Electro-Magnetic Energy Extraction from Rotating Black Holes in AdS

It is widely believed that astrophysical black holes are typically surrounded by magnetospheres composed of an electromagnetic plasma that is force-free to a good approximation. In Blandford:1977ds , Blandford and Znajek used a model of force-free electrodynamics (FFE) for the plasma surrounding a spinning black hole to suggest a mechanism for energy extraction from the black hole. They argued that rotational energy escapes via plasma currents along magnetic field lines that thread the horizon. The Blandford Znajek (BZ) process is considered to provide the basic picture behind astrophysical observations of jets, e.g. in active galactic nuclei and quasars. For more recent discussions on the consistency of the BZ mechanism, see Grignani:2018ntq ; Grignani:2019dqc ; Armas:2020mio .

The FFE solution discussed in the original paper Blandford:1977ds is an approximate analytical solution for a slowly rotating Kerr black hole. Despite a lot of work on numerical as well as analytical solutions (1973MichelCurtis ; 1976Blandford ; Menon:2005mg ; Tanabe:2008wm ; Brennan:2013kea ; Gralla:2015vta , see also Gralla:2014yja and references therein), it remains a challenge to find a physically realistic, exact and energy extracting solution of the FFE equations in a Kerr black hole background.

One successful strategy for generating families of solutions has been to exploit the enhanced amount of symmetry in the near-horizon region of an extremal, i.e. maximally spinning, Kerr black hole Lupsasca:2014pfa ; Lupsasca:2014hua ; Zhang:2014pla . In this paper we will continue this effort by proposing a previously overlooked symmetric ansatz for the electromagnetic field strength of the plasma. We discuss in detail the corresponding energy outflux. By making use of a toy model of a rotating electromagnetic configuration, we argue the correct measure for energy extraction is the outflux measured by a rotating observer in the near-horizon geometry, corresponding to an observer at rest in Kerr. The new symmetric ansatz has the main advantage (over previous energy-extracting, symmetric FFE solutions Zhang:2014pla ) that it gives rise to an energy outflux at the boundary of the near-horizon region, as measured by a Kerr observer, that is finite.

The FFE solutions that can be produced with the new ansatz in the Near Horizon Extremal Kerr geometry – known as NHEK – do not obey the assumed physical boundary conditions presented in this paper. However, when we generalize the background to the Near Horizon Extremal AdS-Kerr geometry – introduced later as AdS-NHEK or SE-NHEK for super-entropic Near Horizon Extremal AdS-Kerr – we do find such a solution. For the background in question the topology is in fact that of a cylinder and this metric therefore provides a close resemblance to our toy model of a rotating conducting cylinder.

The generalization to AdS-Kerr was motivated by the discovery in Jacobson:2017xam of an exotic application of the BZ process that does not involve the presence of plasma, realized in a BTZ background. While not immediately relevant astrophysically, the study of FFE in AdS backgrounds may prove useful for obtaining a better understanding of remaining conceptual questions regarding the BZ process, such as where and how the negative energy originates. Indeed we hope the presented toy model and FFE solutions in near-horizon AdS-Kerr may contribute to this goal in the future.

The toy model and new symmetric log ansatz in NHEK were first presented in the unpublished ThesisLibby . In this paper we additionally apply these ideas in the context of AdS-Kerr black holes.

The paper is organized as follows. We begin in section 2 with a review where we explain how a 2-dimensional (2D) action naturally describes toroidal force-free plasma systems surrounding a spinning black hole. At this stage we already assume a system that is toroidally invariant and therefore effectively 2-dimensional. We present in section 3 all the near-horizon geometries of spinning black holes that we will be using in the paper. Next, in section 4, we apply the technique of imposing the extra scaling symmetry of the near-horizon regions on the field strength of the plasma. This gives rise to the scaling ansatz in 4.1.1 and the new log ansatz in 4.1.2. These ansatze fix the analytic behavior of the field strength as a function of the radius $r$ of the background. The scaling ansatz is known in the literature for NHEK. We employ the same scaling ansatz in AdS-NHEK and SE-NHEK to find a new scaling solution of FFE. The log ansatz, to our knowledge, is new in all three backgrounds.

