Luca Comisso - Extraction of Black Hole Energy via Magnetic Reconnection

Slides of a talk delivered at the Institute for Theoretical Physics (Frankfurt, Germany) on a novel mechanism for energy extraction from black holes by Comisso and Asenjo. The energy extraction rate is calculated and shown that, under certain conditions, it is competitive with the power extracted via the Blandford-Znajek process for jetted outflows.

1. In collaboration with Felipe Asenjo
Luca Comisso
Institute for Theoretical Physics, Frankfurt
February 9, 2021 Image Credit: Interstellar
Extraction of
Black Hole Energy
via Magnetic Reconnection

2. ▸Quick overview of basic mechanisms of energy extraction
▸Connectivity of the magnetic field lines
(and their break up: magnetic reconnection)
▸How a process that involves magnetic reconnection can
drive the extraction of black hole energy
▸Connection with numerical simulations and important
astrophysical applications
OUTLINE 2
Note: we assume the Kerr metric for describing astrophysical black holes

3. WHY SHOULD WE CARE? 3
Mass/Spin Evolution
Emission of Radiation
Credit: EHT Collaboration
Jet Formation
Credit: NASA, NRAO, Biretta and Owen
Credit: NASA/JPL-Caltech

4. 4QUANTUM MECHANICAL SPONTANEOUS EMISSION
Heuristic “visualization” of the Hawking effect.
Image Credit: Hawking 1988
Hawking radiation [Hawking 1974]
▸Radiated Power (non-rotating BH):
τ
BH
=
5120πG
2
ℏc
4
M
3
∼10
67
(
M
M
⊙)
3
years
▸All BH energy can be removed
▸Lifetime of the BH:
P=
ℏc
6
15360πG
2
1
M
2
∼10
−53
(
M
⊙
M)
2
L
⊙

5. 5
Static Limit
Ergosphere
Event
Horizon
Image Credit: Ruffini and Wheeler 1971
E
(0)
E
(1)
E
(2)
PENROSE PROCESS
p
(0)μ
=p
(1)μ
+p
(2)μ
E
(2)
=E
(0)
−E
(1)
can be
since in the ergosphere
E
(1)
=−ξ
μp
(1)μ
<0
ξ
μ
ξ
μ>0
▸Particle splitting inside the ergosphere (purely
mechanical process) [Penrose 1969]
▸Four-momentum is conserved:
▸Then the escaping particle has energy:
▸The gained energy comes from the BH
In that case E
(2)
>E
(0)

6. 6
Christodoulou 1970
MAXIMUM EXTRACTABLE ENERGY
M
irr
=
A
H
16π
=M
1
2
(
1+1−a
2
)
▸Classical physics dictates that part of the black
hole mass is “irreducible” [Christodoulou 1970]
▸Then, the Penrose process can extract up to
of BH energy∼29%
▸That’s a lot of energy, but…
E
rot
=(1−1/2)Mc
2
≃0.29Mc
2
(a→1)
▸The two newborn particles must separate with
[Bardeen, Press, Teukolsky 1972]v>c/2
▸Expected rate of such events is low

7. 7
Image Credit: Dobbie at al. 2009
▸Another important point: the Penrose Process is purely mechanical.
“TYPICAL” BLACK HOLES ARE NOT ISOLATED SYSTEMS
What about magnetic fields?

8. 8
The plasma is attached to the magnetic field linesImage Credit: Thorne 1987
BLANDFORD-ZNAJEK PROCESS
▸Energy can be extracted electromagnetically
through a magnetic field that threads the BH
event horizon [Blandford & Znajek 1977]
P≃
1
6π
Ω
2
H
Φ
2
BH
▸Extracted Power (assuming max efficiency):
▸Analogously to the Penrose process, energy
is extracted by feeding the BH with negative
electromagnetic energy of magnetic fields
Φ
BH
=
1
2∫
θ
∫
ϕ
|B
r
|−gdθdϕΩ
H
=
a
2r
H
(c=1)

