Energy extraction from boosted black holes: Penrose process, jets, and the membrane at infinity The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Penna, Robert F. "Energy extraction from boosted black holes: Penrose process, jets, and the membrane at infinity." Phys. Rev. D 91, 084044 (April 2015). © 2015 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.91.084044 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/96722 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 91, 084044 (2015) Energy extraction from boosted black holes: Penrose process, jets, and the membrane at infinity Robert F. Penna* Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 March 2015; published 20 April 2015) Numerical simulations indicate that black holes carrying linear momentum and/or orbital momentum can power jets. The jets extract the kinetic energy stored in the black hole’s motion. This could provide an important electromagnetic counterpart to gravitational wave searches. We develop the theory underlying these jets. In particular, we derive the analogues of the Penrose process and the Blandford-Znajek jet power prediction for boosted black holes. The jet power we find is ðv=2MÞ2Φ2=ð4πÞ, where v is the hole’s velocity, M is its mass, and Φ is the magnetic flux. We show that energy extraction from boosted black holes is conceptually similar to energy extraction from spinning black holes. However, we highlight two key technical differences: in the boosted case, jet power is no longer defined with respect to a Killing vector, and the relevant notion of black hole mass is observer dependent. We derive a new version of the membrane paradigm in which the membrane lives at infinity rather than the horizon and we show that this is useful for interpreting jets from boosted black holes. Our jet power prediction and the assumptions behind it can be tested with future numerical simulations. DOI: 10.1103/PhysRevD.91.084044 PACS numbers: 04.40.Nr, 04.70.-s I. INTRODUCTION (where the black hole carries linear momentum), there are negative energy trajectories. We use these trajectories to Recent numerical simulations [1–8] and analytic esti- derive the analogue of the Penrose process. This gives a mates [9–12] suggest black holes carrying linear and orbital simple example of energy extraction from boosted black momentum can power jets. The jets are driven by electro- holes. It may be useful for describing the interactions of magnetic fields tapping the kinetic energy stored in the stars with moving black holes. black hole’s motion. The power of the simulated jets scales 2 Our second goal is to develop the analogue of the approximately as v , where v is the hole’s velocity [5]. Blandford-Znajek (BZ) model [15,16]. In the original Such jets could be an important electromagnetic counter- BZ model, electromagnetic fields tap a spinning black part to gravitational wave signals because v ∼ 1 in the final hole’s rotational energy and drive jets. The BZ jet power stages of black hole-neutron star and black hole-black hole prediction is currently being tested against astrophysical mergers. This paper develops the theory underlying observations of spinning black holes [17,18]. these jets. We develop the analogue of the BZ jet power prediction Our first goal is to develop the analogue of the Penrose for boosted black holes in Sec. III. For small v, we find process [13,14] for boosted black holes. The original Penrose process is a simple mechanism for extracting 1 v 2 rotational energy from Kerr black holes. It relies on the P ¼ Φ2; ð1Þ fact that certain geodesics near spinning black holes have jet 4π 2M negative energy (with respect to global time). In the original Penrose process, a particle with positive energy travels where Φ is the magnetic flux at infinity and M is the black toward the black hole and decays into two daughter hole’s rest mass. This is similar to the BZ prediction for particles. One of the daughter particles falls into the black spinning black holes but with v=ð2MÞ in place of the hole with negative energy and the other returns to infinity. horizon angular velocity ΩH, the flux evaluated at infinity The final particle has more energy than the original and the rather than the horizon, and a slightly different normali- ’ 2 black hole s mass decreases. zation constant. The v scaling is consistent with earlier We derive the analogous process for boosted simulations [3–5,8] and estimates [9–11]. Our formula Schwarzschild black holes in Sec. II. In the rest frame predicts jets from boosted black holes and spinning black of a Schwarzschild black hole there are no negative energy holes have comparable strength when v=ð2MÞ ∼ Ω . ’ H trajectories and it is impossible to lower the black hole s Numerical simulations suggest the true power of jets from mass via the Penrose process. However, in a boosted frame boosted black holes is lower by as much as a factor of 100 [5]. We discuss possible reasons for this discrepancy in *[email protected] Sec. III but save a detailed comparison for the future. 