General Relativistic Magnetohydrodynamic Equations Research Articles

Cold Plasma Gravitomagnetic Waves in a Kerr Planar Analogue

In this research, we investigated the wave properties of the Kerr black hole magnetosphere near the pair production region for a cold plasma. For this purpose, we obtain the general relativistic magnetohydrodynamical equations in the 3 + 1 formalism for the Kerr spacetime in a planar analogue. The Fourier analyzed form is considered for a rotating background fllled with a magnetized cold plasma. The dispersion relations are obtained and give the values of the components of the wave vector k. These are used to investigate the behavior of gravitomagnetic waves under the in∞uence of gravity and the magnetospheric wind. This is used to detect the response of the black hole magnetospheric plasma oscillations to gravitomagnetic perturbations near the pair-production region. We conclude that energy extraction is possible for the cold plasma case. Further, we check the existence of Veselago medium in the black hole region and recover the results of Mackay et al.

Dispersion Relations for Isothermal Plasma around the Horizon of Reissner–Nordström–de Sitter Black Hole

We formulate the general relativistic magnetohydrodynamic equations for isothermal plasma in spatially flat Reissner–Nordström–de Sitter metric by using 3+1 split of spacetime. Respective perturbed equations are linearized for rotating magnetized surroundings. These are then Fourier analyzed and the corresponding dispersion relations are obtained. These relations are discussed both analytically and numerically in order to investigate the nature of waves with positive angular frequency around the horizon.

An analysis of wave properties of isothermal plasma around Reissner-Nordström black hole

The 3+1 spacetime split is used in a preliminary investigation of waves propagating in a medium of isothermal plasma in the vicinity of event horizon of Reissner-Nordstrom planar analogue. The general relativistic magnetohydrodynamic equations for isothermal have been investigated there. The dispersion relations are obtained by using linear perturbation and Fourier analyze techniques for non-rotating and rotating, non-magnetized and magnetized environment. The wave number, phase velocity, group velocity etc. are shown to discuss the nature of the waves.

Wave properties of isothermal magneto-rotational fluids

In this paper, the isothermal plasma-wave properties in the neighborhood of the pair-production region for the Kerr black-hole magnetosphere are discussed. We consider the Fourier-analyzed form of the perturbed general relativistic magnetohydrodynamical equations whose determinant leads to a dispersion relation. For the special scenario, the x-components of the complex wave vectors are calculated numerically. Respective components of the propagation vector, attenuation vector, phase and group velocities are shown in graphs. We have investigated, in particular, the existence of a Veselago medium and wave behavior (modes of waves dispersion).

Dispersion Relations for Cold Plasmas Around Reissner-Nordström Black Holes

We investigate the general relativistic magnetohydrodynamic (GRMHD) equations for cold plasma around the Reissner-Nordström black hole. Applying 3+1 spacetime split we linearize the perturbed equations for non-magnetized/magnetized plasma in both rotating and non-rotating background. By Fourier analyze we then derive dispersion relations and investigate the existence of waves with positive angular frequency in the vicinity of the black hole horizon. The analysis finds propagation of negative phase and group velocities for rotating magnetized surroundings.

Wave properties of plasma surrounding the event horizon of a nonrotating black hole

We study the wave properties of a cold isothermal plasma in the vicinity of a Schwarzschild black-hole event horizon. The Fourier-analyzed perturbed 3+1 general relativistic magnetohydrodynamics equations are examined such that the complex dispersion relations are obtained for nonrotating, rotating nonmagnetized, and rotating magnetized backgrounds. The propagation and attenuation vectors along with the refractive index are obtained (shown graphically) to study the dispersive properties of the medium near the event horizon. The results show that no information can be obtained from the Schwarzschild magnetosphere. Further, the pressure stops the existence of normal dispersion of waves.PACS Nos.: 95.30.Sf, 95.30.Qd, 04.30.Nk

