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arXiv:astro-ph/0507383v1 16 Jul 2005

Magnetohydrodynamics in full general relativity: Formulation and tests

Masaru Shibata and Yu-ichiou Sekiguchi

Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8902, Japan

A new implementation for magnetohydrodynamics (MHD) simulations in full general relativity

(involving dynamical spacetimes) is presented. In our implementation, Einstein’s evolution equations

are evolved by a BSSN formalism, MHD equations by a high-resolution central scheme, and induction

equation by a constraint transport method. We perform numerical simulations for standard test

problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic

magnetized Bondi flow in stationary spacetime, and a longterm evolution for self-gravitating system

composed of a neutron star and a magnetized disk in full general relativity. In the final test, we

illustrate that our implementation can follow winding-up of the magnetic field lines of magnetized

and differentially rotating accretion disks around a compact object until saturation, after which

magnetically driven wind and angular momentum transport inside the disk turn on.

04.25.Dm, 04.40.Nr, 47.75.+f, 95.30.Qd

I. INTRODUCTION

Hydrodynamics simulation in general relativity is probably the best theoretical approach for investigating dynamical

phenomena in relativistic astrophysics such as stellar core collapse to a neutron star and a black hole, and the

merger of binary neutron stars. In the past several years, this field has been extensively developed (e.g., [1–7]) and,

as a result, now it is feasible to perform accurate simulations of such general relativistic phenomena for yielding

scientific results (e.g., [6–9] for our latest results). For example, with the current implementation, radiation reaction

of gravitational waves in the merger of binary neutron stars can be taken into account within ∼ 1% error in an

appropriate computational setting [6,7]. This fact illustrates that the numerical relativity is a robust approach for

detailed theoretical study of astrophysical phenomena and gravitational waves emitted.

However, so far, most of the scientific simulations in full general relativity have been performed without taking into

account detailed effects except for general relativistic gravity and pure hydrodynamics. For example, simplified ideal

equations of state have been adopted instead of realistic ones (but see [7]). Also, the effect of magnetic fields has

been neglected although it could often play an important role in the astrophysical phenomena (but see [10]). In the

next stage of numerical relativity, it is necessary to incorporate these effects for more realistic simulations. As a step

toward a more realistic simulation, we have incorporated an implementation for ideal magnetohydrodynamics (MHD)

equations in fully general relativistic manner. In this paper, we describe our approach for these equations and then

present numerical results for test problems computed by the new implementation.

Magnetic fields indeed play an important role in determining the evolution of a number of relativistic objects. In

the astrophysical context, the plasma is usually highly conducting, and hence, the magnetic fields are frozen in the

matter. This implies that a small seed field can wind up and grow in the complex motion of the matter, resulting

in a significant effect in the dynamics of the matter such as magnetically driven wind or jet and angular momentum

redistribution. Specifically, in the context of the general relativistic astrophysics, the magnetic fields will play a role in

the following phenomena and objects: Stellar core collapse of magnetized massive stars to a protoneutron star [11] or a

black hole, stability of accretion disks (which are either non-self-gravitating or self-gravitating) around black holes and

neutron stars, magnetic braking of differentially rotating neutron stars [10] which are formed after merger of binary

neutron stars [6,7] and stellar core collapse [14–16,8,9], and magnetically induced jet around the compact objects

(e.g., [17]). To clarify these phenomena, fully general relativistic MHD (GRMHD) simulation (involving dynamical

spacetimes) is probably the best theoretical approach.

In the past decade, numerical implementations for GRMHD simulation in the fixed gravitational field have been

extensively developed (e.g., [18,17,19–24]). In particular, it is worth to mention that Refs. [19–23] have recently

presented implementations for which detailed tests have been carried out for confirmation of the reliability of their

computation, in contrast with the attitude in an early work [17]. They are applied for simulating magnetorotational

instability (MRI) of accretion disks and subsequently induced winds and jets around black holes and neutron stars.

On the other hand, little effort has been paid to numerical implementations of fully GRMHD (in the dynamical

gravitational field). About 30 years ago, Wilson performed a simulation for collapse of a magnetized star in the

presence of poloidal magnetic fields in general relativity. However, he assumes that the three-metric is conformally

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flat [25], and hence, the simulation is not fully general relativistic, although recent works have indicated that the

conformally flat approximation works well in the axisymmetric collapse (e.g., compare results among [15], [26], and

[27]). The first fully GRMHD simulation for stellar collapse was performed by Nakamura about 20 years ago [28]. He

simulated collapse of nonrotating stars with poloidal magnetic fields to investigate the criteria for formation of black

holes and naked singularities. Very recently, Duez et al. have presented a new implementation capable of evolution

for the Einstein-Maxwell-MHD equations for general cases [10]. They report successful results for test simulations.

