Extracting black-hole rotational energy: the generalized Penrose process

Jean-Pierre LasotaIAP & 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

Tapping 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

Tchekhovskoy, McKinney, Blandford 2012

MAD simulation

Tchekhovskoy, 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 obeying 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.

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Physical view

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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

or

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

sothere 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, Tchekhovskoy, 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 1.0θH/π

0.0

0.5

1.0

1.5

2.0Various,in

unitsmax(E

2)

ωHFµνEµξν − EµEµ

E2≡ EµEµ

0.0 0.2 0.4 0.6 0.8 1.0θH/π

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

Various,f/max

|TEMµ νη µℓν|

TEMµνηµℓ

ν

TEMrt(r

2H+ a2 cos2 θ)/(2mrH)

−ωHFµνEµξν + EµEµ

0.0 0.2 0.4 0.6 0.8 1.0θH/π

0.1

0.2

0.3

0.4

0.5

0.6

ωF,in

unitsofωH

0.0 0.2 0.4 0.6 0.8 1.0θH/π

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

Various,

inunitsofmax|e|

e eEM eMA ωHȷ

Force-free at horizon

MAD

0.0 0.2 0.4 0.6 0.8 1.0θH/π

0.0

0.5

1.0

1.5

2.0Various,in

unitsmax(E

2)

ωHFµνEµξν − EµEµ

E2≡ EµEµ

0.0 0.2 0.4 0.6 0.8 1.0θH/π

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

Various,f/max

|TEMµ νη µℓν|

TEMµνηµℓ

ν

TEMrt(r

2H+ a2 cos2 θ)/(2mrH)

−ωHFµνEµξν + EµEµ

0.0 0.2 0.4 0.6 0.8 1.0θH/π

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Various,

inunitsofmax|e|

e eEM eMA ωHȷ

MAD at horizon

Noether current in GRMHDMHD:

Magnetic field vector

Hence the energy-momentum tensor

Noether current

>0 in the ergosphere

Noether current: force-free

Noether current: MAD

ConclusionsThe 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.