9. 9MECHANISM UNDERPINNING JET FORMATION?
Credits: NRAO/Walker et al. (2018)
Tchekhovskoy 2015 Tchekhovskoy et al. 2011
Image credit: A. Tchekhovskoy

10. 10BEHAVIOR OF THE MAGNETIC FIELD LINES
In a plasma that satisfies the ideal Ohm’s
law, two plasma elements connected by
a magnetic field line at a given time will
remain connected by a field line for all
subsequent times [Newcomb 1958]
▸The Blandford-Znajek process is derived from force-free electrodynamics (ideal MHD
with vanishing plasma inertia)
▸The magnetic field lines preserve their connectivity (i.e., do not “reconnect”) in the
Blandford-Znajek process

11. 11MAGNETIC FIELD LINE CONSERVATION
▸If at , then it remains null for all timesdl×B=0t=0
Using , , :E+v×B=0
∂B
∂t
=−∇×E
dl
dt
=(dl⋅∇)v
d
dt
(dl×B)=−(dl×B)(∇⋅v)−[(dl×B)×∇]×v
d
dt
(dl×B)=0implies conservation of the field line connectivity

12. 12MAGNETIC FIELD LINE CONSERVATION IN GR
Following Asenjo & Comisso 2017, Comisso & Asenjo 2020:
▸Define an infinitesimal spacelike 4-vector (distance between two close
events) such that
dl
μ
dl
μ
F
μν
=0
▸It can be shown that if , thenU
ν
F
μν
=0
d
dτ
(dl
μ
F
μν
)=−(∇
ν
U
α
)(dl
μ
F
μα
)
(
d
dτ
=U
α
∇
α)
▸In curved spacetime, can be interpreted in terms of magnetic field lines
alone if we adopt a 3+1 foliation of spacetime
dl
μ
F
μν
=0
(F
μν
=E
μ
n
ν
−E
ν
n
μ
−ϵ
μνρσ
B
ρ
n
σ
)
Then for dl
μ
F
μν
=0⇒ϵ
0ijk
dl
j
B
k
=0 dl
0
=0
If , one can always restore simultaneity by performing the transformation
such that (see Pegoraro 2012)
dl
0
≠0
dl
μ
→dl′
μ
=dl
μ
+U
μ
dλ dl′
0
=0

13. 13HOWEVER, IN “REAL” PLASMAS…
▸Magnetic reconnection results in conversion of magnetic energy into bulk kinetic energy,
thermal energy and nonthermal particle energy
▸The connectivity of the magnetic field lines is modified due to the presence of a localized
diffusion region

14. 14DOCUMENTED IN MYRIAD NUMERICAL SIMULATIONS
Zenitani & Hoshino 2001
Guo+ 2015
..and many other works not shown here
Sironi+ 2016

15. SO HOW DOES MAGNETIC
RECONNECTION ENTERS THE PICTURE?
15

16. 16
ϵ
∞
0<
ϵ
+
∞
0>Δ
(b)
(a)
BH
BH
B
B
v
r
outv
r
out
β
ϕ
>1
β
ϕ
<1
ALTERNATIVE ENERGY EXTRACTION MECHANISM
ϵ
∞
0<
ϵ
+
∞
0>Δ
(b)
(a)
BH
BH
B
B
v
r
outv
r
out
β
ϕ
>1
β
ϕ
<1
meridional view equatorial view
▸Reconnection occurs in the equatorial current sheet and extracts BH energy if the
decelerated plasma acquires negative energy while the accelerated plasma escapes
Comisso & Asenjo 2021

17. 17ENERGY-AT-INFINITY DENSITY IN RECONNECTION
▸Energy-momentum tensor in one-fluid approx:
e
∞
=−αg
μ0
T
μ0
=α̂e+αβ
ϕ̂P
ϕ
▸The energy-at-infinity density is given by
T
μν
=pg
μν
+wU
μ
U
ν
+F
μ
δF
νδ
−
1
4
g
μν
F
ρδ
F
ρδ
̂e=ŵγ
2
−p+
̂B
2
+̂E
2
2
̂P
ϕ
=ŵγ
2
̂v
ϕ
+(
̂B×̂E)
ϕ
hats (^) denote quantities observed in the
zero-angular-momentum-observer (ZAMO) frame
▸Assuming that most of magnetic energy is converted during reconnection, we have
e
∞
hyd
=α̂γ
[
(̂γ+β
ϕ
̂γ̂v
ϕ
)w−
p
̂γ
]

