# Binary black hole in a double magnetic monopole field

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## Abstract

Ambient magnetic fields are thought to play a critical role in black hole jet formation. Furthermore, dual electromagnetic signals could be produced during the inspiral and merger of binary black hole systems. In this paper, we derive the exact solution for the electromagnetic field occurring when a static, axisymmetric binary black hole system is placed in the field of two magnetic or electric monopoles. As a by-product of this derivation, we also find the exact solution of the binary black hole configuration in a magnetic or electric dipole field. The presence of conical singularities in the static black hole binaries represent the gravitational attraction between the black holes that also drag the external two monopole field. We show that these off-balance configurations generate no energy outflows.

## 1 Introduction

Dazzling electromagnetic signals can be produced during the inspiral and merger of a binary supermassive black hole system as a natural outcome of galaxy mergers [1, 2]. The highly collimated jets of energy originate from a localized central region of the galaxies – generally believed to be spinning black holes – and are thought to play a fundamental role in the formation and evolution of massive galaxies.

In addition to the first direct detection of gravitational waves [3], the first observation of the collision and merger of a pair of stellar mass black holes was recently reported by the Laser Interferometer Gravitational-wave Observatory (LIGO) signaling a major scientific breakthrough [4]. The inspiral of binary black hole systems is therefore of utmost astrophysical importance for future observations in the gravitational wave band as well as in the various electromagnetic bands.

Alongside these substantial developments in the observation of binary black hole signals, there have been dramatic advancements in our theoretical understanding of electromagnetic emissions from black holes [5, 6, 7, 8, 9]. The basic picture of the electromagnetic (force-free) model for black holes was developed by Blandford and Znajek (BZ) [10] and entails an external magnetic field that is *dragged* by a rotating black hole background inducing electric fields, allowing currents to flow.

Extending the success of the process described by BZ for a single spinning black hole, the picture that emerges in [6] for a binary black hole array with jets suggests that the dynamical behavior of the system induces collimation of the electromagnetic field around each intervening black hole, distilling energy from them and ultimately merging into a single black hole with the standard BZ scenario.

In spite of all the progress, a detailed understanding of the jets of binary black holes remains elusive. In part, due to the inability of detecting clean electromagnetic signals from the central region close to the black holes and to the lack of theoretical models dealing with the interaction of electromagnetic fields and possible emissions in binary black hole systems.

As one step towards a systematic theoretical modeling of the electromagnetic emission from an inspiralling black hole binary, we consider the more simplistic scenario in which the binary is static and embedded in an external two magnetic monopole field. In this approximation, the electromagnetic field may be treated as a ”test” field on the unperturbed, asymptotically flat, neutral binary black hole solution. Specifically, we focus on a static binary black hole configuration – a pair of black holes at a finite distance with arbitrary masses – and find exact solutions for the two magnetic(electric) monopoles external field. Our starting point is a class of solutions to Einstein-Maxwell theory [11, 12] that represent two *charged* black holes. Employing a linearized analysis on the (fully back-reacted) charged black hole solutions we find a new exact *neutral* static neutral binary black hole solution embedded in an external two monopole magnetic field. In [13] the authors employed the linearized analysis (starting with the Kerr-Newman solution) to recover Wald’s solution [14]. The features of the binary black hole in a two magnetic monopole field provide a connection to several of the above mentioned issues. In particular, we will study these new exact electromagnetic fields to test the results of energy outflows in head-on collisions of non-spinning black holes [15, 16]. The exact electromagnetic field solutions that we construct are very general and are valid for arbitrary black hole masses and charges.

## 2 Setup

*A*, that solve Maxwell’s equations in vacuum

*z*.

We begin with a review of the classic Michel magnetic monopole solution [17], in polar and canonical Weyl coordinates, which illustrates the basic form of the electromagnetic test field for a single black hole. The exact binary static black hole solution [18], that is the central focus of this paper, is then discussed. We present our results for the black hole binaries in different double monopole electromagnetic fields, while the last section includes our assessment of the new configurations and conclusions.

## 3 Black hole in a monopole field

This section briefly reviews the exact solution of a Schwarzschild black hole in an external magnetic monopole field.

### 3.1 Schwarzschild in an external monopole field

*q*and has a constant flux integral \(4\pi q\) for any radius.

### 3.2 Schwarzschild in an external monopole field: Weyl coordinates

*rods*along the

*z*-axis with \(\rho =0\). For the Schwarzschild black hole the rod that represents the event horizon, has a of length equal to 2

*m*and is located at \(\{\rho =0, -m\le z\le m\}\).

