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# Boosting jet power in black hole spacetimes

Edited by Roger D. Blandford, Stanford University, Menlo Park, CA, and approved June 21, 2011 (received for review January 7, 2011)

## Abstract

The extraction of rotational energy from a spinning black hole via the Blandford–Znajek mechanism has long been understood as an important component in models to explain energetic jets from compact astrophysical sources. Here we show more generally that the kinetic energy of the black hole, both rotational and translational, can be tapped, thereby producing even more luminous jets powered by the interaction of the black hole with its surrounding plasma. We study the resulting Poynting jet that arises from single boosted black holes and binary black hole systems. In the latter case, we find that increasing the orbital angular momenta of the system and/or the spins of the individual black holes results in an enhanced Poynting flux.

Enormously powerful events illuminate the universe and challenge our understanding of the cosmos. Both gamma ray bursts and active galactic nuclei are among the most distant objects seen—a testament to the tremendous amounts of energy these sources can tap and radiate. Intense observational and theoretical efforts are ongoing in order to unravel these fascinating phenomena. Although the full details remain elusive, one of the natural ingredients in theoretical models is the inclusion of a rotating black hole which serves to convert binding and rotational energy of the system to electromagnetic radiation in a highly efficient manner. The starting point for these theoretical models can be traced back to ideas laid out by Penrose (1) and Blandford and Znajek (BZ) (2), which explain the extraction of energy from a rotating black hole. These seminal studies, along with subsequent work (see, e.g., refs. 3–5), have provided a basic understanding of highly energetic emissions from single black hole systems interacting with their surroundings. However, despite important theoretical and observational advances, we still lack a thorough understanding of these systems (e.g., refs. 6 and 7).

Recent work has indicated that related systems can also tap kinetic energy and lead to powerful jets (8). This work concentrated on nonspinning black holes moving through a plasma and highlighted that relative black hole motion alone (with respect to a stationary electromagnetic field topology at far distances from it) can induce the production of jets. Furthermore, subsequent work (9) demonstrated that even black holes with spins misaligned with respect to the asymptotic magnetic field direction induce strong emissions with power comparable to the aligned case. These studies suggested that, independent of their inclination, astrophysical jets might be powered by the efficient extraction of both rotational and translational kinetic energy of black holes, becoming even more powerful than the standard BZ mechanism would suggest.

Galactic mergers provide a likely scenario for yielding both the binary black hole and single black hole systems considered here (10, 11). In such a merger, the supermassive black holes associated with each galaxy will ultimately form a binary surrounded by a circumbinary disk in the resulting merged galaxy. A variety of interactions will tighten the black hole binary and eventually its dynamics will be governed by gravitational radiation reaction which drives the binary to merge. The circumbinary disk will likely be magnetized and thereby anchor magnetic field lines, some of which will traverse the central region containing the binary and eventual final black hole (see Fig. 1). Furthermore, as a result of the merger process, this final black hole will likely acquire a recoil velocity and will move relative to the circumbinary disk until the disk relaxes to the black hole’s motion. Thus, the system will have, at different stages, both binary and single black holes moving through an ambient magnetic field sourced by the surrounding disk. Preliminary observational evidence for supermassive black hole binaries resulting from galactic mergers has already been presented (12–14). Another example of a system where plasma could tap kinetic energy from a moving black hole corresponds to black-hole–neutron star binary (15). In such a system, the black hole would move through the star’s magnetic field. As discussed by BZ (2), an ambient magnetic field threading the black hole region populates a low density plasma surrounding the black hole. Even for black holes with no spin, it was recently shown that the orbiting binary interacting with the surrounding plasma can lead to a collimated Poynting flux (8). In this work, we consider this basic paradigm of energy extraction from black holes with the additional complexity of intrinsic black hole spin. Binaries consisting of spinning black holes demonstrate similar, albeit energetically enhanced, phenomenology. Furthermore, we investigate the dependence of the energy flux on the black hole velocity and highlight a resulting strong boost in the emitted power.