18. ▸ and observed by the ZAMO can be expressed in terms of a Keplerian velocity in
the ZAMO frame (but note that the expression for is more general) and the outflow
velocities in the local rest frame of reconnection. Then we get:
̂γ ̂v
ϕ
ϵ
∞
±
x
μ
′
ϵ
∞
±=
E
∞
hyd,±
H
=α̂γ
K
[
(1+̂v
K
β
ϕ
)γ
out
±cosξ(̂v
K
+β
ϕ
)γ
out
v
out
−
p/w
(1±cosξ̂v
Kv
out)γ
out̂γ
2
K
]
̂v
K
=
A
Δ
1/2
[
(M/r)
1/2
−a(M/r)
2
r
3
−a
2
M
3
]
−β
ϕ
Δ=r
2
−2Mr+(aM)
2
A=[r
2
+(aM)
2
]
2
−(aM)
2
Δsin
2
θ
v
out
≃
σ
0
1+σ
0
ξ≡arctan
(
v
out
1′
v
out
3′)
σ
0
≡B
2
0
/w
0
ENERGY-AT-INFINITY (PER ENTHALPY) IN RECONNECTION

19. ϵ
∞
±=α̂γ
K
[
(1+β
ϕ
̂v
K)1+σ
0
±cosξ(β
ϕ
+̂v
K)σ
1/2
0
−
1
4
1+σ
0
∓cosξ̂v
K
σ
1/2
0
̂γ
2
K
(1+σ
0−cos
2
ξ̂v
2
K
σ
0)]
ϵ
∞
+
≃3σ
0
ϵ
+
∞
�σ�
ϵ
-
∞
-σ�/�
� � � � � ��
-�
�
�
�
�
σ�
ϵ
+∞
�
ϵ
-∞
▸BH energy extraction if
▸ and requires ϵ
∞
−
<0 ϵ
∞
+
>0 σ
0
>1/3
▸For , and :a,r/M→1ξ→0 σ
0
≫1
ϵ
∞
−
≃−σ
0
/3
ϵ
∞
−
<0,Δϵ
∞
+
=ϵ
∞
+
−
(
1−
Γ
Γ−1
p
w)
>0
▸Energy at infinity per enthalpy:
ENERGY EXTRACTION CONDITION AND INTERESTING LIMITS

20. 20ENERGY EXTRACTION ASSESSMENT IN PHASE SPACE
ξ=π12/
0.80 0.85 0.90 0.95 1.00
1.0
1.2
1.4
1.6
1.8
2.0
r
/
M
a
ϵ
∞
0001σ )( <=
0
ϵ
∞
003σ)( <=
0
ϵ
∞
001σ)( <=
0
ϵ
∞
03)( <=
0σ
ϵ
∞
01
0σ)( <=
0.80 0.85 0.90 0.95 1.00
1.0
1.2
1.4
1.6
1.8
2.0
r
/
M
a
=1000σ
ϵ
∞
60ξ )( <=�/
ϵ
∞
201ξ )( <=�/
ϵ
∞
40ξ )( <=�/
ϵ
∞
200ξ )( <=�/

21. 21
ξ=π/� ξ=π/� ξ=π/��
ξ=π/�� ξ=�
��� ��� ��� ��� ��� ���
����
����
�
��
���
�/�
�
����
/
�
�
�=����σ�=��
�
��� ��� ��� ��� ��� ���
����
����
�
��
���
�/�
�
����
/
�
�
�=����ξ=π/��
σ�=��
σ�=��
�
σ�=��
�
σ�=��
�
σ�=��
�
▸Maximum power that can be extracted:
P
extr
=−ϵ
∞
−
w
0
A
in
U
in
ENERGY EXTRACTION RATE
P
max
extr
≃σ
0
/3w
0
A
in
U
in
∼0.1M
2
σ
0
w
0
U
in
=
{
∼10
−1
∼10
−2
A
in
∼(r
2
E
−r
2
ph
)
▸Power extracted from the black hole:
collisionless reconnection
collisional reconnection