A new general relativistic magnetohydrodynamics code for dynamical spacetimes

We present a new numerical code which solves the general relativistic magneto-hydrodynamics (GRMHD) equations coupled to the Einstein equations for the evolution of a dynamical spacetime within the conformally-flat approximation. This code has been developed with the main objective of studying astrophysical scenarios in which both, high magnetic fields and strong gravitational fields appear, such as the magneto-rotational collapse of stellar cores, the collapsar model of GRBs, and the evolution of neutron stars. The code is based on an existing and thoroughly tested purely hydrodynamics code and on its extension to accommodate weakly magnetized fluids (passive magnetic field approximation). The numerical code we present here is based on high-resolution shock-capturing schemes to solve the GRMHD equations together with the flux constraint transport method to ensure the solenoidal condition of the magnetic field. Since the astrophysical applications envisaged do not deviate much from spherical symmetry, the conformal flatness condition approximation is used for the formulation of the Einstein equations. In addition, the code can handle several equations of state, from simple analytical expressions to microphysical tabulated ones. In this paper we present stringent tests of our new GRMHD numerical code, which show its ability to handle all aspects appearing in the astrophysical scenarios for which the code is intended, namely relativistic shocks, highly magnetized fluids, and equilibrium configurations of magnetized neutron stars. As an application, magneto-rotational core collapse simulations of a realistic progenitor are presented, comparing the results with our previous finding in the passive magnetic field approximation.

Effects of Schwarzschild Geometry on Isothermal Plasma Wave Dispersion

The behavior of isothermal plasma waves has been analyzed near the Schwarzschild horizon. We consider a non-rotating background with non-magnetized and magnetized plasmas. The general relativistic magnetohydrodynamical equations for the Schwarzschild planar analogue spacetime with an isothermal state of the plasma are formulated. The perturbed form of these equations is linearized and Fourier analyzed by introducing simple harmonic waves. The determinant of these equations in each case leads to a complex dispersion relation, which gives complex values of the wave number. This has been used to discuss the nature of the waves and their characteristics near the horizon.

An introduction to relativistic hydrodynamics

We review formulations of the equations of (inviscid) general relativistic hydrodynamics and (ideal) magnetohydrodynamics, along with methods for their numerical solution. Both systems can be cast as first-order, hyperbolic systems of conservation laws, following the explicit choice of an Eulerian observer and suitable fluid and magnetic field variables. During the last fifteen years, the so-called (upwind) high-resolution shock-capturing schemes based on Riemann solvers have been successfully extended from classical to relativistic fluid dynamics, both special and general. Nowadays, general relativistic hydrodynamical simulations in relativistic astrophysics are routinely performed, particularly within the test-fluid approximation but also for dynamical spacetimes. While such advances also hold true in the case of the MHD equations, the astrophysical applications investigated so far are still limited, yet the field is bound to witness major developments in the near future. The article also presents a brief overview of numerical techniques, providing state-of-the-art examples of their applicability to general relativistic fluids and magneto-fluids in characteristic scenarios of relativistic astrophysics.

Cosmos++: relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement

A code and methodology are introduced for solving the fully general relativistic magnetohydrodynamic (GRMHD) equations using time-explicit, finite-volume discretization. The code has options for solving the GRMHD equations using traditional artificial-viscosity (AV) or non-oscillatory central difference (NOCD) methods, or a new extended AV (eAV) scheme using artificial-viscosity together with a dual energy-flux-conserving formulation. The dual energy approach allows for accurate modeling of highly relativistic flows at boost factors well beyond what has been achieved to date by standard artificial viscosity methods. It provides the benefit of Godunov methods in capturing high Lorentz boosted flows but without complicated Riemann solvers, and the advantages of traditional artificial viscosity methods in their speed and flexibility. Additionally, the GRMHD equations are solved on an unstructured grid that supports local adaptive mesh refinement using a fully threaded oct-tree (in three dimensions) network to traverse the grid hierarchy across levels and immediate neighbors. Some recent studies will be summarized.

Cold plasma dispersion relations in the vicinity of a Schwarzschild black hole horizon

We apply the ADM 3 + 1 formalism to derive the general relativistic magnetohydrodynamic equations for cold plasma in spatially flat Schwarzschild metric. Respective perturbed equations are linearized for non-magnetized and magnetized plasmas both in non-rotating and rotating backgrounds. These are then Fourier analyzed and the corresponding dispersion relations are obtained. These relations are discussed for the existence of waves with positive angular frequency in the region near the horizon. Our results support the fact that no information can be extracted from the Schwarzschild black hole. It is concluded that negative phase velocity propagates in the rotating background whether the black hole is rotating or non-rotating.

