Journal ArticleDOI

# Accurate Simulations of Binary Black Hole Mergers in Force-free Electrodynamics

20 Jul 2012-The Astrophysical Journal (University of Chicago Press for the American Astronomical Society)-Vol. 754, Iss: 1, pp 36

AbstractWe provide additional information on our recent study of the electromagnetic emission produced during the inspiral and merger of supermassive black holes when these are immersed in a force-free plasma threaded by a uniform magnetic field. As anticipated in a recent letter, our results show that although a dual-jet structure is present, the associated luminosity is ~100 times smaller than the total one, which is predominantly quadrupolar. Here we discuss the details of our implementation of the equations in which the force-free condition is not implemented at a discrete level, but rather obtained via a damping scheme which drives the solution to satisfy the correct condition. We show that this is important for a correct and accurate description of the current sheets that can develop in the course of the simulation. We also study in greater detail the three-dimensional charge distribution produced as a consequence of the inspiral and show that during the inspiral it possesses a complex but ordered structure which traces the motion of the two black holes. Finally, we provide quantitative estimates of the scaling of the electromagnetic emission with frequency, with the diffused part having a dependence that is the same as the gravitational-wave one and that scales as L^(non-coll)_(EM) ≈ Ω^((10/3)–(8/3)), while the collimated one scales as L^(coll)_(EM) ≈ Ω^((5/3)–(6/3)), thus with a steeper dependence than previously estimated. We discuss the impact of these results on the potential detectability of dual jets from supermassive black holes and the steps necessary for more accurate estimates.

## Summary (5 min read)

### 1. INTRODUCTION

• In contrast, magnetic fields generated by the circumbinary accretion disk could play an important role and the dynamics of the plasma in the inner region can then be described within the force-free (FF) approximation.
• These physical conditions are indeed similar to those considered in the seminal investigations of BH electrodynamics of Blandford and Znajek (Blandford & Znajek 1977), who addressed the question of whether the rotational energy of an isolated BH can be extracted efficiently by a magnetic field.

### 2. EVOLUTION EQUATIONS

• The authors solve the combined system defined by the Einstein and Maxwell equations and model either an isolated rotating BH or a BH binary inspiralling in quasi-circular orbits.
• More specifically, the authors solve the Einstein equations Rμν − 1 2 Rgμν = 8πTμν, (1) where Rμν , gμν , and Tμν are the Ricci, the metric, and the stress-energy tensors, respectively.
• Such scalar fields are initialized to zero, but are driven into evolution as soon as violations of the EM constraints are produced.
• In the rest of their discussion the authors will use the expression “electrovacuum” to denote the case when currents and charges of the Maxwell equations are zero.

### 2.1. The Einstein Equations

• For the solution of the Einstein equations the authors make use of a three-dimensional finite-differencing code that adopts a conformal-traceless “3 + 1” BSSNOK formulation of the equations (see Pollney et al. 2007 for the full expressions in vacuum and Baiotti et al. 2008 for the case of a spacetime with matter).
• The code is based on the CactusComputational Toolkit (Allen et al. 2000) and employs adaptive mesh-refinement techniques via the Carpet-driver (Schnetter et al. 2004).
• For compactness the authors will not report here the details regarding the adopted formulation of the Einstein equations and the gauge conditions used, which can, however, be found in Pollney et al. (2007, 2011).
• The authors also note that recent developments, such as the use of eighth-order finite-difference operators or the adoption of a multiblock structure to extend the size of the wave zone, have been recently presented in Pollney et al. (2009, 2011).
• Here, however, in order to limit the computational costs and because a very high accuracy in the waveforms is not needed, the multiblock structure was not used and the authors have used a fourth-order finite-difference operator with a third-order Implicit-Explicit Runge–Kutta integration in time (see Section 2.3).

### 2.2. The Maxwell Equations

• More specifically, these evolution equations describe damped wave equations and have the effect of dynamically controlling the possible growth of the violations of the constraints and of propagating them away from the problematic regions of the computational domain where they are produced.
• The charge density q can be computed either through the evolution (Equation (13)) or by inverting the constraint (Equation (7)).

### 2.3. Numerical Treatment of the Force-free Conditions

• As noted before, within an FF approximation the stressenergy tensor is dominated by the EM part and the contribution coming from the matter can be considered zero.
• From the numerical point of view, specific strategies must be adopted in order to enforce the FF constraints expressed by Equations (25) and (26).
• The authors strategy, however, differs from both the previous ones and follows the same philosophy behind the choice of the driver defined by Equation (29).

### 3. ANALYSIS OF RADIATED QUANTITIES

• The calculation of the EM and gravitational radiation generated during the inspiral, merger, and ringdown is an important aspect of this work as it allows us to measure the amount correlation between the two forms of radiation.
• Any measure of these quantities in the strong-field region is therefore subject to ambiguity and risks producing misleading results.
• The term Φ0 in Equation (36) has been maintained (it disappears at null infinity) to account for the possible presence of an ingoing component in the radiation at finite distances.
• In particular, Equation (36) shows that the net flux is obtained by adding (with the appropriate sign) the respective contributions of the outgoing and ingoing fluxes.
• The second approach that the authors have followed for the computation of the emitted luminosity is the evaluation of the flux of the Poynting vector across a 2-sphere at large distances in terms of the more familiar 3+1 fields Ei and Bi in Equation (19).

### 4. ASTROPHYSICAL SETUP AND INITIAL DATA

• More specifically, the authors consider the astrophysical conditions during and after the merger of two supermassive BHs, each of which is surrounded by an accretion disk.
• During this phase, the binary evolves on the dynamical viscous timescale τd of the circumbinary accretion disk, which is regulated by the ability of the disk to transport its angular momentum outward (either via shear viscosity or magnetically mediated instabilities).
• Because τGW and τd have a very different scaling with D, more specifically, τGW ∼ D4, while τd ∼ D2, at a certain time the timescale τGW becomes smaller than τd.
• This represents the astrophysical scenario in which their simple model is then built.