22. 22
▸Process energization efficiency:
η=
ϵ
∞
+
ϵ
∞
++ϵ
∞
−
BH SPINDOWN & PROCESS EFFICIENCY
�=�
�=�����
�=�����
�=�����
�=�����
��� ��� ��� ��� ��� ���
���
���
���
���
���
���
�/�
η
σ�=���ξ=π/��
▸Reduction of BH spin:
dE
rot
dt
≃−
M
4ϖ
dϖ
dt
=ϵ
∞
−
w
0
A
in
U
in(ϖ=1−a≪1)
e.g., the BH spin would decrease from a = 0.999 to a = 0.99 int
sd
∼1/(σ
0
w
0
M)
η
max
≃
3σ
0
3σ
0−σ
0/3
=3/2

23. 23
�=����
�=����
�=����
�=����
�=����
� �� ���������
�
��
�
��
�
���
���
�
�
��
��
���
σ�
�
����
/
�
��
�=����ξ=π/��
P
BZ
≃κΦ
2
BH(Ω
2
H
+χΩ
4
H
+ζΩ
6
H)
COMPARISON WITH BLANDFORD-ZNAJEK PROCESS
P
extr
P
BZ
∼
−ϵ
∞
−
A
in
U
in
κΩ
2
H
r
4
H
σ
0sin
2
ξ(1+χΩ
2
H
+ζΩ
4
H
)
▸Power extracted via BZ process
κ≈0.044−0.053,χ≈1.38,ζ≈−9.2
[Blandford & Znajek 1977, Tchekhovskoy+ 2010]
▸Assuming Φ
BH
∼|B
r
|r
2
H
∼B
0
sinξr
2
H
(this is very rough; numerical simulations
are needed to determine the magnetic
field configuration at all latitudes)

24. SO IT LOOKS LIKE AN INTERESTING
SCENARIO FOR BH ENERGY EXTRACTION
24
BUT DO WE SEE RECONNECTION IN
NUMERICAL SIMULATIONS OF BLACK HOLES?

25. 25RECONNECTION IN BLACK HOLE SIMULATIONS
Ripperda+ 2020
Parfrey+ 2019
Komissarov 2005
found also
negative-energy
particles

26. 26EXTREME RESOLUTION 3D MAD SIMULATIONS
▸Plasmoid formation can be captured only at high-resolution (Ripperda+, in prep)
Movie courtesy of Bart Ripperda
(5400x2304x2304)

27. 27
-150
-100
-50
0
50
100
0 10 20 30
Jul 22 2018 flare, MJD=58321.9954
-100
0
100
-1000100
x-offset(µas)
y
-
o
f
f
s
e
t

(
µ
a
s
)
-100
0
100
-1000100
R=7 R
g
a=0 i=160
o
W=160
o
c
r
2
=1.2
-200
-100
0
100
0 10 20 30
0
1
2
time (mins)
x
(
b
lu
e
)

a
n
d

y
(
r
e
d
)

o
f
f
s
e
t

(
µ
a
r
c
s
e
c
)
f
lu
x

d
e
n
s
it
y

(
in

u
n
it
s

o
f

S
2
)
-100
0
100
-1000100-150
-100
-50
0
50
100
0 10 20 30
Jul 22 2018 flare, MJD=58321.9954
-100
0
100
-1000100
x-offset(µas)
y
-
o
f
f
s
e
t