Valencia group has also developed a GRMHD implementation very recently [29].

In this paper, we present our new implementation for fully GRMHD which is similar to but in part different from

that in [10]∗. As a first step toward scientific simulations, we have performed simulations in standard test problems

including special relativistic magnetized shocks, general relativistic Bondi flow in stationary spacetime, and long term

evolution of fully general relativistic stars with magnetic fields. We here report the successful results for these test

problems.

Before proceeding, we emphasize that it is important to develop new GRMHD implementations. In the presence of

magnetic fields, matter motion often becomes turbulence-like due to MRIs in which a small scale structure often grows

most effectively [30]. Furthermore in the presence of general relativistic self-gravity which has a nonlinear nature,

the matter motion may be even complicated. Perhaps, the outputs from the simulations will contain results which

have not been well understood yet, and thus, are rich in new physics. Obviously high accuracy is required for such

frontier simulation to confirm novel numerical results. However, because of the restriction of computational resources,

it is often very difficult to get a well-resolved and completely convergent numerical result in fully general relativistic

simulation. In such case, comparison among various results obtained by different numerical implementations is crucial

for checking the reliability of the numerical results. From this point of view, it is important to develop several numerical

implementations in the community of numerical relativity.

implementations, reliability of the numerical results will be improved each other. Our implementation presented

here will be useful not only for finding new physics but also for checking numerical results by other implementations

such as that very recently presented in [10,29].

In Sec. II, we present formulations for Einstein, Maxwell, and GRMHD equations. In Sec. III, numerical methods

for solving GRMHD equations are described. In Sec. IV, methods for a solution of initial value problem in general

relativity is presented. In Secs. V and VI, numerical results for special and general relativistic test simulations are

shown. In the final subsection of Sec. VI, we illustrate that our implementation can follow growth of magnetic fields of

accretion disks in fully general relativistic simulation. Sec. VII is devoted to a summary and a discussion. Throughout

this paper, we adopt the geometrical units in which G = c = 1 where G and c are the gravitational constant and the

speed of light. Latin and Greek indices denote spatial components and spacetime components, respectively. ηµν and

δij(= δij) denote the flat spacetime metric and the Kronecker delta, respectively.

By comparing several results computed by different

II. BASIC EQUATIONS

A. Definition of variables

Basic equations consist of the Einstein equations, general relativistic hydrodynamic equations, and Maxwell equa-

tions. In this subsection, we define the variables used in these equations. The fundamental variables for geometry are

α: lapse function, βk: shift vector, γij: metric in three-dimensional spatial hypersurface, and Kij: extrinsic curvature.

The spacetime metric gµνis written as

gµν= γµν− nµnν, (1)

where nµis a unit normal to a spacelike spatial hypersurface Σ and is written as

nµ=

?1

α,−βi

α

?

,ornµ= (−α,0). (2)

In the BSSN formalism [35], one defines γ ≡ ηe12φ= det(γij): determinant of γij, ˜ γij= e−4φγij: conformal three-

metric, K = Kk

k: trace of the extrinsic curvature, and˜Aij ≡ e−4φ(Kij− Kγij/3): a tracefree part of the extrinsic

∗For instance, our formulation for Einstein’s evolution equations, gauge conditions, and our numerical scheme for GRMHD

equations are different from those in [10] as mentioned in Secs. II and III.

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curvature. Here, η denotes the determinant of flat metric; in the Cartesian coordinates, η = 1, and in the cylindrical

coordinates (̟,ϕ,z), η = ̟2. In the following, ∇µ, Di, and˜Didenote the covariant derivatives with respect to gµν,

γij, and ˜ γij, respectively. ∆ and˜∆ denote the Laplacians with respect to γij and ˜ γij. Rij and˜Rij denote the Ricci

tensors with respect to γij and ˜ γij, respectively.

The fundamental variables in hydrodynamics are ρ: rest-mass density, ε : specific internal energy, P : pressure, and

uµ: four velocity. From these variables, we define the following variables which often appear in the basic equations:

ρ∗≡ ρwe6φ,

vi≡dxi

dt

h ≡ 1 + ε +P

w ≡ αut.

