δA � 0 THIRD LAW It is impossible to achieve κ = 0 in any process Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 2 / 34

Fourty Years of BH Thermodynamics [Bekenstein 1972, Hawking 1976] Using arguments involving a piece of matter with entropy thrown into a BH, Bekenstein derived the BH entropy S BH = α A κ This would require T BH = 8 πα but the thermodynamic temperature of a classical BH is absolute zero since a BH is a perfect absorber However Hawking proved that quantum particle creation effects near a BH κ result in a black body temperature T BH = 2 π . This leads to the famous Bekenstein-Hawking entropy of a stationary black hole S BH = c 3 k A � G 4 The analogy between BH dynamics and the laws of thermodynamics is complete Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 3 / 34

Toward a Generalized First Law for a System of BHs S r Σ Σ r H The mass and the angular momentum of the BH are given by Komar surface integrals at spatial infinity � − 1 ∇ µ t ν d S µν M = 8 π lim r →∞ S r � 1 ∇ µ φ ν d S µν J = 16 π lim r →∞ S r where t µ and φ µ are the two stationary and axi-symmetric Killing vectors Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 4 / 34

Toward a Generalized First Law for a System of BHs congruence The first law of BH dynamics expresses the change of horizon's generators K µ δQ = δM − ω H δJ in the Noether charge Q between two nearby BH configurations, where Q is associated with the Killing vector K µ = t µ + ω H φ µ which is the null generator of the BH horizon Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 5 / 34

Toward a Generalized First Law for a System of BHs A generalized First Law valid for systems of BHs can be obtained when the geometry admits a Helical Killing Vector (HKV) K µ ∂ µ = ∂ t + Ω ∂ ϕ where ∂ t is time-like and ∂ ϕ is space-like (with closed orbits), even when ∂ t and ∂ ϕ are not separately Killing vectors This applies to the case of two Kerr BHs moving on exactly circular orbits with orbital frequency Ω The two BHs should be in corotation, so that ω H should approximately be equal to Ω . In particular the spins should be aligned with the orbital angular momentum Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 6 / 34

Toward a Generalized First Law for a System of BHs L S 1 Ω ω H m S 1 2 CM ω H m 2 = ω Ω H Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 7 / 34

Toward a Generalized First Law for a System of BHs With the Helical Killing Vector K µ ∂ µ = ∂ t + Ω ∂ ϕ the change in the 1 associated Noether charge is given by δQ = δM − Ω δJ provided that the space-time is asymptotically flat [Friedman, Ury¯ u & Shibata 2002] However exact solutions of the Einstein field equations with Helical Killing 2 symmetry cannot be asymptotically flat since they are periodic which contradicts the decrease of the Bondi mass at J + [Gibbons & Stewart 1983] Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 8 / 34

Toward a Generalized First Law for a System of BHs + + J J standing waves I 0 I 0 at infinity no incoming radiation condition - - J J Situation with the HKV Physical situation Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 9 / 34

Looking at the Conservative Part of the Dynamics One way to deal with the problem is to look at approximate solutions which are asymptotically flat. A possible solution is to suppress radiation degrees of freedom by imposing a condition of conformal flatness for the spatial metric [Isenberg & Nester 1980; Wilson & Mathews 1989] Here we follow a different route which is to consider only the conservative part of the dynamics in a post-Newtonian (PN) expansion, neglecting the dissipative effects due to the emission of gravitational radiation Thus we derive the First Law for a class of conservative PN space-times admitting a HKV and describing point particles (possibly with spins) moving on an exactly circular orbit Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 10 / 34

Two Point Particles on an Exactly Circular Orbit µ u 1 µ K µ µ 1 K K light cylinder time space particle's trajectories Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 11 / 34

Conservative versus Dissipative Dynamics in PN theory Internal acceleration of a matter system is written as a formal PN expansion d v A N + 1 c 2 A 1PN + 1 c 4 A 2PN + 1 = c 5 A 2 . 5PN d t � 1 � + 1 c 6 A 3PN + 1 c 7 A 3 . 5PN + 1 c 8 A 4PN + O c 9 Naive split would be to say that conservative effects are those which carry an even power of 1 /c , while dissipative effects, linked to gravitational radiation reaction, are those which carry an odd power of 1 /c Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 12 / 34

Conservative versus Dissipative Dynamics in PN theory This is correct at leading 2.5PN order where the force derives from a scalar in an appropriate gauge, A 2 . 5PN = ∇ V 2 . 5PN with [Burke & Thorne] V 2 . 5PN ( x , t ) = − 1 5 x i x j I (5) ij ( t ) This term would change sign if we change the prescription of retarded potentials to the advanced potentials This is still correct at sub-leading order 3.5PN where the force involves both scalar and vector potentials given by [Blanchet 1997] 1 ijk ( t ) − 1 189 x i x j x k I (7) 70 x 2 x i x j I (7) V 3 . 5PN = ij ( t ) 1 jk ( t ) − 4 21 x � i x j x k � I (6) 45 ε ijk x j x l J (5) V i = kl ( t ) 3 . 5PN which also change sign from retarded to advanced potentials Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 13 / 34

Dissipative Tail Effect in the PN Dynamics However the naive split fails starting at 4PN order because of the appearance of tails in the radiation reaction force [Blanchet & Damour 1988] � t � t − t ′ � V 4PN = − 4 M d t ′ I (7) x i x j ij ( t ′ ) ln 5 2 r −∞ This term is not invariant when we go from retarded to advanced potentials Tail of GW Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 14 / 34

Logarithms at 4PN order in the Conservative Dynamics With the HKV we have at our disposal the binary’s orbital period P = 2 π/ Ω . We split � r � t − t ′ � � t − t ′ � � ln = − ln + ln 2 r P 2 P Tails produce a conservative 4PN logarithmic term � t � r � � t − t ′ � � V 4PN = − 4 M 2 � d t ′ I (7) − I (6) x i x j ij ( t ′ ) ln ij ( t ) ln + 5 P 2 P −∞ � �� � � �� � conservative 4PN log term dissipative term (neglected) We shall see appearing at 4PN and higher orders like 5PN some logarithmic contributions in the conservative part of the dynamics of binary black holes Luc Blanchet ( G R ε C O ) First law of binary black hole dynamics JGRG 2012, Tokyo 15 / 34