In addition to exquisite and powerful electromagnetic detectors, soon gravitational waves will be added to the arsenal of phenomena employed to understand our cosmos. These studies suggest excellent prospects for the coincident detection of both electromagnetic and gravitational signals from binary black hole systems. Certainly dual detection of electromagnetic and gravitational wave signals would transform our understanding of these systems and lead to the refinement of theoretical models (e.g., refs. 16–19).

## Results

In this work, we discuss the basic phenomenology arising from the interaction of black hole systems immersed within a plasma environment. We study both boosted and orbiting black holes moving through an initially constant magnetic field and find that the interaction with the magnetic field induces jets of strong electromagnetic radiation. We find that, remarkably, both the spin and the translational motion of the black hole contribute to this electromagnetic luminosity. Indeed, the luminosity *L* of the jets is naturally decomposed in terms of each contribution as [1]Here the contribution from the spin, *L*_{spin}, is consistent with the expected BZ luminosity, and *L*_{speed} represents the energy extracted from the black hole’s translational kinetic energy. In what follows, we conduct numerical experiments to detail how each component scales.

To fix ideas, we envision a magnetic field anchored in the disk with its associated magnetic dipole aligned, in the binary case, with the orbital angular momentum (chosen along the direction). In the single black hole case, we consider propagation velocities orthogonal to . Our resulting initial configuration thus has the magnetic field perpendicular to the velocity of the black holes and the electric field is set to zero. Because the electromagnetic field is affected by the spacetime curvature, it is dynamically distorted from its initial configuration and generates a transient burst of electromagnetic waves as it settles into a physically relevant and dynamical configuration. The initial magnitude of the magnetic field is *B*_{0} = 10^{4} G. This value is chosen to be consistent with possible astrophysical magnetic field strengths (20, 21). We present our results for a field strength which is bounded by the Eddington magnetic field strength *B* ≃ 6 × 10^{4}(*M*/10^{8} *M*_{⊙})^{-1/2} G (22). For these values, the plasma’s energy is several orders of magnitude smaller than that of the gravitational field. Thus, although the plasma is profoundly affected by the black holes, it has a negligible influence on the black holes.

For simplicity, we introduce the notation *L*_{43} = 10^{43} erg/s and compute fully dimensional quantities with respect to a representative system with total mass 10^{8} *M*_{⊙} immersed in a magnetic field with strength *B*_{0} = 10^{4} G. The luminosities that we calculate scale as *L* ∝ *M*^{2}*B*^{2}, and so luminosities for other masses and magnetic field strengths can be easily obtained from our results. We provide proportionality factors for other quantities of interest. Quantities in geometric units are calculated by setting Newton’s gravitation constant and the speed of light to unity, *G* = *c* = 1. For example, the Schwarzschild radius in geometric units, *R*_{S} = 2*M*, is converted to other unit systems using the constants *G* and *c* as *R*_{S} = 2*GM*/*c*^{2}. Thus, *R*_{S} ≃ 2.95 km for a solar mass black hole, and 2.95 × 10^{8} km for the 10^{8} *M*_{⊙} black hole considered here. Time in geometric units is measured in terms of a mass, and the conversion factor is *G*/*c*^{3}. A time interval of 1 *M*_{⊙} corresponds to about 4.9 × 10^{-6} s, or 4.9 × 10^{2} s for an interval of 10^{8} *M*_{⊙}.

The result captured in Eq. **1** has important consequences for black hole binaries because the luminosity that results from the motion, *L*_{speed}, can be significantly higher than that resulting from the spin, *L*_{spin}. In addition, eccentric orbits and spin–orbit interactions driving orbital plane precession can induce important variabilities in the luminosity, which can also aid in the detection of these systems.

### Single Black Holes.

We first consider the radiation from a single black hole immersed in a magnetic field anchored by currents in a distant circumbinary disk. Such a black hole can be spinning and can also have some constant velocity with respect to the initially uniform magnetic field. We calculate the resulting electromagnetic collimated flux energy and its dependence on both velocity and black hole spin. For these runs, we adopt a computational domain defined by [-160*R*_{S},160*R*_{S}]^{3} (*R*_{S} is the Schwarzschild radius of a black hole). We set the initial linear velocity of the black hole to (with *n* an integer that we vary between 0 and 4), and we set the intrinsic angular momentum parameter to either 0 or (recall that black hole spins must satisfy |*a*| ≤ 1 according to the Kerr bound).

