Multipole Moments

Much research has been done on extensions of the general relativistic gravitational field to internal momenta (space-time metrics and curvature). In this decade, Phinney20 developed the GRMHD analysis of the Blandford-Znajek process in Kerr geometry. The response of the magnetosphere is characterized by the amplitudes and energy fluxes of the waves they excite.

MULTIPOLE EXPANSIONS OF THE GENERAL RELATIVISITC GRAVITATIONAL FIELD OF THE EXTERNAL UNIVERSE

The linearized, stationary solution of the vacuum Einstein field equations can be expressed in terms of these multipole moments. III we will study the structure of our Thunder coordinate external multipole expansion of the metric density. These length scales are formally defined, in terms of the multipolar moments of Eq.

HIGHER-ORDER CORRECTIONS TO THE LAWS OF MOTION AND PRECESSION FOR BLACK HOLES AND OTHER BODIES

The interpretation of the laws of motion (2) and (4) involves a number of subtleties which are discussed by Thome and Hartle. The rest of this paper consists of two derivations of the corrections (4) to the Jaws of motion and precession. See Thome and Hartle1 for a discussion of the applicability of these formulas to this problem.).

3+ 1 FORMULATION OF GENERAL-RELATIVISTIC PERFECT MAGNETOHYDRODYNAMICS

3+1 formulation of general-relativistic perfect magnetohydrodynamics

INTRODUCTION

Recently, Phinney7 has developed and applied to the Kerr geometry a (non-3+ 1) formulation of GRMHD theory and has used it in an improved MHD analysis of the Blandford-Znajek process. As a basis for this study, a 3+1 version of the GRMHD has been developed and presented in this paper. II of this paper, the full and general set of GRMHD equations is given in .

GENERAL-RELATIVISTIC MAGNETOHYDRODYNAMIC EQUATIONS

However, in Kerr geometry we have a set of preferred FIDOs [the zero angular momentum observers 13 (ZAM 0s)]; and our general formalism can easily be specialized in the Kerr geometry, the ZAMOs playing the role of the FIDOs. Vectors and tensors living in four-dimensional space-time will be denoted by bold italics, such as FIDO's four-speed n; vectors living in three-dimensional absolute space are denoted by bold Roman or Greek letters, such as the shift function l three-. All vector analysis notations such as the gradient, the curl, and the vector cross product will be those of the three-dimensional absolute space whose trimetric is y, unless otherwise specified.

Evolution of the magnetic field

Here T fluid is the four-dimensional energy-momentum tensor of the fluid and T BM is that of the electromagnetic field. If we think of the coefficient of DVj ld't on the left-hand side (LHS) of (2.22) as an "effective inertia", we can clearly see this anisotropy in the fluid's inertia caused by the magnetic field. The curl of the avxiJ term is the coupling of the induced electric field to the fluid velocity and the magnetic field.

Thermodynamic variables

To summarize, in our 3+ 1 equations for perfect MHD, the basic variables are the FIDO-measured magnetic field B and fluid velocity v, and the rest mass density Po as measured in the fluid's rest frame. Furthermore, we also insist (as is the case for a rotating Kerr black hole) that the gravitomagnetic potential points along the synunetry direction ni,. We shall see how 3+1 electrodynamics can be simplified under these conditions (Sec.IDA) and how the conservation laws associated with these KVFs can be used to simplify the analysis of equilibrium configurations (Sec.IDB and IDC).

Electrodynamics

Here 'f', considered as a scalar field in absolute space, has an upper value at any point 7 lying on the curve c. Since nt=

MHD flow

Thus, the poloidal parts of the fluxes also satisfy the conservation laws (3.29), and we will focus on them. (henceforth, "generalized momentum") and the energy at infinity carried by the unit mass of the fluid at rest. Let us assume that S "P is located in an almost flat region, so the total fluxes of generalized momentum and energy at infinity along it between S 1 and S 2 can be considered as the rates of extraction of these quantities from the strong region of gravity; e.g., leaving a Sdown surface whose normal is also orthogonal to ;;t within that region. For a Kerr black hole we shrink S 1 to the hole's axis of symmetry and let S2 t approaches the equatorial plane.

The wind equation and its solutions

CONCLUSION

To derive the wind equation, we first use the definitions for I and f [Eq. 3.45) which can be manipulated in shape. Thus, the wind equation (3.46) can be solved for 0', and the remaining flow structure can be calculated algebraically from Eqs. This type of equation of state is also of interest because it includes the cold flow limit, p=O.

For continuous flows (no shocks), the solutions to the wind equation (3.45) should extend smoothly from the region of interest to spatial infinity. However, for an arbitrary set of stream functions, D will generally become singular at critical surfaces where the flow speed is equal to one of the disturbance propagation speeds within the stationary flow. Either we have no solutions outside these critical surfaces, or if energy and momentum conservation allows, we have shocks.