### 4.1. Initial Data and Grid Setup

• The authors construct consistent BH initial data via the “puncture” method as described in Ansorg et al. (2004).
• The authors use these two configurations to best isolate the effects due to the binary orbital motion from those related to the spins of the two BHs.
• The authors note that similar initial data were considered by Koppitz et al. (2007), Pollney et al. (2007), and Rezzolla et al. (2008a, 2008b, 2008c) but they have recalculated them here using a higher resolution and improved initial orbital parameters.
• Shorter, higher-resolution simulations have also been carried out to perform consistency checks.
• Note that the initial Arnowitt, Deser, Misner (ADM) mass of the spacetime is not exactly 1 due to the binding energy of the BHs.

### 5. ACCURATE FORCE-FREE ENFORCEMENT

• As mentioned in Section 2.3, several different approaches are possible to enforce the FF conditions (25) and (26) in the plasma.
• In fact, since this approach acts “by hand” on the EM fields and converts them to values which would yield an FF regime, one is guaranteed that the constraints (25), (26), and (30) are satisfied.
• Denotes the second step of the “discrete” approach of Palenzuela et al. (2010a), which amounts to the modification of the electric field according to Equation (32), also known as 3. discrete2.
• On the other hand, the corresponding currents when the prescriptions discrete1 and discrete2 are used (right column) do not show evident signs of descending currents and, rather, they show unphysical features around the BH and discontinuities along the ∼ ±45◦ diagonals when seen in the (x, z) plane.
• As a final remark the authors also note that their prescriptions (29) and (33) also provide a saving in computational costs.

### OF BBH MERGERS

• After having discussed the details of their implementation of the FF conditions and having shown its higher accuracy with respect to alternative suggestions in the literature, in the following the authors concentrate their discussion on the FF electrodynamics accompanying the inspiral and merger of BH binaries.
• In particular, the authors will discuss the subtleties that emerge with the subtraction of the background radiation, the spatial distribution of the charge density, the EM and GW zones, and the scaling of the EM luminosity with frequency.

### 6.1. Subtraction of Background Radiation

• Hence, a proper identification of this background radiation is essential for the correct measure of the emitted luminosity and to characterize its properties.
• The generic expression (37) for the EM luminosity can be evaluated in combination with Equation (39), that is, by setting as background values those of the Newman-Penrose scalars Φ2 and Φ0 at the initial time.
• As a result, the background choice (39) represents by far the most convenient one.
• Note that the only modes that have a regular time modulation, and are therefore radiative, are (Φ2)22 and (Φ0)22, while the real parts of the (Φ2)20 and (Φ0)20 are essentially constant in time, indicating that these are not radiative modes, and could represent a way to measure the background radiation.

### 6.2. Properties of the EM Luminosity

• Evolution, measured in hours before the merger, of the luminosities as computed with expression (37) and either the prescriptions (39) or (40) for the background subtraction.
• More specifically, the thick lines refer to the total luminosity, while the thin ones to the luminosity in a polar cap of 5◦ semi-opening angle, measured using either expression (39) (red solid line), expression (40) (blue dotted line), or through the expression in terms of the Poynting vector (41) (black dashed line).
• A few comments should be reserved about the different spatial distributions of the EM fluxes that come with the different prescriptions for the subtraction of the background radiation and that are erased when computing the luminosities as integral quantities.
• The differences introduced by the spin are reported in the right panel of Figure 4, which refers to the binary s6 and thus with BHs having a dimensionless spin of J/M2 0.6.
• Less obvious, however, is the fact that the wave zones can be different whether one is considering the gravitational or the EM radiation, with the latter starting at considerably larger distances than the former.

### 6.3. Frequency Scaling

• As remarked already in Paper I, an accurate measure of the evolution of the collimated and non-collimated contributions of the emitted energies is crucial to predict the properties of the system when the two BHs are widely separated.
• The scaling ∼Ω2/3 is clearly incompatible with their data and the authors suspect the accelerated motion of the BHs to be behind this difference and longer simulations will be useful to draw robust conclusions.
• The right panel of Figure 6, which is the same as the left one but where the authors extrapolate the scaling back in frequency.
• The authors rough estimate is therefore that the collimated emission will be larger than the diffused one at an orbital frequency Ω = (1/2)ΩGW 3.2 × 10−5.
• While this is an exciting possibility, the authors should also bear in mind that, when extrapolated back to the time when it becomes dominant, the collimated emission has also decreased by almost one order of magnitude and to luminosities that are only of the order of ∼1042 erg s−1.

### 6.4. Charge-density Distribution

• Providing information which is complementary to the one already presented by Palenzuela et al. (2010b, 2010c) and Neilsen et al. (2011).the authors.
• To further limit the amount of information that can be extracted directly from their simulation is the fact that an FF code does not allow for an unambiguous calculation of the plasma velocity, which can only be estimated a posteriori based on a certain number of assumptions.
• Be appreciated from the first two columns of Figure 7, while the second contribution is the only one responsible for the charge distribution in the last column.
• Note that since they both refer to isolated spinning BHs (although with different spins), the right column of Figure 7 should be compared with the right column of Figure 2, which shows instead the electric currents.
• Within an FF approach, the consequences of this regular and alternate distribution of positive and negative charges, it is clear that it can lead to rather intriguing particle acceleration processes along the surfaces separating regions of different charges.