(3)

=ui

ut= −βi+ γijuj

ut,(4)

ρ,

(5)

(6)

Here, ρ∗is a weighted baryon rest mass density from which the conserved baryon rest mass can be computed as

M∗=

?

ρ∗η1/2d3x.(7)

The fundamental variable in the ideal MHD is only bµ: magnetic field. The electric field Eµin the comoving frame

Fµνuνis assumed to be zero, and electric current jµis not explicitly necessary for evolving the field variables. Using

the electromagnetic tensor Fµν, bµis defined by [31]

bµ≡ −1

2ǫµναβuνFαβ,(8)

where ǫµναβis the Levi-Civita tensor with ǫt123=√−g and ǫt123= −1/√−g. Equation (8) implies

bµuµ= 0. (9)

Using Eq. (8), Fµνin the ideal MHD is written as

Fµν= ǫµναβuαbβ, (10)

and thus, it satisfies the ideal MHD condition

Fµνuν= 0. (11)

The dual tensor of Fµν is defined by

F∗

µν≡1

2ǫµναβFαβ= bµuν− bνuµ.(12)

For rewriting the induction equation for the magnetic fields into a simple form (see Sec. II D), we define the three-

magnetic field as

Bi≡ −e6φγi

jF∗jµnµ= e6φ(wbi− αbtui). (13)

Here, we note that Bt= 0 (i.e., Bµnµ= 0), and thus, Bi= γijBj. Equations (13) and (9) lead to

bt=Bµuµ

αe6φ

and bi=

1

we6φ

?

Bi+ Bjujui

?

(14)

Using the hydrodynamic and electromagnetic variables, energy-momentum tensor is written as

Tµν= TFluid

µν

+ TEM

µν. (15)

TFluid

µν

and TEM

µν

denote the fluid and electromagnetic parts defined by

TFluid

µν

= (ρ + ρε + P)uµuν+ Pgµν= ρhuµuν+ Pgµν,(16)

TEM

µν = FµσFσ

ν −1

4gµνFαβFαβ=

?1

2gµν+ uµuν

?

b2− bµbν,(17)

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where

b2= bµbµ=B2+ (Biui)2

w2e12φ

.(18)

Thus, Tµνis written as

Tµν= (ρh + b2)uµuν+

?

P +1

2b2?

gµν− bµbν. (19)

For the following, we define magnetic pressure and total pressure as Pmag= b2/2 and Ptot≡ P + b2/2, respectively.

The (3+1) decomposition of Tµνis

ρH≡ Tµνnµnν= (ρh + b2)w2− Ptot− (αbt)2,

Ji≡ −Tµνnµγν

Sij≡ Tµνγµ

(20)

i= (ρh + b2)wui− αbtbi,

j= (ρh + b2)uiuj+ Ptotγij− bibj.

(21)

iγν

(22)

Using these, the energy-momentum tensor is rewritten in the form

Tµν= ρHnµnν+ Jiγi

µnν+ Jiγi

νnµ+ Sijγi

µγj

ν. (23)

This form of the energy-momentum tensor is useful for deriving the basic equations for GRMHD presented in Sec. II

C. For the following, we define

S0≡ e6φρH,

Si≡ e6φJi.

(24)

(25)

These variables together with ρ∗and Biare evolved explicitly in the numerical simulation of the ideal MHD (see Sec.

II C).

B. Einstein’s equation

Our formulation for Einstein’s equations is the same as in [6] in three spatial dimensions and in [34] in axial

symmetry. Here, we briefly review the basic equations in our formulation. Einstein’s equations are split into constraint

and evolution equations. The Hamiltonian and momentum constraint equations are written as

Rk

k−˜Aij˜Aij+2

Di˜Ai

3K2= 16πρH, (26)

j−2

3DjK = 8πJj, (27)

or, equivalently

˜∆ψ =ψ

8

˜Rk

k− 2πρHψ5−ψ5

j) −2

8

?˜Aij˜Aij−2

3K2?

, (28)

˜Di(ψ6˜Ai

3ψ6˜DjK = 8πJjψ6, (29)

where ψ ≡ eφ. These constraint equations are solved to set initial conditions. A method in the case of GRMHD is

presented in Sec. IV.

