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 ?

Ruffini & Wilson 1975, Damour 1978, Blandford & Znajek 1977

T apping black-hole rotational energy by unipolar induction

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 = Pjet/Mc2 > 100%

.

Sądowski et al. (2013)

For

Penrose process

• timelike (at ∞ ) stationarity Killing vector

• 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

For a matter distribution or a nongravitational field

• beying the null energy condition, a necessary and

sufficient condition for energy extraction from a rotating black hole is that it absorbs negative energy ΔEH and negative angular momentum ΔJH .

Energy « gain »: can be positive, if and only if We refer to any such process as a Penrose process.

s

! "

!

!"#

!"#$#!%

!%

!&'($)

!"*#+,-.&#&-&/01#,2#.3-2&/4&56#7/38#!"9!%#:-5#!&'($)#,(##73;;3<2#(=:(##!>#?#)

!"#9#!%

"#*#+,-.&#&-&/01#,2#.3-2&/4&56#7/38#!"$!%#:-5#!&'(9)#,(##73;;3<2#(=:(##!>#$#@!&'(

!&'(9)

!"#

A#

;,0=(#.3-&2

!>#?)B

BC!>#$#@!&'(D

"# !"

Physical view

• Numerical view

Mechanical Penrose process

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

General electromagnetic field

Therefore the integrand in is: since

• pseudoelectric field 1-form on H

Hence

• r

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

( )

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

and is a pure gradient.

Force free case (Blandford-Znajek)

• electric 4-current. From stationarity

so there exists a function ω(Ψ) such that

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

Blandford-Znajek = Penrose

MAD SANE

(Magnetically Arrested discs) (Standard And Normal Evolution)

General Relativistic MagnetoHydroDynamics (GRMHD)

(McKinney, T chekhovskoy, Narayan, Blandford)

GRMHD HARM (Gammie, McKinney, Tóth 2003)

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.

Flux densities:

Energy-momentum tensor etc.

At horizon

Force-free

0.0 0.2 0.4 0.6 0.8 1.0 θH/π 0.0 0.5 1.0 1.5 2.0 Various, in units max(E2) ωHFµνEµξν − EµEµ E2 ≡ EµEµ 0.0 0.2 0.4 0.6 0.8 1.0 θH/π −1.0 −0.8 −0.6 −0.4 −0.2 0.0 Various, f/ max |TEM

µ νηµℓν|

TEM

µ νηµℓν

TEM

r t(r2 H + a2 cos2 θ)/(2mrH)

−ωHFµνEµξν + EµEµ 0.0 0.2 0.4 0.6 0.8 1.0 θH/π 0.1 0.2 0.3 0.4 0.5 0.6 ωF, in units of ωH 0.0 0.2 0.4 0.6 0.8 1.0 θH/π −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 Various, in units of max | ˙ e| ˙ e ˙ eEM ˙ eMA ωH ˙ ȷ

Force-free at horizon

MAD

0.0 0.2 0.4 0.6 0.8 1.0 θH/π 0.0 0.5 1.0 1.5 2.0 Various, in units max(E2) ωHFµνEµξν − EµEµ E2 ≡ EµEµ 0.0 0.2 0.4 0.6 0.8 1.0 θH/π −1.0 −0.8 −0.6 −0.4 −0.2 0.0 Various, f/ max |TEM

µ νηµℓν|

TEM

µ νηµℓν

TEM

r t(r2 H + a2 cos2 θ)/(2mrH)

−ωHFµνEµξν + EµEµ 0.0 0.2 0.4 0.6 0.8 1.0 θH/π −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Various, in units of max | ˙ e| ˙ e ˙ eEM ˙ eMA ωH ˙ ȷ

MAD at horizon

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.