WhiskyMHD: a new numerical code for general relativistic magnetohydrodynamics

The accurate modelling of astrophysical scenarios involving compact objects and magnetic fields, such as the collapse of rotating magnetized stars to black holes or the phenomenology of γ-ray bursts, requires the solution of the Einstein equations together with those of general-relativistic magnetohydrodynamics. We present a new numerical code developed to solve the full set of general-relativistic magnetohydrodynamics equations in a dynamical and arbitrary spacetime with high-resolution shock-capturing techniques on domains with adaptive mesh refinements. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code and assess its accuracy. Such tests range from the solution of relativistic Riemann problems in flat spacetime, over to the stationary accretion onto a Schwarzschild black hole and up to the evolution of oscillating magnetized stars in equilibrium and constructed as consistent solutions of the coupled Einstein–Maxwell equations.

Numerical models of black hole accretion flows

Black hole accretion flows are the hot, magnetized, turbulent plasma that surround black holes in astrophysical settings and in some cases make them the most luminous objects in the universe. I review recent developments in the numerical modeling of these plasmas. For weakly radiating flows it is now possible to follow the dynamics of the plasma by integrating the equations of general relativistic magnetohydrodynamics using numerical methods borrowed from nonrelativistic hydrodynamics. Ray tracing, along the curved trajectories followed by light close to the black hole event horizon, can then be used to create simulated observations of these weakly radiating flows. These models are relevant to the faint source at the galactic center, which may soon be directly imaged by millimeter interferometry.

Relativistic MHD with adaptive mesh refinement

This paper presents a new computer code to solve the general relativistic magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh refinement (AMR). The fluid equations are solved using a finite difference convex ENO method (CENO) in 3 + 1 dimensions, and the AMR is Berger–Oliger. Hyperbolic divergence cleaning is used to control the ∇ ⋅ B = 0 constraint. We present results from three flat space tests, and examine the accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel solution. The AMR simulations substantially improve performance while reproducing the resolution equivalent unigrid simulation results. Finally, we discuss strong scaling results for parallel unigrid and AMR runs.

General relativistic magnetohydrodynamic simulations of the jet formation and large-scale propagation from black hole accretion systems

The formation and large-scale propagation of Poynting-dominated jets produced by accreting, rapidly rotating black hole systems are studied by numerically integrating the general relativistic magnetohydrodynamic equations of motion to follow the self-consistent interaction between accretion disks and black holes. This study extends previous similar work by studying jets till $t\approx 10^4GM/c^3$ out to $r\approx 10^4GM/c^2$, by which the jet is super- fast magnetosonic and moves at a lab-frame bulk Lorentz factor of $\Gamma\sim 10$ with a maximum terminal Lorentz factor of $\Gamma_\infty\lesssim 10^3$. The radial structure of the Poynting-dominated jet is piece-wise self-similar, and fits to flow quantities along the field line are provided. Beyond the \alf surface at $r\sim 10$--$100GM/c^2$, the jet becomes marginally unstable to (at least) current-driven instabilities. Such instabilities drive shocks in the jet that limit the efficiency of magnetic acceleration and collimation. These instabilities also induce jet substructure with $3\lesssim\Gamma\lesssim 15$. The jet is shown to only marginally satisfy the necessary and sufficient conditions for kink instability, so this may explain how astrophysical jets can extend to large distances without completely disrupting. At large distance, the jet angular structure is Gaussian-like (or uniform within the core with sharp exponential wings) with a half-opening angle of $\approx 5^\circ$ and there is an extended component out to $\approx 27^\circ$. Unlike in some hydrodynamic simulations, the environment is found to play a negligible role in jet structure, acceleration, and collimation as long as the ambient pressure of the surrounding medium is small compared to the magnetic pressure in the jet.

Numerical 3+1 General Relativistic Magnetohydrodynamics: A Local Characteristic Approach

We present a general procedure to solve numerically the general relativistic magnetohydrodynamics (GRMHD) equations within the framework of the 3+1 formalism. The work reported here extends our previous investigation in general relativistic hydrodynamics (Banyuls et al. 1997) where magnetic fields were not considered. The GRMHD equations are written in conservative form to exploit their hyperbolic character in the solution procedure. All theoretical ingredients necessary to build up high-resolution shock-capturing schemes based on the solution of local Riemann problems (i.e. Godunov-type schemes) are described. In particular, we use a renormalized set of regular eigenvectors of the flux Jacobians of the relativistic magnetohydrodynamics equations. In addition, the paper describes a procedure based on the equivalence principle of general relativity that allows the use of Riemann solvers designed for special relativistic magnetohydrodynamics in GRMHD. Our formulation and numerical methodology are assessed by performing various test simulations recently considered by different authors. These include magnetized shock tubes, spherical accretion onto a Schwarzschild black hole, equatorial accretion onto a Kerr black hole, and magnetized thick accretion disks around a black hole prone to the magnetorotational instability.

