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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
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Relativistic jets in Active Galactic Nuclei
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Relativistic jets in compact binaries (microquasars)
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Common source of energy ?
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Ruffini & Wilson 1975, Damour 1978, Blandford & Znajek 1977
Tapping black-hole rotational energy by unipolar induction
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Controversy: • is the BH surface an analogue of a Faraday
disc (causality) • is the Blandford-Znajek mechanism efficient
(rotation of black-hole or disc) ?
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Recent (2011-2013) GRMHD simulations clearly showed BH rotational energy extraction in a particular (MAD) magnetic field configuration
Tchekhovskoy, McKinney, Blandford 2012
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MAD simulation
Tchekhovskoy, McKinney, Narayan 2011
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MAD
BH Jet in MAD state has a large efficiency: η = Pjet/Mc2 > 100%
.
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Sądowski et al. (2013)
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For
Penrose process
- timelike (at ∞ ) stationarity Killing vector
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- timelike (at ∞ ) stationarity Killing vector
- spacelike axisymmetry Killing vector
-ZAMO,
Energy measured by ZAMOs always non-negative:
Hence for .Since
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Horizon
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T - energy moment tensor
- null energy condition
• Energy conservation
Noether current (« energy momentum density vector »)
by Stoke’s theorem
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***********************************************
angular-momentum density vector
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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
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Mechanical Penrose process
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so it is timelike and past-directed(possible only in the ergosphere)
is collinear to
because is negative.
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General electromagnetic field
Therefore the integrand in is:
since
• pseudoelectric field 1-form on H
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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
( )
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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.
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Force free case (Blandford-Znajek)
- electric 4-current. From stationarity
sothere exists a function ω(Ψ) such that
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therefore on H and
(Blandford & Znajek 1977)
One gets
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Blandford-Znajek = Penrose
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MAD SANE(Magnetically Arrested discs) (Standard And Normal Evolution)
General Relativistic MagnetoHydroDynamics (GRMHD)
(McKinney, Tchekhovskoy, Narayan, Blandford)
GRMHD HARM (Gammie, McKinney, Tóth 2003)
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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.
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Flux densities:
Energy-momentum tensor
etc.
At horizon
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Force-free
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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
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MAD
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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
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Noether current in GRMHDMHD:
Magnetic field vector
Hence the energy-momentum tensor
Noether current
>0 in the ergosphere
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Noether current: force-free
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Noether current: MAD
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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.
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