### 7. PROSPECTS AND CONCLUSIONS

• Assessing the detectability of the EM emission from merging BH binaries is much more than an academic exercise.
• Nevertheless, relying on a number of assumptions with varying degree of realism, several investigations have been recently carried out to investigate the properties of these EM counterparts either during the stages that precede the merger or in those following it.
• The authors have therefore provided the first quantitative estimates of the scaling of the EM emission with frequency and shown that the diffused part has a dependence that is very close to the one exhibited by the GW luminosity and therefore of the type Lnon-coll EM ≈ Ω10/3−8/3.
• Unfortunately, however, their use of an FF condition (and their ability to maintain it essentially to machine precision) prevents us from producing such electric fields and hence the corresponding accelerations.
• Even in the optimistic case in which the majority of the Poynting flux is converted into radio emission via synchrotron processes, the EM radiation (either collimated or diffused) will eventually exit the evacuated central region around the binary and penetrate in the ambient medium.

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The Astrophysical Journal, 754:36 (17pp), 2012 July 20 doi:10.1088/0004-637X/754/1/36
C
2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
ACCURATE SIMULATIONS OF BINARY BLACK HOLE MERGERS IN FORCE-FREE ELECTRODYNAMICS
Daniela Alic
1
, Philipp Moesta
1,2
, Luciano Rezzolla
1,3
, Olindo Zanotti
4
,andJos
´
e Luis Jaramillo
1
1
Max-Planck-Institut f
¨
ur Gravitationsphysik, Albert-Einstein-Institut, Potsdam, Germany
2
TAPIR, MC 350-17, California Institute of Technology, Pasadena, CA 91125, USA
3
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, USA
4
Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, 38123 Trento, Italy
Received 2012 April 7; accepted 2012 May 9; published 2012 July 3
ABSTRACT
We provide additional information on our recent study of the electromagnetic emission produced during the inspiral
and merger of supermassive black holes when these are immersed in a force-free plasma threaded by a uniform
magnetic ﬁeld. As anticipated in a recent letter, our results show that although a dual-jet structure is present, the
associated luminosity is 100 times smaller than the total one, which is predominantly quadrupolar. Here we discuss
the details of our implementation of the equations in which the force-free condition is not implemented at a discrete
level, but rather obtained via a damping scheme which drives the solution to satisfy the correct condition. We show
that this is important for a correct and accurate description of the current sheets that can develop in the course of the
simulation. We also study in greater detail the three-dimensional charge distribution produced as a consequence of
the inspiral and show that during the inspiral it possesses a complex but ordered structure which traces the motion
of the two black holes. Finally, we provide quantitative estimates of the scaling of the electromagnetic emission
with frequency, with the diffused part having a dependence that is the same as the gravitational-wave one and that
scales as L
non-coll
EM
Ω
10/38/3
, while the collimated one scales as L
coll
EM
Ω
5/36/3
, thus with a steeper dependence
than previously estimated. We discuss the impact of these results on the potential detectability of dual jets from
supermassive black holes and the steps necessary for more accurate estimates.
Key words: binaries: close galaxies: jets gravitation magnetic ﬁelds plasmas relativistic processes
Online-only material: color ﬁgures
1. INTRODUCTION
The gravitational interaction among galaxies, most of which
are supposed to host a supermassive black hole (BH), with
M 10
6
M
(Shankar et al. 2004; Lou & Jiang 2008), is a
well-established observational fact (Gopal-Krishna et al. 2003;
Ellison et al. 2011; Mohamed & Reshetnikov 2011; Lambas
et al. 2012). Moreover, in a few documented astrophysical cases,
strong indications exist to believe that a binary merger among
supermassive BHs has occurred or is ongoing (Rodriguez et al.
2006; Komossa et al. 2003; Dotti et al. 2009).
A strong motivation for studying supermassive binary black
holes (SMBBHs) comes from the fact that their gravitational sig-
nal will be detected by the planned Laser Interferometric Space
Antenna (eLISA/NGO; Amaro-Seoane et al. 2012;Bin
´
etruy
et al. 2012). When combined with the usual electromagnetic
(EM) emission, the detection of gravitational waves (GW) from
these systems will provide a new tool for testing a number of fun-
damental astrophysical issues (Cornish & Porter 2007;Haiman
et al. 2009; Phinney 2009). For this reason, SMBBHs are cur-
rently attracting widespread interest, both from an observational
and a theoretical point of view (Rezzolla 2009; Reisswig et al.
2009; Kesden et al. 2010; Kocsis et al. 2011; Tanaka et al. 2012;
Sesana et al. 2012; Barausse 2012). According to the simplest
picture that has gradually emerged through a series of semi-
analytical studies and numerical simulations (Milosavlje
´
c&
Phinney 2005; MacFadyen & Milosavljevi
´
c 2008; Roedig et al.
2011; Bode et al. 2012), the accretion disk formed around the
two merging BHs, commonly referred to as the “circumbinary”
accretion disk, can follow the dynamical evolution of the system
up until the dynamical timescale for the emission of GWs, which
scales like D
4
, where D is the separation of the binary, be-
comes shorter than the viscous timescale, which instead scales
like D
2
. When this happens, the circumbinary accretion disk is
essentially decoupled from the binary, which rapidly enters the
ﬁnal stages of the inspiral. Under these conditions, neglecting
the inertia of the accreting ﬂuid can be regarded as a very good
approximation. In contrast, magnetic ﬁelds generated by the cir-
cumbinary accretion disk could play an important role and the
dynamics of the plasma in the inner region can then be described
within the force-free (FF) approximation. These physical con-
ditions are indeed similar to those considered in the seminal
investigations of BH electrodynamics of Blandford and Zna-
jek (Blandford & Znajek 1977), who addressed the question of
whether the rotational energy of an isolated BH can be extracted
efﬁciently by a magnetic ﬁeld. After the ﬁrst two-dimensional
investigations of Komissarov and Barkov (Komissarov 2004;
Komissarov & Barkov 2009), the numerical study of BH mag-
netospheres has now entered a mature phase in the context of
SMBBHs evolution.
In an extensive analysis, but still in the absence of currents
and charges, i.e., in electrovacuum, M
¨
osta et al. (2010)showed
that, even though the EM radiation in the lowest = 2 and
m = 2 multipole reﬂects the gravitational one, the energy
emitted in EM waves is 13 orders of magnitude smaller than
that emitted in GWs for a reference binary with mass M =
10
8
M
and a magnetic ﬁeld B = 10
4
G, thus casting serious
doubts about a direct detection of the two different signals.
However, a series of more recent numerical simulations in which
currents and charges are taken into account have suggested the
intriguing possibility that a mechanism similar to the original
one proposed by Blandford and Znajek may be activated in the
case of binaries (Palenzuela et al. 2009a, 2010a, 2010b, 2010c;
Moesta et al. 2012; note that Palenzuela et al. 2010a, 2010b;
1