To avoid such situations, we must constrain the flow functions in such a way that either these critical surfaces are pushed to or beyond spatial infinity, or the solutions pass through the critical surfaces smoothly.7•17 To make our discussion more concrete , let us assume that we have parameterized the streamlines with a parameter yP, which can be thought of as the coordinate length along the streamlines. In a future paper, the author will use this formalism to construct stationary, syncentric MHD model magnetospheres and linearize the evolution equations to study the dynamical perturbations of those magnetospheres in order to gain insight into the dynamical effects of coupling magnetic field with gravitomagnetic field. The generalized momentum and energy are transported from one region to another between flux tubes S 1 and S 2 • On a two-dimensional surface Sup• whose normal is orthogonal to m, these flux tubes are bounded by integral curves C 1 and C 2 from m .

S af is another two-dimensional surface whose normal is also orthogonal to m. b) When the length of integral curves of m is unbounded, e.g. when iit corresponds to a translation, we consider a section of flux sheets S 1 and S 2 with width oL and the general momentum and energy at infinity transported within oL .

THE INTERACTION OF MAGNETOHYDRODYNAMIC WAVES WITH GRA VITOMAGNETIC FIELDS, AND THEIR POSSIBLE ROLES

IN BLACK-HOLE MAGNETOSPHERE

ABSTRACT

MODEL SPACETIMES IN WHICH TO STUDY INTERACTIONS OF PLASMA WITH RELATIVISTIC GRAVITY

One of the main sources of difficulty is that the analysis must be 2+1 dimensional (two spatial dimensions plus a temporal dimension). Our chosen aspect will, of course, be the displacement function J3 with its conservation and release of space-time rotational (or translational) energy. We will isolate it by working with a modified version of the planar space (2.3) in which the transition function ex.

This fluid velocity is related to the components U~t of the fluid velocity in the coordinate system (2.4). Here B, the z component of B, is a constant as guaranteed by the vanishing divergence of the magnetic field [Eq. This quartic equation will be solved to obtain a stationary configuration: the quantity yu as a function of the renormalized shift C and the constant parameters s, I, f (which are considered pre-specified); and then through EQs.

In this procedure, the z-dependence of the equilibrium configuration will arise solely from the renormalized shift function C(z. There is another singular point hidden in the denominator of the derivative of yu with respect to C. Instead, we will look at the wind equation and the constraints it imposes on stationary configurations in regions of the current where C is very small and where it is very large.

Focus first on the flow region (near z =0 in the models of Section IV) where C « l.

STATIONARY MUD "MAGNETOSPHERE"

The junction conditions at the interface are really just a special form of the MHD equations there. MHD WAVES IN THE "MAGNETOSPHERE": ANALYTICAL ANALYSIS We now turn to disturbances of the steady "magnetospheres" discussed in Art. In this section we will develop the linearized, analytical theory of the perturbations, and then in Sec.

In the fluid's rest frame (denoted by prime numbers) with a uniform magnetic field along the z' direction, and the velocity and the magnetic field confined to the x'-z' plane, the Fourier components of the magnetosonic modes satisfy. In this section we will present the results of our numerical solutions of the Fourier- Details of our numerical methods and checks, and of the numerical accuracy, are discussed in the appendix.

By studying disturbances at different frequencies co and wave number kx, we will be able to deduce the response of the MHD system to disturbances in the ro-kx plane. This driving force results in a magnetospheric response in the plane of plasma production, which can be characterized by the amplitude <>s x (z =0) of the modulations of the FIDO-measured upward energy flux. By comparison, the amplification of the upward energy flux in the undisturbed magnetosphere is, because Sz=pyu (j +cl) at all altitudes,.

In this appendix, we discuss in some detail our numerical implementation of the MHD perturbation equations [Eqs.

111th z-coordinate z0, shown in the table, is the place where the two solutions begin to differ by more than 0.1%, i.e. when. The table shows results for four initial conditions, starting in turn from one of b, v. The "light plane" where C =1 lies outside the Alfven point CA, the location where the two positive branches intersect. 4 Local wavenumbers kz of MHD perturbations as functions of z for the stationary background shown in Fig.

The perturbations are only induced by f x=I0--6 at the interface z=O (i.e. f ff.=N~O), and there are generally no incoming waves I z I. The perturbations in the MHD background current act with the background very close to z = O, and do not have a simple waveform. Fixed curves correspond to perturbations on the z > 0 side; dotted curves correspond to perturbations on the z < 0 side. A). 15 Disturbances generated on both sides of the interface by a delta function source fx=10-6 there alone.

16 Perturbations in the magnetic field bz excited on both sides of the interface by a source with delta function p. Perturbations in the fluid variables do not show much variation and are therefore not given here. a) excited perturbations on the side z <0; (b) perturbations excited on the z >0 side. 18 Amplification factors (ratios of perturbation amplitudes evaluated at I z I and at z=O) for perturbations excited only by f x=I0-6 at z=O.

20 Local wavenumbers kz of MHD perturbations as a function of z for the stationary background shown in the figure. Solid curves correspond to perturbations on the side with >0; dashed curves correspond to perturbations on the side with <0. Note that on the z > 0 side the oscillation starts at about kx?: 5.5 as opposed to kx = 4 in Fig.