1550-7998=2015=91(8)=084044(15) 084044-1 © 2015 American Physical Society ROBERT F. PENNA PHYSICAL REVIEW D 91, 084044 (2015) Our third goal is to develop a new version of the where pvffiffiffiffiffiffiffiffiffiffiffiffiffiis a constant parameter, 0 <v<1, and 2 0 0 0 0 membrane paradigm in which the membrane lives at future γ ¼ 1= 1 − v . In the boosted frame, ðdτ ;dx;dy;dzÞ, null infinity, I þ. In the usual membrane paradigm, the the black hole is moving in the þz direction. The boosted membrane lives at the black hole horizon [16,19] and metric is a solution of the vacuum Einstein equations energy extraction is driven by torques acting on the because it is related to the Schwarzschild solution by a membrane [16,20]. However, the energy flux at the horizon coordinate transformation (3)–(6). The horizon is at of a boosted black hole is not expected to match the energy r ¼ 2M. Its area is invariant under the boost but its shape flux at Iþ in our jet model. So it is more natural to place the is distorted: it becomes squashed along the direction of membrane at infinity. We derive this new version of the motion [28]. membrane paradigm in Sec. IV. Energy extraction from If spacetime is foliated with respect to τ, then the black boosted black holes may be formulated in terms of hole’s ADM 4-momentum is interactions with the membrane at infinity. Ordinary BZ ADM jets and other processes involving black holes may also be Pμ ¼ð−M; 0; 0; 0Þ: ð7Þ reinterpreted using this formalism. The idea of reformulating black hole physics in terms of If spacetime is foliated with respect to τ0, then its ADM a fluid at infinity (or perhaps a “screen” some finite distance 4-momentum is outside the horizon) is not new. The idea has been – ADM developed extensively for asymptotically anti de Sitter Pμ0 ¼ð−γM; 0; 0; γMvÞ: ð8Þ black holes [21–24] and it has also been applied to asymptotically flat black holes [25]. The main novelties That is, the black hole has linear momentum γMv in the of our approach are to emphasize the connection with the boosted frame. These are standard calculations, see for classical black hole membrane paradigm and to develop the example [27]. electromagnetic properties of the membrane at infinity The black hole’s energy in the boosted frame, γM,is which are important for describing jets. larger than its energy in the unboosted frame by a factor of To summarize, in Sec. II we derive the analogue of the γ. However, in both frames the black hole’s irreducible Penrose process for boosted black holes, in Sec. III we ¼ðA 16πÞ1=2 ¼ mass is Mirr = M. This follows from boost derive the analogue of the BZ model, and in Sec. IV we invariance of the horizon area, A (see Appendix A for a derive a new version of the membrane paradigm in which proof). So in the black hole’s rest frame its ADM energy the membrane lives at infinity. We use this formalism to and irreducible mass coincide, but in the boosted frame give an alternate interpretation of jets from boosted black they do not. The difference, holes. We summarize our results and discuss open prob- lems in Sec. V. Supporting calculations are collected in γ − ¼ðγ − 1Þ ð Þ M Mirr M; 9 Appendices A–E. is the energy that can be extracted from the boosted II. BOOSTED BLACK HOLES AND black hole. PENROSE PROCESS Energy extraction from a boosted black hole is an ADM observer-dependent process because −Pτ0 is not a A. Arnowitt-Deser-Misner (ADM) 4-momentum Lorentz invariant. What one observer interprets as energy The Schwarzschild metric in Kerr-Schild (KS) coordi- transfer from black hole to matter, another observer nates, ðτ;x;y;zÞ,is interprets as energy transfer from matter to black hole. However, the boosted picture is more natural for astro- gμν ¼ ημν þ 2Hlμlν; ð2Þ physical problems involving kicked and orbiting black pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi holes. It is also conceptually interesting. Rotational energy 2 2 2 where H ¼ M=r, r ¼ x þ y þ z , and lμ ¼ extraction from Kerr black holes is an observer independent ð1 Þ − 2 ; x=r; y=r; z=r .