The Astrophysical Journal, 754:36 (17pp), 2012 July 20 Alic et al.
Moesta et al. 2012 also make use of an FF approximation).
In particular, the Blandford–Znajek mechanism is likely to be
valid under rather general conditions, namely even if stationarity
and axisymmetry are relaxed and even if a non-spinning BH is
simply boosted through a uniform magnetic ﬁeld. Moreover,
for such uniform magnetic ﬁeld, the emitted EM ﬂux shows a
high degree of collimation, making the EM counterpart more
easily detectable. A less optimistic view has emerged recently
in Moesta et al. (2012, hereafter Paper I), where we have shown,
through independent calculations in which the EM emission
was extracted at much larger radii, that the dual-jet structure
is indeed present but energetically subdominant with respect to
the non-collimated and predominantly quadrupolar emission. In
particular, even if the total luminosity at merger is 100 times
larger than in Palenzuela et al. (2010b), the energy ﬂux is
only 8–2 times larger near the jets, thus yielding a collimated
luminosity that is 100 times smaller than the total one. As a
result, Paper I indicated that the detection of the dual jets at the
merger is difﬁcult if not unlikely.
Here we provide additional information on the results pre-
sented in Paper I and discuss the details of our implementation of
the equations in which the FF condition is obtained via a damp-
ing scheme which drives the solution to satisfy the correct con-
dition. We show that this is important for a correct and accurate
description of the current sheets that can develop in the course
of the simulation. We also study in greater detail the three-
dimensional charge distribution produced as a consequence of
the inspiral and show that during the inspiral it has a complex
structure tracing the motion of the two BHs. Finally, we provide
quantitative estimates of the scaling of the EM emission with
frequency, with the diffused part having a dependence that is
the same as the GW one and that scales as L
non-coll
EM
Ω
10/38/3
,
while the collimated one scales as L
coll
EM
Ω
5/36/3
, thus with
a steeper dependence than previously estimated by Palenzuela
et al. (2010b).
This paper is organized as follows. In Section 2 we describe
the system of equations considered in our analysis, with par-
ticular emphasis on the treatment of the FF condition, while in
Section 3 we discuss the different routes to the calculation of the
EM radiated quantities. In Section 4 we present the astrophysical
setup of a BH binary merger, while Section 5 compares different
approaches for the enforcement of the FF condition. Section 6
is devoted to the presentation of the results, and, in particular,
to the computation of the luminosity. Finally, Section 7 presents
the conclusion of our work and the prospects for the detection
of an EM counterpart to SMBBHs.
In the rest of the paper, we set c = G = 1, adopt the standard
convention for the summation over repeated indices with Greek
indices running from 0 to 3, Latin indices from 1 to 3, and make
use of the Lorentz–Heaviside notation for the EM quantities, in
which all
4π factors disappear.
2. EVOLUTION EQUATIONS
We solve the combined system deﬁned by the Einstein and
Maxwell equations and model either an isolated rotating BH or
a BH binary inspiralling in quasi-circular orbits. In both cases
we assume that there is an external FF magnetic ﬁeld. More
speciﬁcally, we solve the Einstein equations
R
μν
1
2
Rg
μν
= 8πT
μν
, (1)
where R
μν
, g
μν
, and T
μν
are the Ricci, the metric, and the
stress-energy tensors, respectively. In addition, we solve the
following extended set of Maxwell equations (Komissarov
2007; Palenzuela et al. 2009b):
μ
(F
μν
+ g
μν
Ψ) = I
ν
κn
ν
Ψ, (2)
μ
(
F
μν
+ g
μν
Φ) =−κn
ν
Φ, (3)
where F
μν
is the Faraday tensor,
F
μν
is its dual, I
μ
is the four-
current, and we have introduced a 3+1 slicing of spacetime, with
n
μ
being the unit (future oriented) timelike vector associated
with a generic normal observer to the spatial hypersurfaces.
The set of Maxwell Equations (2) and (3)isreferredtoas
“extended” because it incorporates the so-called divergence-
cleaning approach, originally presented in Dedner et al. (2002)
in ﬂat spacetime, and which amounts to introducing two
additional scalar ﬁelds, Ψ and Φ, that propagate away the
deviations of the divergences of the electric and of the magnetic
ﬁelds from the values prescribed by Maxwell equations. Such
scalar ﬁelds are initialized to zero, but are driven into evolution
as soon as violations of the EM constraints are produced. The
total stress-energy tensor is composed of a term corresponding
to the EM ﬁeld:
T
μν
f
F
μ
λ
F
νλ
1
4
(F
λκ
F
λκ
)g
μν
, (4)
and of a term due to matter, T
μν
m
. However, because the EM ﬁeld
is assumed to be FF, T
μν
f
T
μν
m
, and the total stress-energy
tensor is then assumed to be given entirely by Equation (4),
namely T
μν
T
μν
f
. In the rest of our discussion we will use the
expression “electrovacuum” to denote the case when currents
and charges of the Maxwell equations are zero. Such a scenario
was extensively studied in M
¨
osta et al. (2010) and it will be used
here as an important reference. In the following we discuss in
more detail our strategy for the solution of the Einstein equations
and of the Maxwell system in an FF regime.
2.1. The Einstein Equations
For the solution of the Einstein equations we make use
of a three-dimensional ﬁnite-differencing code that adopts
a conformal-traceless “3 + 1” BSSNOK formulation of the
equations (see Pollney et al. 2007 for the full expressions in
vacuum and Baiotti et al. 2008 for the case of a spacetime with
matter). The code is based on the Cactus Computational Toolkit
(Allen et al. 2000) and employs adaptive mesh-reﬁnement
techniques via the Carpet-driver (Schnetter et al. 2004). For
compactness we will not report here the details regarding the
adopted formulation of the Einstein equations and the gauge
conditions used, which can, however, be found in Pollney et al.
(2007, 2011).
We also note that recent developments, such as the use of
eighth-order ﬁnite-difference operators or the adoption of a
multiblock structure to extend the size of the wave zone, have
been recently presented in Pollney et al. (2009, 2011). Here,
however, in order to limit the computational costs and because
a very high accuracy in the waveforms is not needed, the multi-
block structure was not used and we have used a fourth-order
ﬁnite-difference operator with a third-order Implicit-Explicit
Runge–Kutta (RKIMEX) integration in time (see Section 2.3).
2