## 4 Binary black hole

### 4.1 Binary black hole geometry

*z*-distance separating these sources (when \(0<z_1<z_2\)), \( \delta = m_1 m_2/(\ell ^2-m_1^2-m_2^2)\) the conical angle on the

*z*-axis and, \(f_0=(1+2 \delta )^{-2}\) guarantees the asymptotic flatness of the solution. Here \((x_i,y_i)\) with \(i=1,2\) are the bi-polar coordinates centered on the symmetry axis \(\rho =0\) at \(z=z_1\) and \(z=z_2\) defined by the expressions

*i*-th rod mid point. The black holes are of course expected to attract each other. Hence, for any choice of these parameters the solution has conical singularities which indicate the presence of pressure, a

*strut*, along the symmetry axis between the sources. Note that when \(\ell =m_1+m_2\) , the rods overlap and the solution for an individual Schwarzschild black hole of mass \(m_1+m_2\) can be retrieved. The Israel-Khan solution [18] of two identical static (neutral) black holes as found in [11] is recovered considering \(m_1=m_2=m\) positioned at \(-z_1=z_2=k\). Finally, it is worth noticing that the irreducible mass in e.g. the latter case is \(M_{irr}=(\mathcal {A}/16\pi )^{1/2}\) where \(\mathcal {A}=16\pi m^2 (1+m/k)\) is the event horizon area of a single black hole. Thus, while from a static neutral black hole energy cannot be extracted, the maximum extractable energy for the binary black hole configuration that we are studying yields \(M-M_{irr}=(2-\sqrt{1+m/k})m\).

## 5 Binary black hole in a double monopole field

### 5.1 Exact double electric monopole field solution

*charged*black holes. We are interested in finding the two monopole electric field of the solution for two

*neutral*black holes (without any back-reaction) which are relevant in the construction of the so called BZ models. Henceforth, employing a linearized analysis starting with a the two charged black hole solution, we are able to find the exact binary black hole in an external double electric monopole field solution of (1). The metric becomes precisely the neutral static binary black hole metric (2) with functions (11), and

With these solutions in hand, we will turn to the magnetic case.

### 5.2 Exact double magnetic monopole field solution

The new solutions that we find support two magnetic monopole fields of any magnitude for black holes of all sizes. As in the classical two magnetic monopole configurations (without black holes) the magnetic field lines radiate away(toward) such a positive(negative) magnetic monopole which can be thought of as an isolated magnetic north(south) pole. Another interesting situation is the limiting case in which one of the magnetic sources is zero e.g. \(q_1=0\). This remains physically sensible and corresponds to two Schwarzschild black holes hovering freely in a one magnetic monopole field of charge \(q_2\). The magnetic field patterns generated by both types of monopole are sketched in Fig. 3. Finally, one can recover the electromagnetic field of two point-like monopoles in Minkowski vacua when \(m_1=m_2=0\).

The apparent singularities at the event horizons are simply discontinuities in physical coordinates. For the solutions (18) we find that \(F^2\) is smooth on and outside the event horizons. In particular, for black hole binaries of the same mass (\(m_1=m_2=m\)) and equal external field (\(q_1=q_2=q\)) located at \(z_1=-z_2=-k\) we find that \(F^2|_{\rho =0,z=z_i\pm m_i}=k^2 q^2/8m^4(k+m)^2\).

*z*-axis. It was shown in [11, 21] that this gravitational pressure can be identified with the interaction energy. In our case this is

*z*-direction in Weyl coordinates and outflows in the \(\rho \)-direction. Nevertheless, since the time evolution of the system in the numerical simulations presumably happen through highly-dynamical geometries, it is very likely that these have little to do with the conically-deformed ones that we have studied.

Let us now make a final comment about the magnetic field (18) with (11). On the *z*-axis along the black holes, the lines of force are perpendicular and also vary. One can interpret this in terms of the standard magnetic pressure \(\mathcal {P}_z (z)=(F^2/8\pi )|_{\rho =0}\) (which applies since the electromagnetic field is degenerate \(*F^{\mu \nu }F_{\mu \nu }\)=0) and check that the conical singularity in the metric has no effect in the magnetic pressure. This can be quickly verified by noting that the pressures at the ends of the rods are equivalent \(\mathcal {P}_z(z=z_i- m_i)=\mathcal {P}_z(z=z_i+ m_i)\) causing no effect on the balance of the system. At the origin, the magnetic pressure vanishes \(\mathcal {P}_z(z=0)=0\).

## 6 Discussion

In this paper, we have obtained explicit analytic solutions describing the electric/magnetic field of two monopole fields in a static binary black hole configuration. The geometry is conically-deformed, and captures the dragging of the electromagnetic fields but not the energy outflows of the dual jets found in [15, 16]. A closer approximation to magnetospheres with non-trivial energy fluxes could be constructed by not simply splitting the monopole field, as we also did in this work, but also considering spinning black holes. We have not attempted here this generalization but hope to return to this subject in the future.

## Notes

### Acknowledgements

I would like to thank the participants of the *”Joint Columbia-USU Strings and Black Holes Workshop”*, 1-3 May 2017, in particular, Eric Hirschmann, Robert Penna and Bob Wald for inspiring discussions. I am grateful to Luis Lehner and Oscar Varela for various insightful comments. This work was supported by the Max Planck Gesellschaft through the *Gravitation and Black Hole Theory* Independent Research Group and by NSF grant PHY-1707571 at Utah State University.

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