Near-horizon extreme Kerr magnetospheres Near-horizon extreme Kerr magnetospheres Roberto Oliveri Universit´ e Libre de Bruxelles V Postgraduate Meeting On Theoretical Physics 17th November 2016 [mainly based on hep-th:1509.07637 with G. Comp` ere, w/ ref to hep-th:1602.01833 by Lupsasca, Gralla, Strominger and astro-ph:1401.6159 by Gralla, Jacobson]
Near-horizon extreme Kerr magnetospheres Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres Introduction and motivation Physical setup Active galactic nuclei (AGN) are the brightest regions at the center of a galaxy. Spinning supermassive black holes are believed to be hosted at the center of the AGN. AGN is a wonderful playground of high-energy physics phenomena in strong gravity regime, yet to be fully understood. Among them: 1. matter accretion onto the black hole; 2. collimated jets; 3. magnetospheres with different field lines topologies: radial, vertical, parabolic, hyperbolic.
Near-horizon extreme Kerr magnetospheres Introduction and motivation Theoretical setup Some important facts to know: ◮ a rotating black hole immersed in an external magnetic field induces an F µν F µν � = 0 [Wald (’74)]; electric field with Lorentz invariant ˜ ◮ a pair-production mechanism operates to produce a plasma-filled F µν F µν = 0; magnetosphere until ˜ ◮ the magnetosphere is force-free. It means that the plasma rest-mass density is negligible with respect to the electromagnetic energy density; ◮ force-free magnetosphere extracts electromagnetically energy and angular momentum from the rotating black hole [Blandford, Znajek (’77)].
Near-horizon extreme Kerr magnetospheres Force-free electrodynamics (FFE) Derivation of FFE equations Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres Force-free electrodynamics (FFE) Derivation of FFE equations Derivation of FFE equations (1) Let g µν be the background spacetime metric and A µ be the gauge potential. The Maxwell field is F µν = ∇ µ A ν − ∇ ν A µ . It obeys Maxwell’s equations: ∇ ν F µν = j µ ∇ [ σ F µν ] = 0 , with j µ being the electric current density. The full energy-momentum tensor is T µν = T µν em + T µν matter Assumption 1 : we neglect any backreaction to the spacetime geometry. The energy-momentum conservation 0 = ∇ ν T µν em + ∇ ν T µν matter = − F µν j ν + ∇ ν T µν matter , governs the transfer of energy and momentum between the electromagnetic field and the matter content.
Near-horizon extreme Kerr magnetospheres Force-free electrodynamics (FFE) Derivation of FFE equations Derivation of FFE equations (2) Assumption 2 : the exchange of energy and momentum from the EM field and the matter is negligible. Then, energy-momentum conservation implies that F µν j ν = 0 , the Lorentz force density is zero. Thus, FFE equations are ∇ ν F µν = j µ , F µν j ν = 0 , ∇ [ σ F µν ] = 0 , or, eliminating j µ , F µν ∇ σ F νσ = 0 . ∇ [ σ F µν ] = 0 ,
Near-horizon extreme Kerr magnetospheres Force-free electrodynamics (FFE) Some properties Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres Force-free electrodynamics (FFE) Some properties Some properties FFE equations: ∇ ν F µν = j µ , F µν j ν = 0 , ∇ [ σ F µν ] = 0 , or, in differential form, dF = 0 , d ⋆ F = ⋆ J , J ∧ ⋆ F = 0 , 1. Any vacuum Maxwell solution ( j µ = 0) is trivially force-free; 2. Assume j µ � = 0. Because F µν j ν = 0, then F [ µν F σρ ] j ρ = 0 ⇒ F [ µν F σρ ] = 0 ( F ∧ F = 0) In other words, force-free fields are degenerate: F µν = α µ β ν − α ν β µ ; 3. FFE is nonlinear ⇒ no general superposition principle. A sufficient condition for linear superposition of two solutions F 1 and F 2 is to have collinear currents J 1 ∝ J 2 , up to an arbitrary function.