Making use of the ansatze, the problem reduces to solving an ODE for a function $\Phi(\theta)$ describing the dependence of the field strength on the polar angle. This is the part that will be treated numerically. The numerical problem is to solve the EOM summarized in section 4.2 with physical boundary conditions provided in section 4.3.

Before presenting the numerical solutions, in section 5 we consider the energy flux of electrodynamic fields in rotating frames, making use of a ‘toy model’, to explain how the fluxes at the boundary of the near-horizon throats connect with non-rotating observers outside the throat. We argue it is the latter observer’s flux that determines whether a near-horizon solution is energy extracting. The main results of this section are equation (5.15) for the energy flux in the toy model, equation (5.29) for the Kerr energy flux and (5.40) for the AdS-Kerr energy flux. The general relation (5.48) highlights the equivalence with the toy model. These formulas are applied to our energy-extracting semi-analytical solutions in section 6. The energy extraction is also discussed in section 5.5 for the full BTZ solution of Jacobson:2017xam . This provides a check on the proposed procedure for obtaining the flux outside the near-horizon throat.

Finally we present a scaling solution in NHEK, a scaling solution in AdS-NHEK and a log solution in SE-NHEK, with their corresponding Poynting fluxes, in Figures 2-4. Final remarks are discussed in section 7. Some mathematical details of our analysis have been relegated to Appendices A-D.

We adopt the metric signature $(-,+,+,+)$, the units $c=G=1$, and orientation of $dt\wedge d\phi\wedge dr\wedge d\theta$ for defining the Hodge dual $\star$.

In this section we review the set-up of the problem, hereby setting our notation. In particular, we will make use of an available action principle for the description of a force-free plasma in black holes. This has several advantages in general, such as allowing for the easy identification of conserved quantities through Noether’s theorem.

As first studied by Blandford and Znajek Blandford:1977ds , the equations for the force-free plasma around rotating black holes (described by a fixed background metric $g_{\mu\nu}$) consists of Maxwell’s equations

and the force-free electromagnetic (FFE) condition

with source current $J_{\nu}$ and field strength $F_{\mu\nu}$. Solutions to these equations involve finding a field strength $F_{\mu\nu}$ that generates a current $J^{\mu}$ through (2.1), and satisfies the FFE constraint (2.3). The Bianchi identity (2.2) follows straightforwardly by writing the field strength as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ in terms of the gauge field $A_{\mu}$.

The criterion for the FFE condition (2.3) to hold is that in local inertial frames the matter source contribution $T^{matter}_{\mu\nu}$ becomes negligible. The stress tensor can therefore be defined as

where $T_{\mu\nu}^{EM}$ is the contribution from the electric and magnetic field.¹¹1The correspondence with the force-free condition $(\ref{eq:ffe})$ follows directly from stress-energy conservation $\nabla_{\mu}T^{\mu\nu}=0$, combined with $\nabla^{\nu}T^{EM}_{\mu\nu}=-F_{\mu\nu}J^{\nu}$ from Maxwell’s equations. In this FFE regime, the Einstein-Maxwell action for the interaction between the charged matter and the gauge field $A_{\mu}$ must be nearly independent of $g_{\mu\nu}$, hence

Maxwell’s inhomogeneous equations $(\ref{eom})$ follow as the Euler-Lagrange equations of this action.