Fig. 2 illustrates the qualitative features of the electromagnetic emission for a boosted black hole with and without spin. A collimated emission is clearly induced along the asymptotic magnetic field direction. An even stronger emission is obtained for the spinning black hole, as expected.

To achieve a more quantitative understanding, we compute a measure of the collimated electromagnetic luminosity, *L*_{collimated}, by integrating the flux over a 15° cone of a spherical surface centered along the moving black hole with a radius *r* = 40*R*_{S}. We present our results with respect to the velocity, *v* (the measured *coordinate velocity* of the black hole), and plot in Fig. 3 the collimated luminosity achieved once the system reaches a quasi-stationary state (in other words, after the initial transient stage) for the spinning and nonspinning cases.

Several key observations are evident from the figure. For *v* = 0, the electromagnetic energy luminosity does not vanish for the spinning black hole, although it does vanish for the nonspinning black hole. That there is radiation in the spinning case is expected, as the spinning black hole interacts with a surrounding plasma and radiates by the BZ mechanism (2). This luminosity results from the plasma’s ability to extract rotational energy from the black hole and power a jet with an energy luminosity scaling as (9, 23) [with Ω_{H} ≡ *a*/(2*R*_{H}) the rotation frequency associated with the black hole].

For *v* ≠ 0, both the nonspinning and spinning black holes have a nonzero energy flux. In the former case, this flux arises solely from the ability of the system to tap translational kinetic energy from the black hole, whereas the latter results from the extraction of both translational and rotational kinetic energies.

The nonspinning black hole demonstrates an electromagnetic luminosity that increases with *v*^{2} (see Fig. 3). This behavior is consistent with treating the black hole within the membrane paradigm of ref. 24. A spinless black hole moving with speed *v* through an ambient magnetic field behaves as a conductor and acquires an induced charge proportional to the speed (8, 9, 25, 26). In this regard, the charge separation on the surface of the black hole is completely analogous to the classical Hall effect. With this observation and the induction equation, it is straightforward to show that the electromagnetic energy flux will increase as ∝ *v*^{2}*B*^{2}. This quadratic dependence on speed is apparent in the figure. These results are also consistent with the work of Drell et al. (27), who studied the Poynting flux associated with a moving conductor in a magnetized plasma as applied to the motion of artificial satellites in orbit. They found that the flux obeys *L*_{v} ≈ (*v*/*v*_{alf})^{2}*B*^{2} (27), where *v*_{alf}, the propagation speed of the Alfvén modes, is the speed of light in a force-free environment. Furthermore, a misalignment is expected between the collimated energy flux and the original magnetic field orientation such that tan(*α*) = *v*/*v*_{alf}. With the cautionary note that measuring this angle is ambiguous in the curved spacetime around the black hole, such a relation is indeed manifested in our results. For instance, for *a* = 0, *v*_{x} = 0.10, the measured angle is tan(*α*) = 0.07, in good agreement with the predicted value of 0.08 (27).

When we allow the black hole to spin, we once again find the same quadratic dependence on velocity, with an additional contribution due to the hole’s spin. Notice that the difference between the obtained luminosities in the spinning and spinless cases remains fairly constant for the different values of *v*. The roughly constant offset between spinning and nonspinning cases supports the decomposition given in Eq. **1** in which the luminosity receives two contributions, one from the black hole spin and another from its velocity.

We have seen that , and that *L*_{speed} ∝ *v*^{2}. In order to be more definite, we introduce two fitting constants, *L*_{1} and *L*_{2}, such that the general luminosity dependence of Eq. **1** can be expressed as [2]A fit of the luminosity for the nonspinning black hole (*a* = 0) gives *L*_{2} = 127. Similarly, a fit for a spinning black hole with *a* = 0.6 provides the constant *L*_{1} = 0.87. In the above equation, we express the spin contribution in a general way so that, to obtain the expected luminosity for an arbitrary black hole, one need only provide the ratio of the rotational frequency to that of our fiducial *a* = 0.6 black hole.