The Astrophysical Journal, 754:36 (17pp), 2012 July 20 Alic et al.
2.2. The Maxwell Equations
The Maxwell Equations (2) and (3) take a more familiar form
when expressed in terms of the standard electric and magnetic
ﬁelds as deﬁned by the following decomposition of the Faraday
tensor in a 3+1 foliation:
F
μν
= n
μ
E
ν
n
ν
E
μ
+
μναβ
B
α
n
β
, (5)
F
μν
= n
μ
B
ν
n
ν
B
μ
μναβ
E
α
n
β
, (6)
where the vectors E
μ
and B
μ
are purely spatial (i.e., E
μ
n
μ
=
B
μ
n
μ
= 0) and correspond to the electric and magnetic ﬁelds
measured by the normal (Eulerian) observers. The two extra
scalar ﬁelds Ψ and Φ introduced in the extended set of Maxwell
equations lead to two evolution equations for the EM constraints,
which, we recall, are given by the divergence equations
i
E
i
= q, (7)
i
B
i
= 0, (8)
where the electric current has been decomposed in the electric
charge density q ≡−n
μ
I
μ
and the spatial current J
i
I
i
.
More speciﬁcally, these evolution equations describe damped
wave equations and have the effect of dynamically controlling
the possible growth of the violations of the constraints and of
propagating them away from the problematic regions of the
computational domain where they are produced.
In terms of E
μ
and B
μ
, the 3 + 1 formulation of Equations (2)
and (3) becomes (Palenzuela et al. 2010a)
D
t
E
i
ij k
j
( αB
k
)+αγ
ij
j
Ψ = αKE
i
αJ
i
, (9)
D
t
B
i
+
ij k
j
( αE
k
)+αγ
ij
j
Φ = αKB
i
, (10)
D
t
Ψ + α
i
E
i
= αq ακ Ψ, (11)
D
t
Φ + α
i
B
i
=−ακ Φ, (12)
D
t
q +
i
( αJ
i
) = αKq, (13)
where
D
t
(
t
L
β
) and L
β
is the Lie derivative along the
shift vector β and K is the trace of the extrinsic curvature. The
charge density q can be computed either through the evolution
(Equation (13)) or by inverting the constraint (Equation (7)). For
simplicity, we choose the latter approach, which ensures that the
constraint (12) is automatically satisﬁed if Ψ = 0 initially and
effectively removes the need for the potential Ψ.
Exploiting now that the covariant derivative in the second term
of Equations (10) and (11) reduces to a partial derivative, i.e.,
ij k
j
B
k
=
ij k
(
j
B
k
+ Γ
l
jk
B
l
) =
ij k
j
B
k
, (14)
and using a standard conformal decomposition of the spatial
3-metric
˜γ
ij
= e
4φ
γ
ij
=
1
12
ln γ, (15)
we obtain the ﬁnal expressions for the extended Maxwell equa-
tions that we actually evolve
D
t
E
i
ij k
e
4φ
[(
j
α ) ˜γ
ck
B
c
+ α (4 ˜γ
ck
j
φ +
j
˜γ
ck
) B
c
+ α ˜γ
ck
j
B
c
] = αKE
i
αJ
i
, (16)
D
t
B
i
+
ij k
e
4φ
[(
j
α ) ˜γ
ck
E
c
+ α (4 ˜γ
ck
j
φ +
j
˜γ
ck
) E
c
+ α ˜γ
ck
j
E
c
]+αe
4φ
˜γ
ij
j
Φ = αKB
i
, (17)
D
t
Φ + α
i
B
i
=−ακ Φ. (18)
Clearly, the standard Maxwell equations in a curved background
are recovered for Φ = 0, so that the Φ scalar can then be con-
sidered as the normal-time integral of the standard divergence
constraint (8), which propagates at the speed of light and is
damped during the evolution.
As mentioned above, the coupling of the Einstein to the
Maxwell equations takes place via the inclusion of a nonzero
stress-energy tensor for the EM ﬁelds which is built in terms of
the Faraday tensor as dictated by Equation (4). More speciﬁcally,
the relevant components of the stress-energy tensor can be
obtained in terms of the electric and magnetic ﬁelds, that is as
τ n
μ
n
ν
T
μν
=
1
8π
(E
2
+ B
2
), (19)
S
i
≡−n
μ
T
μ
i
=
1
4π
ij k
E
j
B
k
, (20)
S
ij
T
ij
=
1
4π
E
i
E
j
B
i
B
j
+
1
2
γ
ij
(E
2
+ B
2
)
, (21)
where E
2
E
k
E
k
and B
2
B
k
B
k
. The scalar function τ can
be identiﬁed with the energy density of the EM ﬁeld, while the
energy ﬂux S
i
is the Poynting vector.
As already discussed in the Introduction, we remark again
that the EM energies that will be considered here are so small
when compared with the gravitational binding ones that the
contributions of the stress-energy tensor to the right-hand side
of the Einstein Equations (1) are effectively negligible and thus
can be set to zero, reducing the computational costs. The fully
coupled set of the Einstein–Maxwell equations was considered
in Palenzuela et al. (2009a, 2010c) and the comparison with the
results obtained here suggests that for the ﬁelds below 10
8
G,
the use of the test-ﬁeld approximation is fully justiﬁed.
2.3. Numerical Treatment of the Force-free Conditions
As noted before, within an FF approximation the stress-
energy tensor is dominated by the EM part and the contribution
coming from the matter can be considered zero. Following
Palenzuela et al. (2010a), the conservation of energy and
momentum,
ν
T
μν
= 0, implies that also the Lorentz force
is negligible, i.e.,
0 =∇
ν
T
μν
≈∇
ν
T
μν
f
=−F
μν
I
ν
, (22)
which can also be written equivalently in terms of quantities
measured by Eulerian observers as
E
k
J
k
= 0, (23)
3

