Thermodynamics of a Black Hole with Moon

Thermodynamics of a Black Hole with Moon Laboratoire Univers et Théories Observatoire de Paris / CNRS In collaboration with Sam Gralla Phys. Rev. D 88 (2013) 044021

Outline ➀ Mechanics and thermodynamics of stationary black holes ➁ Mechanics of a black hole with a corotating moon ➂ Surface area, angular velocity and surface gravity

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 J Black holes have no hair. (J. A. Wheeler) Black hole event horizon characterized by: Angular velocity ω H Surface gravity κ Surface area A A κ ω H r + M,J

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

Analogy with the laws of thermodynamics [Bardeen, Carter & Hawking (1973)] Black Hole (BH) Thermo. System Zeroth law κ const. T const. First law δm ω H δj ` κ 8π δa Second law δa ě 0 δs ě 0 δe δw ` T δs Black holes should have an entropy S BH 9 A [Bekenstein (1973)] Analogy suggests stationary BHs have temperature T BH 9 κ However BHs are perfect absorbers, so classically T BH 0

Thermodynamics of stationary black holes [Hawking (1975)] Quantum fields in a classical curved background spacetime Stationary black holes radiate particles at the temperature T H 2π κ T H ω H Thus the entropy of any black hole is given by S BH A{4 Key results for the search of a quantum theory of gravity: string theory, LQG, emergent gravity, etc.

Going beyond stationarity and axisymmetry A fully general theory of radiating black holes is called for However, even the classical notion of surface gravity for a dynamical black hole is problematic [Nielsen & Yoon (2008)] Main difficulty: lack of a horizon Killing field, a Killing field tangent to the null geodesic generators of the event horizon Objective: explore the mechanics and thermodynamics of a dynamical and interacting black hole Main tool: black hole perturbation theory

Outline ➀ Mechanics and thermodynamics of stationary black holes ➁ Mechanics of a black hole with a corotating moon ➂ Surface area, angular velocity and surface gravity

Rotating black hole + orbiting moon Kerr black hole of mass M and spin J perturbed by a moon of mass m! M: g ab pλq ḡ ab ` λ Dg ab ` Opλ 2 q M,J ω H e,j m Perturbation Dg ab obeys the linearized Einstein equation with point-particle source [Gralla & Wald (2008)] 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. j m φ a u a Physical Dg ab : retarded solution, no incoming radiation, perturbations DM e and DJ j [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 Opλ 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 jq Σ 2π/ω H γ u a χ a

Zeroth law for a black hole with moon 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 pλq t a ` p ω H ` λ Dω H q φ a ` Opλ 2 q The surface gravity κ is defined in the usual manner as κ 2 1 2 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

Zeroth law for a black hole with moon 2π/ω H κ k a γ u a k a Σ H

Smarr formula for a black hole with moon For any spacetime with a Killing field k a, Stokes theorem yields the following identity (with Q ab ε abcd c k d ): ż ż Q ab 2 ε abcd R de k e BΣ Σ Applied to a given BH with moon spacetime, this gives the formula M 2ω H J ` κa 4π ` mz In the limit λ Ñ 0, we recover Smarr s formula [Smarr (1973)] B H H k a i + I + k a Σ k a S i γ u a i 0

First law for a black hole with moon Adapting [Iyer & Wald (1994)] to non-vacuum perturbations of non-stationary, non-axisymmetric spacetimes we find: 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 ω H 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 i γ u a i 0

Outline ➀ Mechanics and thermodynamics of stationary black holes ➁ Mechanics of a black hole with a corotating moon ➂ Surface area, angular velocity and surface gravity

Perturbation in horizon surface area A κ ω H add moon ω H +Dω H e,j κ+dκ A+DA m Application of the first law to this perturbation gives DM ω H DJ ` κ 8π DA ` m z

Perturbation in horizon surface area A κ ω H add moon ω H +Dω H e,j κ+dκ A+DA m Application of the first law to this perturbation gives e ω H j ` κ 8π DA ` pe ω H jq

Perturbation in horizon surface area A κ ω H add moon ω H +Dω H e,j κ+dκ A+DA m Application of the first law to this perturbation gives e ω H j ` κ 8π DA ` pe ω H jq The perturbation in horizon surface area vanishes: DA 0

Particle Hamiltonian first law Geodesic motion of test mass m in Kerr geometry ḡ ab px; M, Jq derives from canonical Hamiltonian [Carter (1968)] Hpy, pq 1 2 ḡ ab py; M, Jq p a p b Varying Hpy, pq yields the Hamiltonian first law [Le Tiec (2013)] δe ω H δj ` z δm ` z `B m M H δ M ` B J H δ J Combining both first laws, we find for the perturbations in horizon angular velocity and surface gravity Dω H z m p ω H B MH ` B JHq and Dκ z m κ B MH

0.5 Perturbation in horizon angular velocity 0.4 0.3 0.2 last stable corotating orbit ω H M 0.1 0 Dω H M 2 / m 0 0.2 0.4 0.6 0.8 1 χ = J / M 2

Perturbation in horizon surface gravity 0.4 0.2 0-0.2-0.4-0.6-0.8 last stable corotating orbit κ M Dκ M 2 / m 0 0.2 0.4 0.6 0.8 1 χ = J / M 2

Cooling a black hole with an orbiting moon Key properties of semi-classical calculation in Kerr preserved: k a t a ` ω H φ a is infinitesimally related to χ a k a is normalized so that k a t a 1 at infinity t a is an asymptotic time translation symmetry The constant surface gravity κ should thus correspond to the Hawking temperature T H of the perturbed black hole: DT H 2π Dκ ă 0 The moon has a cooling effect on the rotating black hole! Both rotationally-induced and tidally-induced deformations of a BH horizon have a cooling effect ÝÑ generic result?

Physical process version ➀ Set moon on circular trajectory with angular frequency ω H ω H ➁ Lower moon adiabatically down corotating cylinder ➂ Release moon on corotating orbit and retract tensile rod ω H m

Main results Summary and prospects We established a zeroth law and a first law of black hole mechanics valid beyond stationarity and axisymmetry We studied a realistic, dynamical and interacting black hole whose surface gravity is well-defined The moon has a cooling effect on the rotating black hole! Future work Recover our results from a direct analysis of Dg ab near H Semi-classical calculation confirming that T H κ{2π Recover our results from a microscopic point of view

Additional Material

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

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 δpdm b 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

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 (reaction) can be turned off (neglected): Conformal Flatness Condition Post-Newtonian approximation Linear perturbation theory For asymptotically flat spacetimes: k a Ñ t a ` Ω φ a and δq δm ADM Ω δj