A method for incorporating the Kerr–Schild metric in electromagnetic particle-in-cell code

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Abstract

An algorithm is presented that incorporates the Kerr–Schild metric in an electromagnetic particle-in-cell code. This paper describes the algorithm and implementation for the simulation of charged particles in the region of a spinning black hole. A description is given of the method used to calculate fields, currents, and particle motion using the tensor formalism. We test the overall model by using a ‘toy’ black hole and accretion disk system in a uniform magnetic field to produce bipolar jets.

Introduction

Relativistic jets have been observed in active galactic nuclei (AGNs) [1], [2], [3], in microquasars in our Galaxy [4], in GRBs [5], [6], as they originate in the regions near accreting (stellar) central mass [7]. It is generally accepted that astrophysical jets are magnetically launched. The magnetic-acceleration mechanism has been proposed not only for AGN jets [8], but also for (nonrelativistic) protostellar jets [7]. The accreting central mass system has been investigated by general relativistic magnetohydrodynamics (GRMHD) simulations. In GRMHD a single fluid is treated, therefore electrons and positrons (ions) are not separated. GRMHD simulations assume the frozen-in condition in which the magnetic fields may be dragged by the fluid. GRMHD codes use ideal fluid conditions and numerical diffusion. Electromagnetic particle-in-cell (EMPIC) code [9] moves individual particles separately thereby self-consistently generating electric and magnetic fields. Daniel and Tajima [10] used the EMPIC algorithm in a 3+1 Schwartzchild metric, where the simulation was a 112 dimension PIC simulation around the event horizon. Kinematic effects may contribute a role in the formation of the jet. Charge separation is developed which in turn grows the electromagnetic field. Special relativistic EMPIC codes have been used to study the formation of jets from an accreting disk [11]. These simulations have been used to understand the nature of the Poynting flux and jet current. This required a modification of the ISIS [12] code to simulate an electron jet streaming from an “anode” disk. The “frozen-in” condition, E+v×B/c=0, was imposed on the disk giving it infinite conductivity. The latest modified EMPIC code [13] was used to study the Poynting flux. This code uses the modified object-oriented EMPIC code (OOPIC) [14] and imposes electric potential conditions on the disk. The mass of the central object is not used to provide gravitational potential. Since the standard EMPIC code does not include the Kerr–Schild metric, it cannot be used to study the frame-dragging effect associated with the Kerr–Schild space–time.

The use of general curvilinear coordinates for EMPIC code has been implemented [15]. We have extended these methods to allow the representation of the relevant physics formulas in a covariant form. In this way this satisfies Einstein's theory of “local covariance”. Therefore, we reformulated the EMPIC algorithm into the tensor form, thus preserving the physics of the plasma in a non-inertial reference frame. The tensor approach provides an elegant and simple representation for incorporation of the Kerr metric in EMPIC code. Eastwood et al. use a body fitted 3DPIC algorithm for use on applications where the plasma simulation is confined in complex geometries. We use the tensor formalism of MTW. This makes the algorithm more accessible to researchers who study the jet formation for spinning black holes. With this paper we present a EMPIC simulation which uses the Kerr–Schild metric for the simulation of bipolar jet formation. The advantage of using EMPIC is that the self-consistent physics is included. However, scalings and stability conditions of the plasma must be considered.

Our algorithm was developed for plasma simulation in an environment around a spinning central mass. The basic equations used for this new code are described in Section 2. In Section 3, the numerical algorithms for the field updates and particle mover are explained. The current deposition method is also described. Jet formation is described as an example application of this new code in Section 4. The concluding remarks are discussed in Section 5.

Section snippets

Tensor and metric equations

Our algorithm is based on the EMPIC algorithm. We incorporated the physical four-vectors (i.e., velocity, current and position) along with the electromagnetic field tensor to simulate the plasma particle and field dynamics. The equations which control the development of the particles and fields are given by the tensor form of the Maxwell and Newton–Lorentz equations [16] and the Kerr metric [17].F;βαβ=4πcJαFαβ;γ+Fβγ;α+Fγα;β=0m(duαdτ+Γrsαdxrdτdxsdτ)=qFβαuβ where Jα is the four-current, Fαβ is

The computational cycle

It should be noted that in EMPIC simulations each particle represents an ensemble of charged particles with the finite shape over the grids [18], [19], [20]. The underlying physics of the particle motion is governed by the tensor form of the Newton–Lorentz equation. This form provides the equation for the acceleration of the particle. The acceleration is a function of the space–time curvature defined by the metric and the Lorentz force due to the electromagnetic field. The local field is

Application

The initial disk geometry of our simulation consists of a free falling corona and a Keplerian disk (see Fig. 5a), which is similar to Nishikawa et al. [27], [28]. The central mass is located at the center of the computational space and is co-rotating with a=.95. The central mass has a value of Mc=300. The Keplerian disk is located at r>rd3rs |cosθ|<δ, where δ=1/8 rd is the disk radius. In this region the particle number is 100 times that of the corona. There are 0.8 million disk particles. The

Conclusion

The algorithm presented here solves a EMPIC algorithm. It incorporates a generalized metric in the numerical calculation of the tensor form of Maxwell's equations and the Newton–Lorentz equation. If sufficient care is taken to satisfy the numerical stability, then this can be a very useful algorithm for using particle-in-cell codes for the simulation of astrophysical regimes where the gravity due to a central mass plays a role in the dynamics of the system.

The results presented in Fig. 5 show

Acknowledgements

M.W. was funded by NASA summer faculty fellowships (2005, 2006). This research was partially funded by a NSF subcontract PHY-0114343 with COSM (Hampton University) and the AAS Small Grant Program. M.W. is supported by UNCF Special Programs Corporation, NASA and the ASA Administrator's Fellowship Program. Simulations were performed at the Columbia facility at the NASA Advanced Supercomputing (NAS), and IBM p690 (Cobalt) at the National Center for Supercomputing Applications (NCSA) which is

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