The Astrophysical Journal, 754:36 (17pp), 2012 July 20 Alic et al.
qE
i
+
ij k
J
j
B
k
= 0. (24)
Computing the scalar and vector product of the equations above
with the magnetic ﬁeld B
i
, we obtain
E
k
B
k
= 0, (25)
J
i
= q
ij k
E
j
B
k
B
2
+ J
B
B
i
B
2
. (26)
The ﬁrst relation (25) implies that the electric and magnetic
ﬁelds are orthogonal, while expression (26) deﬁnes the current,
whose component parallel to the magnetic ﬁeld, namely J
B
J
i
B
i
, needs to be deﬁned via a suitable Ohm law. From the
numerical point of view, speciﬁc strategies must be adopted in
order to enforce the FF constraints expressed by Equations (25)
and (26). In fact, even though such constraints are exactly
satisﬁed at time t = 0, there is no guarantee that they will
remain so during the evolution of the system.
The approach introduced by Palenzuela et al. (2010a)to
enforce the constraints (25) and (26) consists in a modiﬁcation
of the system at the discrete level, by redeﬁning the electric ﬁeld
after each timestep in order to remove any component parallel to
the magnetic ﬁeld. In other words, after each timestep the newly
computed electric ﬁeld is “cleaned” by imposing the following
transformation (Palenzuela et al. 2010a)
E
i
E
i
(E
k
B
k
)
B
i
B
2
. (27)
In addition, the current is computed from Equation (26)af-
ter setting J
B
= 0. An alternative approach, introduced in
Komissarov (2011) and then in Lyutikov (2011), uses the
Maxwell equations to compute
D
t
(E
k
B
k
), which has to van-
ish according to Equation (25). Using Equations (10) and (11)
it is then easy to obtain the following prescription for J
B
:
J
B
=
1
α
[B
i
ij k
j
(αB
k
) E
i
ij k
j
(αE
k
)]. (28)
Without further modiﬁcations, however, this approach leads to
large violations of the FF constraint (25) in long-term numerical
simulations, as it does not provide a mechanism for imposing
the constraint at later times.
As we will show later on, both approaches (27) and (28)
are not fully satisfactory and, as a consequence, here we
present an alternative method, which takes inspiration from the
treatment of currents (and related stiff source terms) in resistive
magnetohydrodynamics. The idea of introducing a suitable Ohm
law was proposed in Komissarov (2004) and then in Palenzuela
et al. (2010a), but it has not been used so far in numerical
simulations, due to the presence of stiff terms which appear as a
result. In practice, our continuum approach is equivalent to the
insertion of suitable driver terms, so that the parallel component
J
B
is computed from an Ohm law of the type
J
B
= σ
B
E
k
B
k
, (29)
where σ
B
is the anisotropic conductivity along the magnetic-
ﬁeld lines. This additional term in the current acts like a damping
term in the evolution
t
(E
k
B
k
), and enforces the constraint (25)
on a timescale 1
B
.Forσ
B
sufﬁciently large, one can ensure
that the FF constraint (25) is always satisﬁed. In the simulations
presented in this paper, we choose σ
B
> 1/Δt, where Δt is the
timestep on the ﬁnest reﬁnement level. The resulting hyperbolic
system with stiff terms is solved using a third-order RKIMEX
time integration method with the technical implementation
following the one discussed in Palenzuela et al. (2009b) and
with additional details presented in the Appendix.
An additional problem in the numerical treatment of the
FF approach is represented by the development of current sheets,
namely of regions where the electric ﬁeld becomes larger than
the magnetic ﬁeld, such that the condition
B
2
E
2
> 0 (30)
is violated. If this happens, and in the absence of a proper Ohm
law responsible for the resistive effects, the Alfv
´
en wave speed
becomes complex and the system of FF equations is no longer
hyperbolic (Komissarov 2004). Under realistic conditions, one
expects that in these regions an anomalous and isotropic
resistivity would restore the dominance of the magnetic ﬁeld.
A solution to this problem was proposed in Komissarov (2006),
where the velocity of the drift current was modiﬁed in order
to ensure that it is always smaller than the speed of light. This
leads to the following prescription for the current:
J
i
= q
ij k
E
j
B
k
B
2
+ E
2
+ J
B
B
i
B
2
, (31)
which should be compared with Equation (26) and has the net
result of underestimating the value of the current.
An alternative solution to the numerical treatment of current
sheets consists in a modiﬁcation of the system again at the
discrete level (Palenzuela et al. 2010a). In practice, after each
timestep a correction is applied “by hand” to the magnitude of
the electric ﬁeld in order to keep it smaller than the magnetic
ﬁeld, i.e.,
E
i
E
i
(1 Θ)+Θ
B
2
E
2
, (32)
with Θ = 1 when B
2
E
2
< 0 and Θ = 0 otherwise.
Our strategy, however, differs from both the previous ones
and follows the same philosophy behind the choice of the driver
deﬁned by Equation (29). We therefore introduce a second
driver in Ohm law, which will act as a damping term for the
electric ﬁeld in those cases when E
2
>B
2
. This additional
term, combined with the prescription for the parallel part of the
current (29), leads to the following effective Ohm law:
J
i
= q
ij k
E
j
B
k
B
2
+ σ
B
(E
k
B
k
)
B
i
B
2
σ
B
(B
2
E
2
)E
i
E
2
B
2
.
(33)
Expression (33) shows therefore that in normal conditions, i.e.,
when B
2
E
2
0, the last term introduces a very small
and negative current along the direction of the electric ﬁeld.
However, should a violation of the condition (30) take place, a
positive current is introduced, which reduces the strength of the
electric ﬁeld and restores the magnetic dominance.
In Section 5 we will compare the different prescriptions for
the enforcement of the FF condition and show that, in contrast to
recipes (27) and (32), our suggestions (29) and (33) yield both
and accurate and a smooth distribution of the EM currents.
4