In the following of this subsection, we assume that Einstein’s equations are solved in the Cartesian coordinates

(x,y,z) for simplicity. Although we apply the implementation described here to axisymmetric issues as well as

nonaxisymmetric ones, this causes no problem since Einstein’s equations in axial symmetry can be solved using the

so-called Cartoon method in which an axisymmetric boundary condition is appropriately imposed in the Cartesian

coordinates [32–34]: In the Cartoon method, the field equations are solved only in the y = 0 plane, and grid points

of y = ±∆x (∆x denotes the grid spacing in the uniform grid) are used for imposing the axisymmetric boundary

conditions.

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We solve Einstein’s evolution equations in our latest BSSN formalism [35,6]. In this formalism, a set of variables

(˜ γij,φ,˜Aij,K,Fi) are evolved. Here, we adopt an auxiliary variable Fi≡ δjl∂l˜ γij that is the one originally proposed

and different from the variable adopted in [10] in which ∂i˜ γijis used. Evolution equations for ˜ γij, φ,˜Aij, and K are

(∂t− βl∂l)˜ γij= −2α˜Aij+ ˜ γikβk

,j+ ˜ γjkβk

,i−2

?

3˜ γijβk

?

,k, (30)

(∂t− βl∂l)˜Aij= e−4φ

?

α

?

Rij−1

3e4φ˜ γijRk

k

−DiDjα −1

,j˜Aki−2

3e4φ˜ γij∆α

??

+α(K˜Aij− 2˜Aik˜Ak

?

?

?˜Aij˜Aij+1

j) + βk

,i˜Akj+ βk

3βk

,k˜Aij

−8παe−4φSij−1

3˜ γijSk

k

?

, (31)

(∂t− βl∂l)φ =1

6

−αK + βk

,k

?

, (32)

(∂t− βl∂l)K = α

3K2?

− ∆α + 4πα(ρH+ Sk

k). (33)

For a solution of φ, the following conservative form may be adopted [6]:

∂te6φ− ∂i(βie6φ) = −αKe6φ. (34)

For computation of Rijin the evolution equation of˜Aij, we decompose

Rij=˜Rij+ Rφ

ij, (35)

where

Rφ

ij= −2˜Di˜Djφ − 2˜ γij˜∆φ + 4˜Diφ˜Djφ − 4˜ γij˜Dkφ˜Dkφ,

˜Rij=1

2

(36)

?

δkl(−hij,kl+ hik,lj+ hjk,li) + 2∂k(fkl˜Γl,ij) − 2˜Γl

kj˜Γk

il

?

. (37)

In Eq. (37), we split ˜ γijand ˜ γijas δij+ hijand δij+ fij, respectively.˜Γk

˜ γij, and˜Γk,ij= ˜ γkl˜Γl

In addition to a flat Laplacian of hij,˜Rijinvolves terms linear in hijas δklhik,lj+ δklhjk,li. To perform numerical

simulation stably, we replace these terms by Fi,j+ Fj,i. This is the most important part in the BSSN formalism,

pointed out originally by Nakamura [28]. The evolution equation of Fiis derived by substituting Eq. (30) into the

momentum constraint as

ijis the Christoffel symbol with respect to

ij. Because of the definition det(˜ γij) = 1 (in the Cartesian coordinates), we use˜Γk

ki= 0.

(∂t− βl∂l)Fi= −16παJi+ 2α

+ δjk?

?

fkj˜Aik,j+ fkj

,j˜Aik−1

2

˜Ajlhjl,i+ 6φ,k˜Ak

i−2

?

3K,i

?

−2α,k˜Aij+ βl

,khij,l+

?

˜ γilβl

,j+ ˜ γjlβl

,i−2

3˜ γijβl

,l

?

,k

.(38)

We also have two additional notes for handling the evolution equation of˜Aij. One is on the method for evaluation

of Rk

kfor which there are two options, use of the Hamiltonian constraint and direct calculation by

Rijγij= e−4φ(˜Rk

k+ Rφ

ij˜ γij).(39)

We always adopt the latter one since with this, the conservation of the relation˜Aij˜ γij= 0 is much better preserved.

The other is on the handling of a term of ˜ γijδklhij,klwhich appears in˜Rk

k. This term is written by

˜ γijδklhij,kl= −δklhij,kfij

,l,(40)

where we use det(˜ γij) = 1 (in the Cartesian coordinates).

As the time slicing condition, an approximate maximal slice condition K ≈ 0 is adopted following previous papers

(e.g., [36]). As the spatial gauge condition, we adopt a hyperbolic gauge condition as in [37,6]. Successful numerical

results for merger of binary neutron stars and stellar core collapse in these gauge conditions are presented in [6,7,26,8].

We note that these are also different from those in [10].

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