Thermodynamics of a Black Hole with Moon Alexandre Le Tiec - PowerPoint PPT Presentation

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

Stationary black holes Black hole with a corotating moon Perturbations 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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) ω H • Black hole event horizon characterized by: κ ˝ Angular velocity ω H r  ˝ Surface gravity κ + ¥ A ˝ Surface area A M,J IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations The laws of black hole mechanics [Hawking (1972); Bardeen et al. (1973)] ω H • Zeroth law of mechanics: κ κ “ const. r  +¥ A M,J • First law of mechanics: δ M “ ω H δ J ` κ 8 π δ A A 1 A 3 ≥ A 1 + A 2 • Second law of mechanics: H A 2 δ A ě 0 time IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Analogy with the laws of thermodynamics [Bardeen, Carter & Hawking (1973)] Black Hole (BH) Thermo. System Zeroth law κ “ const. T “ const. First law δ M “ ω H δ J ` κ 8 π δ A δ E “ δ W ` T δ S Second law δ A ě 0 δ S ě 0 • 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Thermodynamics of stationary black holes [Hawking (1975)] • Quantum fields in a classical curved background spacetime • Stationary black holes radiate ω H particles at the temperature T H T H “ � 2 π κ • 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. IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Rotating black hole + orbiting moon ω H • Kerr black hole of mass ¯ M and spin ¯ J perturbed by a moon of mass m ! ¯ e,j M : M,J g ab ` λ D g ab ` O p λ 2 q g ab p λ q “ ¯ m • Perturbation D g ab obeys the linearized Einstein equation with point-particle source [Gralla & Wald (2008)] ż D G ab “ 8 π D T ab “ 8 π m d τ δ 4 p x , y q u a u b γ • Particle has energy e “ ´ m t a u a and ang. mom. j “ m φ a u a • Physical D g ab : retarded solution, no incoming radiation, perturbations D M “ e and D J “ j [Keidl et al. (2010)] IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 p λ 2 q and neglected ë the spacetime geometry has a helical symmetry • In adapted coordinates, the γ helical Killing field reads u a χ a “ t a ` ¯ ω H φ a 2 π / ω H χ a • Conserved orbital quantity Σ associated with symmetry: z ” ´ χ a u a “ m ´ 1 p e ´ ¯ ω H j q IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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); Chru´ sciel (1997); Friedrich et al. (1999); etc] • The horizon-generating Killing field must be of the form k a p λ q “ t a ` p ¯ ω H ` λ D ω H q φ a ` O p λ 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Zeroth law for a black hole with moon γ κ k a u a 2 π / ω H k a Σ H IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 ε abcd R de k e Q ab “ 2 B Σ Σ i + • Applied to a given BH with moon I + H spacetime, this gives the formula u a k a M “ 2 ω H J ` κ A H k a S 4 π ` mz Σ B i 0 k a • In the limit λ Ñ 0, we recover γ Smarr’s formula [Smarr (1973)] i − IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 ε abcd k d G ef δ g ef p δ Q ab ´ Θ abc k c q “ 2 δ ε abcd G de k e ´ B Σ Σ Σ i + • Applied to nearby BH with moon I + spacetimes, this gives the first law H u a k a δ M “ ω H δ J ` κ H 8 π δ A ` z δ m k a S Σ B i 0 k a • Features variations of the Bondi γ mass and angular momentum i − IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations 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 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Perturbation in horizon surface area ω H ω H + D ω H e,j κ + D κ κ add moon A + D A A m • Application of the first law to this perturbation gives ω H D J ` ¯ κ D M “ ¯ 8 π D A ` m z IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Perturbation in horizon surface area ω H ω H + D ω H e,j κ + D κ κ add moon A + D A A m • Application of the first law to this perturbation gives ω H j ` ¯ κ e “ ¯ 8 π D A ` p e ´ ¯ ω H j q IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Perturbation in horizon surface area ω H ω H + D ω H e,j κ + D κ κ add moon A + D A A m • Application of the first law to this perturbation gives ω H j ` ¯ κ e “ ¯ 8 π D A ` p e ´ ¯ ω H j q • The perturbation in horizon surface area vanishes: D A “ 0 IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec

Stationary black holes Black hole with a corotating moon Perturbations Perturbation in horizon surface area ω H ω H + D ω H e,j κ + D κ κ add moon A + D A A m • Application of the first law to this perturbation gives ω H j ` ¯ κ e “ ¯ 8 π D A ` p e ´ ¯ ω H j q • The perturbation in horizon surface area vanishes: D A “ 0 • The black hole’s entropy is unaffected by the moon IH´ ES Gravity Seminar — October 23, 2014 Alexandre Le Tiec