The Astrophysical Journal, 754:36 (17pp), 2012 July 20 Alic et al.
3. ANALYSIS OF RADIATED QUANTITIES
The calculation of the EM and gravitational radiation gen-
erated during the inspiral, merger, and ringdown is an impor-
tant aspect of this work as it allows us to measure the amount
correlation between the two forms of radiation. We compute
the gravitational radiation via the Newman-Penrose curvature
scalars. In practice, we deﬁne an orthonormal basis in the three-
dimensional space (
ˆ
r,
ˆ
θ,
ˆ
φ), with poles along ˆz. Using the nor-
mal to the slice as timelike vector
ˆ
t, we construct the null or-
thonormal tetrad {l , n, m,
m}:
l =
1
2
(
ˆ
t +
ˆ
r), n =
1
2
(
ˆ
t
ˆ
r), m =
1
2
(
ˆ
θ + i
ˆ
φ), (34)
with the bar indicating a complex conjugate. Adopting this
tetrad, we project the Weyl curvature tensor C
αβγ δ
to obtain
Ψ
4
C
αβγ δ
n
α
¯m
β
n
γ
¯m
δ
, that measures, ideally at null inﬁnity,
the outgoing gravitational radiation. For the EM emission, on the
other hand, we use two equivalent approaches to cross-validate
our measures. The ﬁrst one uses the Newman-Penrose scalars
Φ
0
(for the ingoing EM radiation) and Φ
2
(for the outgoing
EM radiation), deﬁned using the same tetrad (Teukolsky 1973):
Φ
0
F
μν
l
ν
m
μ
, Φ
2
F
μν
m
μ
n
ν
. (35)
By construction, the Newman-Penrose scalars Ψ
4
, Φ
0
, Φ
2
are
dependent on the null tetrad (34), so that truly unambiguous
scalars are measured only at very large distances from the
sources, where inertial observers provide preferred choices. Any
measure of these quantities in the strong-ﬁeld region is therefore
subject to ambiguity and risks producing misleading results. As
an example, the EM energy ﬂux does not show the expected
1/r
2
scaling when Φ
2
and Φ
0
are measured at distances of
r 20 M, as used in Palenzuela et al. (2010a, 2010b), which
is instead reached only for r 100 M. As we will show in
Section 6, this fact is responsible for signiﬁcant differences in
the estimates of the non-collimated EM emission.
The use of a uniform magnetic ﬁeld within the computational
domain has a number of drawbacks, most notably, nonzero
initial values of Φ
2
, Φ
0
. As a result, great care has to be
taken when measuring the EM radiation. Fortunately, we can
exploit the linearity in the Maxwell equations to distinguish the
genuine emission induced by the presence of the BH(s) from
the background one. Following Teukolsky (1973), we compute
the total EM luminosity as a surface integral across a 2-sphere
at a large distance:
L
EM
= lim
r→∞
1
2π
r
2
(|Φ
2
|
2
−|Φ
0
|
2
) dΩ, (36)
which results straightforwardly from the integration of the
component of EM stress-energy tensor (4) along the time-
like vector n
μ
and the normal direction to the large
2-sphere (namely, the ﬂux of the Poynting vector in
Equation (19) through the 2-sphere). The term Φ
0
in
Equation (36) has been maintained (it disappears at null inﬁnity)
to account for the possible presence of an ingoing component
in the radiation at ﬁnite distances. In particular, Equation (36)
shows that the net ﬂux is obtained by adding (with the appropri-
ate sign) the respective contributions of the outgoing and ingoing
ﬂuxes. More speciﬁcally, in terms of the complex scalars Φ
2
and
Φ
0
, the outgoing net ﬂux is obtained by subtracting the square of
their respective moduli. In the speciﬁc scenario considered here,
where a nonzero non-radiative component of the magnetic ﬁeld
extends to large distances, expression (36) must be modiﬁed.
More speciﬁcally we rewrite it as
L
EM
= lim
r→∞
1
2π
r
2
(|Φ
2
Φ
2,B
|
2
−|Φ
0
Φ
0,B
|
2
) dΩ, (37)
where Φ
2,B
and Φ
0,B
are the values of the background scalars
induced by the asymptotically uniform magnetic-ﬁeld solution
in the time-dependent spacetime produced by the binary BHs.
Under the assumption of a vanishing net ingoing radiation, i.e.,
Φ
0
Φ
0,B
and of stationarity of the background ﬁeld, i.e.,
Φ
2,B
Φ
0,B
, expression (37) can also be rewritten as (Neilsen
et al. 2011;Ruizetal.2012)
L
EM
= lim
r→∞
1
2π
r
2
(|Φ
2
Φ
0
|
2
) dΩ. (38)
Although Equation (38) does not represent, at least in a strict
physical and mathematical sense, a valid expression for the
emission of EM radiation in generic scenarios, it can provide a
useful recipe whenever the assumed approximations made above
are actually fulﬁlled. In Section 6 we will assess to what degree
this is the case for the speciﬁc scenario and model considered
here.
The choice of the background values of the Newman-Penrose
scalars Φ
2,B
and Φ
0,B
plays a crucial role in measuring correctly
the radiative EM emission, since these quantities are themselves
time dependent and cannot be distinguished, at least a priori,
from the purely radiative contributions. This introduces an
ambiguity in the deﬁnition of Φ
2,B
and Φ
0,B
, which can,
however, be addressed in at least two different ways. The ﬁrst
one consists in assuming that the background values are given by
the initial values, and further neglecting their time dependence,
namely setting
Φ
2,B
= Φ
2
(t = 0), Φ
0,B
= Φ
0
(t = 0). (39)
Since all the m = 0 multipoles of the Newman-Penrose scalars
are not radiative, a second way to resolve the ambiguity is to
remove those multipole components from the estimates of the
scalars, namely, of deﬁning
Φ
2,B
= (Φ
2
)
,m=0
, Φ
0,B
= (Φ
0
)
,m=0
, (40)
where (Φ
2
)
,m=0
refer to the m = 0 modes of the multipolar
decomposition of Φ
2
( 8 is sufﬁcient to capture most of the
background). Note also that because the m = 0 background
is essentially time independent (after the initial transient),
the choice (40) is effectively equivalent to the assumption that
the background is given by the ﬁnal values of the Newman-
Penrose scalars as computed in an electrovacuum evolution
of the same binary system. While apparently different, ex-
pressions (39) and (40) lead to very similar estimates (see
Section 6.1) and, more importantly, they have a simple interpre-
tation in terms of the corresponding measures that they allow.
The second approach that we have followed for the compu-
tation of the emitted luminosity is the evaluation of the ﬂux of
the Poynting vector across a 2-sphere at large distances in terms
of the more familiar 3+1 ﬁelds E
i
and B
i
in Equation (19). Of
course, such evaluation is adequate only far from the binary.
The purpose of implementing both versions of the luminosity
calculation, which are conceptually equivalent but differ in the
technical details, is precisely to quantify the error introduced by
5