(
µ
a
s
)
-200
-100
0
100
0 10 20 30
0
1
2
time (mins)
x
(
b
lu
e
)

a
n
d

y
(
r
e
d
)

o
f
f
s
e
t

(
µ
a
r
c
s
e
c
)
f
lu
x

d
e
n
s
it
y

(
in

u
n
it
s

o
f

S
2
)
-150
-100
-50
0
50
100
0 10 20 30
Jul 22 2018 flare, MJD=58321.9954
-100
0
100
-1000100
x-offset(µas)
y
-
o
f
f
s
e
t

(
µ
a
s
)
-100
0
100
-1000100
R=7 R
g
a=0 i=160
o
W=160
o
c
r
2
=1.2
-200
-100
0
100
0 10 20 30
0
1
2
time (mins)
x
(
b
lu
e
)

a
n
d

y
(
r
e
d
)

o
f
f
s
e
t

(
µ
a
r
c
s
e
c
)
f
lu
x

d
e
n
s
it
y

(
in

u
n
it
s

o
f

S
2
)
-100
0
100
-1000100-150
-100
-50
0
50
100
0 10 20 30
Jul 22 2018 flare, MJD=58321.9954
-100
0
100
-1000100
x-offset(µas)
y
-
o
f
f
s
e
t

(
µ
a
s
)
-200
-100
0
100
0 10 20 30
0
1
2
time (mins)
x
(
b
lu
e
)

a
n
d

y
(
r
e
d
)

o
f
f
s
e
t

(
µ
a
r
c
s
e
c
)
f
lu
x

d
e
n
s
it
y

(
in

u
n
it
s

o
f

S
2
)
b
a
c
d
-0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80
P
pol
=57±8 min
July 22nd, 2018
MJD58321.9954
I
/
S
2

(
b
lu
e
)
,

Q
/
S
2
(
r
e
d
)
-1.0
-0.5
0
0.5
1.0
0 20 40 60 80
-1
0
1
2
3
July 28th, 2018
MJD 58328.0841
time (min)
I
/
S
2
Q
/
Ö
(
Q
2
+
U
2
)

(
r
e
d
)
,

U
/
Ö
(
Q
2
+
U
2
)

(
g
r
e
e
n
)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 20 40 60 80
P
pol
=73±15 min
May 27th, 2018
MJD58266.3420
time (min)
I
/
S
2

(
b
lu
e
)
,

Q
/
S
2
(
r
e
d
)

M
a
y

2
7
t
h
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
-1.5-1.0-0.500.51.01.5
55
61
42
P
pol
= 46±6 min
17
68
30
t=0 min
Q/Ö(Q
2
+U
2
)
U
/
Ö
(
Q
2
+
U
2
)
a
b
c
d
-0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80
P
pol
=57±8 min
July 22nd, 2018
MJD58321.9954
I
/
S
2

(
b
lu
e
)
,

Q
/
S
2
(
r
e
d
)
-1.0
-0.5
0
0.5
1.0
0 20 40 60 80
-1
0
1
2
3
July 28th, 2018
MJD 58328.0841
time (min)
I
/
S
2
Q
/
Ö
(
Q
2
+
U
2
)

(
r
e
d
)
,

U
/
Ö
(
Q
2
+
U
2
)

(
g
r
e
e
n
)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 20 40 60 80
P
pol
=73±15 min
May 27th, 2018
MJD58266.3420
time (min)
I
/
S
2

(
b
lu
e
)
,

Q
/
S
2
(
r
e
d
)

M
a
y

2
7
t
h
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
-1.5-1.0-0.500.51.01.5
55
61
42
P
pol
= 46±6 min
17
68
30
t=0 min
Q/Ö(Q
2
+U
2
)
U
/
Ö
(
Q
2
+
U
2
)
a
b
c
d
POSSIBLE MANIFESTATIONS OF RECONNECTION
Flares from Sgr A* (GRAVITY Collaboration 2018)
Variability near black holes
(EHT Collaboration 2021)
BH spin limited by reconnection (Reynolds 2019)

28. ▸Reconnection of magnetic field lines can drive the extraction of
large amounts of black hole energy in an efficient way
28
The results covered here can be found in Comisso & Asenjo 2021
SUMMARY
Felipe myself