Cosmos++: Relativistic Magnetohydrodynamics on Unstructured Grids with Local Adaptive Refinement

A new code and methodology are introduced for solving the general relativistic magnetohydrodynamic (GRMHD) equations in fixed background spacetimes using time-explicit, finite-volume discretization. The code has options for solving the GRMHD equations using traditional artificial-viscosity (AV) or non-oscillatory central difference (NOCD) methods, or a new extended AV (eAV) scheme using artificial-viscosity together with a dual energy-flux-conserving formulation. The dual energy approach allows for accurate modeling of highly relativistic flows at boost factors well beyond what has been achieved to date by standard artificial viscosity methods. It provides the benefit of Godunov methods in capturing high Lorentz boosted flows but without complicated Riemann solvers, and the advantages of traditional artificial viscosity methods in their speed and flexibility. Additionally, the GRMHD equations are solved on an unstructured grid that supports local adaptive mesh refinement using a fully threaded oct-tree (in three dimensions) network to traverse the grid hierarchy across levels and immediate neighbors. A number of tests are presented to demonstrate robustness of the numerical algorithms and adaptive mesh framework over a wide spectrum of problems, boosts, and astrophysical applications, including relativistic shock tubes, shock collisions, magnetosonic shocks, Alfven wave propagation, blast waves, magnetized Bondi flow, and the magneto-rotational instability in Kerr black hole spacetimes.

General relativistic hydrodynamics and magnetohydrodynamics and their applications

Upon the choice of an Eulerian observer adapted to a 3 + 1 spacetime foliation and suitable fluid and magnetic field variables, it is possible to cast the equations of both general relativistic (inviscid) hydrodynamics (GRHD) and (ideal) magneto-hydrodynamics (GRMHD) as first-order, hyperbolic systems of conservation laws for state-vectors comprising the densities of mass, momentum, energy and magnetic field components. Hyperbolicity allows the use of the flux-vector Jacobians wave structure to build up stable and accurate numerical schemes for their solution. In recent years, the so-called Godunov-type schemes, based upon approximate Riemann solvers, have been successfully extended from classical to relativistic fluid dynamics (both special and general). While such advances also hold true in the case of the MHD equations, the development still awaits here for a thorough numerical exploration. This paper reports formulations of the GRHD/GRMHD equations amenable to numerical investigations using Godunov-type schemes. A number of relevant applications in the field of relativistic astrophysics is also covered.

Pulsar magnetospheres: a general relativistic treatment

A fully general relativistic description of pulsar magnetospheres is provided. To be more specific, a study of pulsar magnetospheres is performed in the context of general relativistic magnetohydrodynamics (MHD) employing the so-called Grad–Shafranov approach. Not surprisingly, the resulting Grad–Shafranov equations and all the other related general relativistic MHD equations turn out to take essentially the same structures as those for a (rotating) black hole magnetosphere. Other different natures between the two cases including the structure of singular surfaces of MHD flows in each magnetosphere are essentially encoded in the different space–time (metric) contents. In this way, pulsar and black hole magnetospheres can be described in a unified fashion. Particularly, the direction of poloidal currents circulating in the neutron star magnetosphere turns out to be the same as that of currents circulating in a black hole magnetosphere, which in turn leads to pulsar and black hole spin-down via ‘magnetic braking’.

A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole

Some active galactic nuclei, microquasars, and gamma-ray bursts may be powered by the electromagnetic braking of a rapidly rotating black hole. We investigate this possibility via axisymmetric numerical simulations of a black hole surrounded by a magnetized plasma. The plasma is described by the equations of general relativistic magnetohydrodynamics, and the effects of radiation are neglected. The evolution is followed for 2000GM/c3, and the computational domain extends from inside the event horizon to typically 40GM/c2. We compare our results to two analytic steady state models, including the force-free magnetosphere of Blandford & Znajek. Along the way we present a self-contained rederivation of the Blandford-Znajek model in Kerr-Schild (horizon penetrating) coordinates. We find that (1) low-density polar regions of the numerical models agree well with the Blandford-Znajek model, (2) many of our models have an outward Poynting flux on the horizon in the Kerr-Schild frame, (3) none of our models have a net outward energy flux on the horizon, and (4) one of our models, in which the initial disk has net magnetic flux, shows a net outward angular momentum flux on the horizon. We conclude with a discussion of the limitations of our model, astrophysical implications, and problems to be addressed by future numerical experiments.