Notice that, for *a* = 0.6, the nonrotational contribution *L*_{speed} to the emitted power becomes larger than the rotational component for speeds approximately *v* ≳ 0.08*c*. This relationship, for example, would predict a luminosity for an *a* = 0.95, *v* = 0.5*c* black hole to be ≃36*L*_{43}(*M*_{8}*B*_{4})^{2}. This phenomenology strongly suggests that the Poynting flux can tap both rotational and translational kinetic energies from the black hole and that faster and more rapidly spinning black holes have a stronger associated power output.

### Binary Black Holes.

We turn our attention now to orbiting binary black holes. We consider equal-mass black holes with orbital diameter *D* ≈ 8*R*_{S}. We adopt a computational domain of [-320*R*_{S},320*R*_{S}]^{3} and consider black holes with either no spin (abbreviated “0” below), or with a spin aligned (spin up or “*u*”) or antialigned (spin down or “*d*”) with the *z* direction. In particular, we concentrate on three cases: a binary with no individual spin; a binary with zero net spin (by considering equal, but antialigned, individual spins); and a binary with equal, aligned spins. For the spinning black hole cases, all spinning holes have spin parameter |*a*| = 0.515. Where required, we denote these cases by 0/0, *u*/*d*, and *u*/*u*, respectively, and associated measured quantities have this notation as subindices. These configurations are summarized in Table 1.

Before discussing the results from these binaries, we first describe our analysis that enables a quantitative comparison between the evolutions. First, we evaluate the luminosities as functions of gravitational wave frequency, as this is an observable and allows for a direct comparison of the different cases. We obtain frequencies from the dominant *l* = 2, *m* = 2 gravitational mode. Second, we compute three different luminosities for each case: (*i*) the collimated luminosity *L*_{collimated} obtained by integrating the electromagnetic flux over a cone of points within 15° of vertical from the center of mass of the system; (*ii*) the noncollimated, or “isotropic,” luminosity *L*_{isotropic} obtained from the integral over an encompassing sphere minus the collimated luminosity of (*i*); and (*iii*) the gravitational wave luminosity, *L*_{GW}. These different luminosities are displayed for the three binary configurations in Fig. 4.

We first consider the 0/0 and *u*/*d* binaries, which have essentially the same total angular momentum. Their qualitative behavior is illustrated in Fig. 5. Extensive numerical simulations (see, for instance, ref. 28) and simple estimates (29) indicate that both these binaries will merge into a final black hole with essentially the same spin (*a* ≃ 0.67), and so for late times after the merger, the expected jet structure should be quite similar: namely, that described by the standard BZ mechanism.

The binary with individual spins aligned reaches a higher orbital velocity before merger than the previous two cases; thus, the expected maximum power should be higher. Moreover, the resulting final black hole spins faster (*a* ≃ 0.8) and thus its BZ associated power will be higher than that of the previous cases. Fig. 4 illustrates this expected behavior by presenting the Poynting flux energy vs. gravitational wave frequency for the three cases.

As evident from the figure, at low frequencies where the orbital dynamics are the same in all cases, both spinning cases have a higher output than the nonspinning one and the difference is provided by the spin contribution to the jet emission. Furthermore, both spinning cases have equal collimated power output because the spin contribution to the luminosity depends only on the spin magnitude.

