Extracting black-hole rotational energy: the generalized Penrose - PowerPoint PPT Presentation

Extracting black-hole rotational energy: the generalized Penrose process Jean-Pierre Lasota IAP & N.Copernicus Astronomical Center Based on Lasota, Gourgoulhon, Abramowicz, Tchekhovskoy & Narayan ; Phys. Rev. D 89, 024041 (2014) IHES, 6th of February 2014

Relativistic jets in Active Galactic Nuclei

Relativistic jets in compact binaries (microquasars)

Common source of energy ?

T apping black-hole rotational energy by unipolar induction Ruffini & Wilson 1975, Damour 1978, Blandford & Znajek 1977

Controversy: • is the BH surface an analogue of a Faraday disc (causality) • is the Blandford-Znajek mechanism efficient (rotation of black-hole or disc) ?

Recent (2011-2013) GRMHD simulations clearly showed BH rotational energy extraction in a particular (MAD) magnetic field configuration T chekhovskoy, McKinney, Blandford 2012

MAD simulation T chekhovskoy, McKinney, Narayan 2011

MAD BH Jet in MAD state has a large efficiency: . η = P jet /Mc 2 > 100%

S ą dowski et al. (2013)

Penrose process - timelike (at ∞ ) stationarity Killing vector For

- timelike (at ∞ ) stationarity Killing vector - spacelike axisymmetry Killing vector -ZAMO, Energy measured by ZAMOs always non-negative: . Hence for Since

Horizon

T - energy moment tensor - null energy condition • Energy conservation Noether current (« energy momentum density vector ») by Stoke’s theorem

*********************************************** angular-momentum density vector

Energy « gain »: can be positive, if and only if We refer to any such process as a Penrose process. For a matter distribution or a nongravitational field obeying the null energy condition, a necessary and sufficient condition for energy extraction from a rotating black hole is that it absorbs negative energy Δ E H and negative angular momentum Δ J H .

Physical view ;,0=(#.3-&2 ! "# $#! % ! "# 9#! % " s ! > #?) B ! &'( $) ! &'( 9) ! !"# !"# A # !" ! % "# ! BC! > #$#@! &'( D "# *#+,-.&#&-&/01#,2#.3-2&/4&56#7/38#! " $! %# :-5#! &'( 9)#,(##73;;3<2#(=:(##! > #$#@! &'( !" *#+,-.&#&-&/01#,2#.3-2&/4&56#7/38#! " 9! %# :-5#! &'( $)#,(##73;;3<2#(=:(##! > #?#)

Numerical view � �� ��� � � � � ����� ��� � ��� � ��� � � � ��������������������������������� � �� �� ����� ��� ���������������������� � ����� ���

Mechanical Penrose process

(possible only in the ergosphere) is collinear to so it is timelike and past-directed is negative. because

General electromagnetic field is: Therefore the integrand in since • pseudoelectric field 1-form on H

Hence or therefore if This is the most general condition on any electromagnetic field configuration allowing black-hole energy extraction through a Penrose process ( ) Since is tangent to H

Stationary and axisymmetric electromagnetic field therefore Φ , Ψ and I are gauge-invariant. Introducing a 1-form A such that F=dA one can choose A so that is a pure gradient. and

Force free case (Blandford-Znajek) - electric 4-current. From stationarity so there exists a function ω ( Ψ ) such that

One gets therefore on H and (Blandford & Znajek 1977)

Blandford-Znajek = Penrose

General Relativistic MagnetoHydroDynamics (GRMHD) GRMHD HARM ( Gammie, McKinney, Tóth 2003) SANE MAD (Magnetically Arrested discs) (Standard And Normal Evolution) (McKinney, T chekhovskoy, Narayan, Blandford)

Blandford-Znajek efficiency - time average, -normalized magnetic flux Magnetic flux can be accumulated only if the disc is not thin, h/r ~ 1. Here discs are slim, h/r ~ 0.3.

Energy-momentum tensor Flux densities: etc. At horizon

Force-free

Force-free at horizon 2 . 0 ω H F µ ν E µ ξ ν − E µ E µ µ ν η µ ℓ ν T EM 0 . 0 E 2 ≡ E µ E µ H + a 2 cos 2 θ ) / (2 mr H ) r t ( r 2 T EM ν η µ ℓ ν | Various , in units max( E 2 ) − ω H F µ ν E µ ξ ν + E µ E µ 1 . 5 − 0 . 2 µ Various , f/ max | T EM − 0 . 4 1 . 0 − 0 . 6 0 . 5 − 0 . 8 − 1 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 θ H / π θ H / π 0 . 5 0 . 6 0 . 0 e | Various , in units of max | ˙ 0 . 5 ω F , in units of ω H − 0 . 5 0 . 4 − 1 . 0 0 . 3 − 1 . 5 0 . 2 − 2 . 0 0 . 1 − 2 . 5 e EM e MA ˙ ˙ ω H ˙ ȷ e ˙ − 3 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 θ H / π θ H / π

MAD

MAD at horizon 2 . 0 µ ν η µ ℓ ν T EM ω H F µ ν E µ ξ ν − E µ E µ 0 . 0 H + a 2 cos 2 θ ) / (2 mr H ) r E 2 ≡ E µ E µ t ( r 2 T EM ν η µ ℓ ν | − ω H F µ ν E µ ξ ν + E µ E µ Various , in units max( E 2 ) 1 . 5 − 0 . 2 µ Various , f/ max | T EM − 0 . 4 1 . 0 − 0 . 6 0 . 5 − 0 . 8 − 1 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 θ H / π θ H / π 1 . 5 1 . 0 e | Various , in units of max | ˙ 0 . 5 0 . 0 − 0 . 5 − 1 . 0 e EM e MA ˙ ˙ ω H ˙ ȷ e ˙ − 1 . 5 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 θ H / π

Noether current in GRMHD MHD: Magnetic field vector Hence the energy-momentum tensor Noether current >0 in the ergosphere

Noether current: force-free

Noether current: MAD

Conclusions The Blandford-Znajek mechanism is rigorously a Penrose process. GRMHD simulations of Magnetically Arrested Discs correctly (from the point of view of general relativity) describe extraction of black-hole rotational energy through a Penrose process.