Near-horizon extreme Kerr magnetospheres Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties NHEK metric and properties NHEK spacetime describes the region near the horizon of the extreme Kerr. It can be derived from extreme Kerr ( a = M ) metric, performing the scaling λ R → r − M t T → 2 M t , λ M , Φ → φ − 2 M ; In Poincar´ e coordinates, NHEK metric reads − R 2 dT 2 + dR 2 � � ds 2 = 2 M 2 Γ( θ ) R 2 + d θ 2 + γ 2 ( θ )( d Φ + RdT ) 2 , Its main properties are: 1. it has an enhanced isometry group SL (2 , R ) × U (1) generated by: √ Q 0 = ∂ Φ , H + = 2 ∂ T , H 0 = T ∂ T − R ∂ R , √ � 1 � T 2 + 1 � ∂ T − TR ∂ R − 1 � H − = 2 R ∂ Φ , 2 R 2 obeying [ H 0 , H ± ] = ∓ H ± , [ H + , H − ] = 2 H 0 , [ Q 0 , H ± ] = 0 = [ Q 0 , H 0 ]. 2. it has no globally timelike Killing vectors. ∂ T is timelike for γ 2 ( θ ) < 1 and becomes null at the velocity of light surface γ 2 ( θ ) = 1.
Near-horizon extreme Kerr magnetospheres FFE around NHEK Defining the problem Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres FFE around NHEK Defining the problem Defining the problem We want to find solutions to FFE equations around NHEK spacetime, dF = 0 , d ⋆ F = ⋆ J , J ∧ ⋆ F = 0 , further obeying the highest-weight (HW) conditions: L H + F = 0 , L H 0 F = hF , L Q 0 F = iqF , where h ∈ C is the weight of F , while q ∈ Z is the U (1)-charge.
Near-horizon extreme Kerr magnetospheres FFE around NHEK Solving FFE around NHEK Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres FFE around NHEK Solving FFE around NHEK Solving FFE around NHEK 1. Define real SL (2 , R ) covariant basis for: 1-form µ i , such that L H + µ i = 0 = L H 0 µ i , and 2-form w j , such that L H + w j = 0, L H 0 w j = w j ; 2. Consider A , F and J in the HW representation and expand them: A ( h , q ) = Φ ( h , q ) a i ( θ ) µ i , F ( h , q ) = Φ ( h − 1 , q ) f i ( θ ) w i , J ( h , q ) = Φ ( h , q ) j i ( θ ) µ i , where H + Φ ( h , q ) = 0 = ∂ θ Φ ( h , q ) , H 0 Φ ( h , q ) = h Φ ( h , q ) , Q 0 Φ ( h , q ) = iq Φ ( h , q ) . Maxwell’s equations constraint the functions f i and j i in terms of a i . 3. Fix the gauge a 4 = 0, ∀ h . 4. Rewrite the force-free condition J ∧ ⋆ F = 0 to get three nonlinear ODEs in terms of a 1 , a 2 , a 3 . 5. Classify solutions according to their HW representation labeled by ( h , q ).
Near-horizon extreme Kerr magnetospheres FFE around NHEK Potentially physical solutions Outline Introduction and motivation Force-free electrodynamics (FFE) Derivation of FFE equations Some properties Near-horizon extreme Kerr (NHEK) geometry NHEK metric and properties FFE around NHEK Defining the problem Solving FFE around NHEK Potentially physical solutions Two notable solutions Summary and conclusions
Near-horizon extreme Kerr magnetospheres FFE around NHEK Potentially physical solutions Potentially physical solutions (definition) Thus far, we have a list of complex (and therefore) unphysical solutions. A potentially physical solution must be 1. real; 2. magnetically dominated or null, i.e., we demand that the Lorentz scalar invariant ⋆ ( F ∧ ⋆ F ) = − 1 2 F µν F µν ≤ 0; 3. such that the energy and angular momentum flux densities to be finite E ≡ √− γ T µ ν n µ ( ∂ T ) ν ∝ E ( θ ) R 2 − 2 h , ˙ J ≡ √− γ T µ ν n µ ( ∂ Φ ) ν ∝ J ( θ ) R 1 − 2 h , ˙ (with n µ the unit normal and γ induced metric on constant R surface) 3.1. either at the spatial boundary of the NHEK spacetime, or 3.2. with respect to an asymptotically flat observer.
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