We can further examine the basic relation given in Eq. **1** by comparing the fluxes corresponding to the three cases studied. We estimate the black hole *coordinate velocities* *v* and (collimated) luminosities at three representative frequencies Ω_{i} = {1; 1.5; 2}10^{-5} [Hz/*M*_{8}] (*i* = 1,…,3) before the strong nonlinear interaction starts. Because for these frequencies the measured speeds for the 0/0 and *u*/*d* cases are essentially the same, differences in their corresponding fluxes should be due to the contribution from the spin. This contribution can be *estimated* to be [3]where *L*_{u/d} and *L*_{0/0} are the measured luminosities for both cases. The obtained values for the frequencies considered are as follows: Notice that the remains fairly constant through these frequencies, which is evident also in Fig. 4. That this value is essentially the same for all cases further supports the basic relation in Eq. **1**. Furthermore, we can employ this value to estimate the luminosity for the *u*/*u* case (illustrated in Fig. 6) exploiting the one measured for the 0/0 case. To do so, we observe that for a given frequency Ω the difference between the two cases is the contribution due to the spin (absent in the 0/0 case) together with a suitable scaling of *L*_{speed} as [4]The obtained values can be compared with those computed directly from the numerical evolution, *L*_{u/u}. Using our measured orbital velocities, Eq. **4** provides the following estimates which we show next to our computed values: The estimates are within ≃10% of the measured values, when the black holes are sufficiently separated that their jets do not strongly interact. Finally, notice that, at even higher frequencies, Fig. 4 illustrates that the aligned (*u*/*u*) case indeed has a higher associated power.

In all cases, a significant noncollimated emission is induced during the merger phase (evident in the upper-right plot of Fig. 4 and illustrated in the second frames of Figs. 5 and 6). Clearly, the simple-minded picture of a jet produced by the superposition of the orbital and spinning effect cannot fully capture the complete behavior at the merger epoch, although it serves to understand the main qualitative features and provides a means to estimate the power of the electromagnetic emission.

## Discussion

We have studied the impact of black hole motion through a plasma and indicated how the interaction can induce powerful electromagnetic emissions even for nonspinning black holes. Despite having examined a very small subset of the binary black hole parameter space, the results presented both here and in refs. 8 and 9 suggest broad applicability to general black hole binaries. Moreover, as the plasma generally has a negligible effect on the dynamics of the black holes, one needs only to know the dynamics of the black holes, say by numerical solution or other approximate methods, in order to estimate the expected electromagnetic luminosity. A recent example is the work of McWilliams (30), which uses the expected ≃*B*^{2}*v*^{2} scaling for nonspinning black hole binaries coupled with the known distance dependence in time to obtain excellent agreement with the luminosity from fully nonlinear numerical evolutions.

Naturally, spinning black holes produce stronger jets, and these jets, as shown earlier (9), will be aligned with the asymptotic magnetic field direction.* The resulting electromagnetic luminosity can be estimated to be *L* ≃ *L*_{spin} + *L*_{speed}, which can be significant and have associated time variabilities tied to the dynamical behavior of the system. In particular, the luminosity tied to the motion can become significantly higher than that tied to the spin. In addition, eccentric orbits and spin–orbit interactions driving orbital plane precessions can induce important variabilities that can aid in the detection of these systems. Furthermore, a significant pulse of nearly isotropic radiation is emitted during merger, thereby allowing observations of the system along directions not aligned with the jet.

Consequently, binary black hole interactions with surrounding plasmas can yield powerful electromagnetic outputs and allow for observing these systems through both gravitational and electromagnetic radiation. Gravitational waves from these systems corresponding to the last year before the merger could be observed to large distances with the Laser Interferometer Space Antenna (up to redshifts of 5–10, ref. 31, for masses ≃10^{4–7} *M*_{⊙}) or earlier in the orbiting phase (and possibly through merger) via Pulsar Timing Array observations (32), targeting binaries with masses of *M* ≃ 10^{7–10} *M*_{⊙}. As we have indicated here, both scenarios can have strong associated electromagnetic emissions. Our inferred luminosities of several 10^{43}(*M*_{8}*B*_{4})^{2} erg/s correspond to an isotropic bolometric flux of *F*_{x} ≃ 10^{-15} erg (*M*_{8}*B*_{4})^{2}/(cm^{2} s) that could be detected to redshifts of *z* ≈ 1 and beyond depending on anisotropies and on the efficiency of processes which tap this available energy and produce observable signals.