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• ...The first one uses the Newman-Penrose scalars Φ0 (for the ingoing EM radiation) and Φ2 (for the outgoing EM radiation), defined using the same tetrad (Teukolsky 1973): Φ0 ≡ Fµν lνmµ , Φ2 ≡ Fµνmµnν ....

[...]

• ...Following Teukolsky (1973), we compute the total EM luminosity as a surface integral across a 2-sphere at a large distance: L EM = lim r→∞ 1 2π ∫ r2 ( |Φ2|2 − |Φ0|2 ) dΩ , (36) which results straightforwardly from the integration of the component of EM stress-energy tensor (4) along the timelike…...

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Journal ArticleDOI

TL;DR: A new approach to the stabilization of numerical schemes in magnetohydrodynamic processes in which the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same time is developed.
Abstract: In simulations of magnetohydrodynamic (MHD) processes the violation of the divergence constraint causes severe stability problems. In this paper we develop and test a new approach to the stabilization of numerical schemes. Our technique can be easily implemented in any existing code since there is no need to modify the solver for the MHD equations. It is based on a modified system in which the divergence constraint is coupled with the conservation laws by introducing a generalized Lagrange multiplier. We suggest a formulation in which the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same time. This corrected system is hyperbolic and the density, momentum, magnetic induction, and total energy density are still conserved. In comparison to results obtained without correction or with the standard "divergence source terms," our approach seems to yield more robust schemes with significantly smaller divergence errors.

1,053 citations

### "Accurate Simulations of Binary Blac..." refers background in this paper

• ...…Maxwell equations (2) and (3) is referred to as “extended” because it incorporates the so-called divergencecleaning approach, originally presented in Dedner et al. (2002) in flat spacetime, and which amounts to introducing two additional scalar fields, Ψ and Φ, that propagate away the deviations…...

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Book
25 May 2010
Abstract: The physics of black holes is explored in terms of a membrane paradigm which treats the event horizon as a two-dimensional membrane embedded in three-dimensional space. A 3+1 formalism is used to split Schwarzschild space-time and the laws of physics outside a nonrotating hole, which permits treatment of the atmosphere in terms of the physical properties of thin slices. The model is applied to perturbed slowly or rapidly rotating and nonrotating holes, and to quantify the electric and magnetic fields and eddy currents passing through a membrane surface which represents a stretched horizon. Features of tidal gravitational fields in the vicinity of the horizon, quasars and active galalctic nuclei, the alignment of jets perpendicular to accretion disks, and the effects of black holes at the center of ellipsoidal star clusters are investigated. Attention is also given to a black hole in a binary system and the interactions of black holes with matter that is either near or very far from the event horizon. Finally, a statistical mechanics treatment is used to derive a second law of thermodynamics for a perfectly thermal atmosphere of a black hole.

913 citations