In general, the above system of equations (2.1)-(2.3) is highly nonlinear and can only be solved numerically. However, in toroidal spacetimes the symmetries can be exploited to simplify the analysis and obtain semi-analytical solutions Lupsasca:2014pfa ; Lupsasca:2014hua ; Zhang:2014pla . Here, by toroidal spacetimes we mean curved spacetimes that can be described by a coordinate system $(t,\phi,r,\theta)$ in which the metric is independent of time $t$ and angle $\phi$. The coordinates $(t,\phi)$ are then referred to as toroidal coordinates $x^{\alpha,\beta}$, and $(r,\theta)$ as poloidal coordinates $x^{a,b}$. The line-element of such a spacetime with a block-diagonal metric ($g_{a\alpha}=0$), independent of time $t$ and angle $\phi$ ($\partial_{t}g_{\mu\nu}=\partial_{\phi}g_{\mu\nu}$=0) takes the form

with toroidal metric $g_{T}$ and poloidal metric $g_{P}$.

Typically the dynamics around a black hole will capture the black hole’s symmetries. It is natural then to impose the same toroidal symmetry of the background metric on the solutions for the field strength. As we now argue, this will allow to reduce the 4D problem to a well-defined action in the two poloidal coordinates.

We assume a stationary and axisymmetric solution, characterized by a field strength with non-zero components

The analysis of the conditions on $F_{\mu a}$ imposed by the FFE equations is most clear in form notation, with the field strength defined as a two-form $F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$. See e.g. Jacobson:2017xam for a quick review of differential forms in the context of force-free electrodynamics. In this notation, the equations (2.1)-(2.2) are written respectively as

for the 2-form field strength $F=dA$ in terms of the 1-form gauge field $A$, with $\star$ the Hodge dual and $\wedge$ the wedge product. Introducing the notation $A_{T}=A_{\alpha}dx^{\alpha}$ for the toroidal contribution and $A_{P}=A_{a}dx^{a}$ for the poloidal contribution to the gauge field, we can write

and

Similarly, the 1-form current $J$ is

In the expression for the field strength, $d$ is the differential operator in 4D, with $dA=\partial_{\mu}A\,dx^{\mu}$. Imposing toroidal invariance comes down to reinterpreting the differential operator $d$ as a 2D differential operator in the poloidal metric $g_{P}$. That is, $dA=\partial_{a}A\,dx^{a}$. The toroidal invariance is therefore implicitly present in the expressions through the interpretation of $d$.

The toroidal component of the FFE equation (2.3) can be written in form notation as

Here, we use $*$ to refer to the star operator in the poloidal metric, to be distinguished from the star operator $\star$ in the full 4D spacetime (as used in Maxwell’s equation). Its action is defined as $*J_{P}=J_{a}*dx^{a}=J_{a}\epsilon^{a}_{\phantom{a}b}dx^{b}$. Maxwell’s equation implies conservation of current $d\star J=0$ (in 4D). For the toroidally invariant solution this further implies $d*J_{P}=0$ (in 2D), which allows to introduce a 0-form current $I$ by writing $*J_{P}$ as a total derivative

The interpretation of $I$ is that it is proportional to the integrated polar current through a ‘spherical cap’ $S$, stretching over a time interval $\Delta t$, azimuthal angle interval $\Delta\phi=2\pi$ and range $\theta<\theta_{0}$ at constant radius $r$, which by Stokes’ theorem is given by

with $C$ (along $t$ and $\phi$) the boundary of the 3D surface $S$. Following Gralla:2014yja , we will refer to $I$ as the polar current.

We can parametrize the field strength in terms of the polar current $I$. By Maxwell’s equations and again Stokes’ theorem, the integrated polar current $\int_{S}\star J$ is also equal to $\int_{S}d\star F=\int_{C}\star F=\int_{C}\star dA_{P}$, so that $\star dA_{P}=I\,dt\wedge d\phi$ or

in equation (2.11).

The FFE equation (2.13) with (2.14) imposes the conditions

These conditions fix different terms of the field strength to be related to each other by

and

The field strength thus has to be of the form

where $\omega_{F}$ is the angular velocity of the EM fields, and both $\omega_{F}$ and the polar current $I$ are functions of $A_{\phi}$.