## Materials and Methods

The combined gravitational and electromagnetic systems that we consider consist of black hole spacetimes in which the black holes can be regarded as immersed in an external magnetic field. Such fields, as mentioned earlier, will be anchored to a disk. We consider this disk to be outside our computational domain but its influence is realized through the imposition of suitable boundary conditions on the incoming electric and magnetic field modes on the boundaries of our domain (8, 9). These conditions essentially correspond to setting to zero the electric field while constant values for the magnetic field (given by .) With regard to the magnetosphere around the black holes, we assume that the energy density of the magnetic field dominates over its tenuous density such that the inertia of the plasma can be neglected. The magnetosphere is therefore treated within the force-free approximation (2, 33). We note that the contribution of the energy density of the plasma to the dynamics of the spacetime is negligible and we can ignore its back reaction on the spacetime.

We use the Baumgarte–Shapiro–Shibata–Nakamura formulation (34, 35) of the Einstein equations and the force-free equations as described in refs. 8, 9, and 36. We discretize the equations using finite difference techniques on a regular Cartesian grid and use adaptive mesh refinement (AMR) to ensure that sufficient resolution is available where required in an efficient manner. We use the Hybrid Adaptive computational infrastructure, which provides distributed, Berger–Oliger style AMR (37, 38) with full subcycling in time, together with an improved treatment of artificial boundaries (39). Refinement regions are determined using truncation error estimation provided by a shadow hierarchy (40), which adapts dynamically to ensure the estimated error is bounded by a prespecified tolerance. Typically our adopted values result in a grid hierarchy yielding a resolution such that 40 grid points in each direction cover each black hole. We use a fourth-order accurate spatial discretization and a third-order accurate in time Runge–Kutta integration scheme (41). We adopt a Courant parameter of *λ* = 0.4, so that Δ*t*_{ℓ} = 0.4Δ*x*_{ℓ} on each refinement level *ℓ*. In tests performed here for the coupled system (and in our previous works for the force-free Maxwell equations), the code demonstrates convergence while maintaining small constraint residuals for orbiting black holes. Furthermore, we obtain for orbiting black hole evolutions agreement with runs from other codes for the same initial data.

To extract physical information, we monitor the Newman–Penrose electromagnetic (Φ_{2}) and gravitational (Ψ_{4}) radiative scalars (42). These scalars are computed by contracting the Maxwell and the Weyl tensors, respectively, with a suitably defined null tetrad (as discussed in ref. 43), [5]and they allow us to account for the energy carried off by outgoing waves at infinity. The scalar Φ_{2} (whose modulus squared is essentially the radial component of the Poynting vector) provides a measure of the electromagnetic radiation at large distances from an isolated system. However, as the system studied here has a ubiquitous magnetic field, special care must be taken to compute the energy flux. We account for this difficulty by subtracting the scalar Φ_{0} = -*F*_{ab}*l*^{a}*m*^{b} from Φ_{2}. Hence, by Φ_{2}, we mean the difference Φ_{2} → Φ_{2} - Φ_{0}. The luminosities in electromagnetic and gravitational waves are given by the integrals of the fluxes [6][7]

## Acknowledgments

The authors thank J. Arons, P. Chang, B. MacNamara, K. Menou, E. Quataert, and C. Thompson, as well as our long time collaborators Matthew Anderson, Miguel Megevand, and Oscar Reula for useful discussions and comments. We acknowledge support from National Science Foundation Grants PHY-0803629 (to Louisiana State University), PHY-0969811 (to Brigham Young University), PHY-0969827 (to Long Island University), as well as the Natural Sciences and Engineering Research Council through a Discovery Grant. Research at Perimeter Institute is supported through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. Computations were performed at Louisiana Optical Network Initiative, Teragrid, and Scinet.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: llehner{at}perimeterinstitute.ca.

Author contributions: L.L. designed research; D.N., L.L., C.P., and T.G. performed research; L.L., C.P., E.W.H., S.L.L., P.M.M., and T.G. contributed new analytic tools; D.N., C.P., and E.W.H. code development; D.N., L.L., C.P., S.L.L., and P.M.M. analyzed data; and D.N., L.L., C.P., E.W.H., and S.L.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵

^{*}The magnitude of the BZ-associated emission diminishes for nonaligned cases but it is nevertheless significant: The orthogonal case is only half as powerful as the aligned case.

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