As shown originally in Uchida2:1997 , plugging this expression for the field strength into the action (2.21) yields an effective 2D action

with the notation $|X|^{2}=g^{ab}X_{a}X_{b}$ for a 1-form $X$. The integration over $t$ and $\phi$ only produces overall volume factors. This action is a generalization of the Scharlemann-Wagoner action for pulsars WS to curved space-times. Similar results can be obtained from the Einstein-Maxwell action for a field strength of the form (2.20) with the force-free condition enforced through the use of a Lagrange multiplier term Thompson:1998ss ; Buniy:2013sca . Caution has to be taken in the process of eliminating $A_{P}$ from the action using $I(A_{\phi})$ Uchida2:1997 .

The problem is now reduced to solving the remaining FFE equation, also called the stream equation, for the toroidal gauge field $A_{\phi}(x^{a})$. It is obtained as the Euler-Lagrange equation of the action $\mathcal{S}^{eff}[A]=\int\mathcal{L}\,drd\theta$, given by

In the ensuing analysis we will find the action, derive the stream equation and solve this equation for specific choices of the rotating background metrics.

Rather than analyzing the 2D poloidal problem directly in the Kerr black hole geometry, it has proved advantageous to concentrate specifically on the near-horizon regions, which are described by a decoupled metric. These metrics possess a higher amount of symmetry than the black hole geometry they are derived from through a scaling limit. It is this high amount of symmetry that can be exploited in finding ansatze for symmetric, semi-analytic FFE solutions. This approach has been successfully applied in the near horizon of extremal Kerr or NHEK geometry, in the sense that energy-extracting solutions have been obtained Lupsasca:2014hua ; Lupsasca:2014pfa ; Zhang:2014pla . (However, we will comment on an unphysical feature of such solutions in the discussion of the energy flux.) We generalize the search for energy-extracting FFE solutions to include near-horizon geometries of AdS-Kerr black holes. Indeed, a 3D version of such a solution was presented in Jacobson:2017xam for the BTZ black hole. For previous work on the BZ process in AdS-Kerr, see Wang:2014vza .

In this section we collect the near horizon geometries that we will consider in this paper, each of the general rotating form (2.23): NHEK, AdS-NHEK, SE-NHEK, and near horizon extremal BTZ, which we will refer to as NHEBTZ. They are obtained from scaling limits of the respective black hole geometries. These limits are summarized in appendix A. The action and stream equation associated to these specific near horizon geometries are also described.

In this section we construct solutions to the FFE stream equations in (AdS/SE-)NHEK. The stream equation (3.15) for the gauge field component $A_{\phi}(r,\theta)$ is written out explicitly for the (AdS/SE-)NHEK background geometries (see also Appendix B). To obtain the stream equations in the form (3.15) we already have used one aspect of the symmetry of the problem, that is, the gauge field solution is required to have the same independence on time $t$ and azimuthal angle $\phi$ as the stationary and axisymmetric background metrics. This strategy can be applied further by also imposing the scaling symmetry of the background near horizon metrics on the solution. This gives rise to two ansatze for the behavior of the solution as a function of the radius $r$, which we will refer to as the ‘scaling ansatz’ (which was considered earlier in Zhang:2014pla ; Lupsasca:2014hua ) and the ‘log ansatz’. For these ansatze, the stream equations reduce further to a single ODE for the $\theta$-profile of $A_{\phi}$, which can be solved numerically. The ODE’s are summarized in section 4.2.

Before analyzing in this section the energy flux of our obtained solutions in rotating geometries, we first turn to a toy model. The toy model consists of a rotating electromagnetic (EM) field configuration, and serves to illustrate that the direction of the Poynting flux, i.e. whether the set-up has an inwards or outwards pointing flux of energy, depends on the rotation of the observer. The lesson relevant for the discussion of the energy flux of the (AdS/SE-)NHEK solutions, is that it is the Poynting flux measured by a Kerr observer rather than a near horizon observer (which rotate with respect to each other) that determines whether or not a solution is energy extracting.