PHYSICAL REVIEW D 103, 023006 (2021)
Astrophysical hints for magnetic black holes
† ‡ Diptimoy Ghosh ,* Arun Thalapillil , and Farman Ullah Department of Physics, Indian Institute of Science Education and Research Pune, Pune 411008, India
(Received 2 October 2020; accepted 11 December 2020; published 6 January 2021)
We discuss a cornucopia of potential astrophysical signatures and constraints on magnetically charged black holes of various masses. As recently highlighted, being potentially viable astrophysical candidates with immense electromagnetic fields, they may be ideal windows to fundamental physics, electroweak symmetry restoration, and nonperturbative quantum field theoretic phenomena. We investigate various potential astrophysical pointers and bounds—including limits on charges, location of stable orbits, and horizons in asymptotically flat and asymptotically de Sitter backgrounds, bounds from galactic magnetic fields and dark matter measurements, characteristic electromagnetic fluxes, and tell-tale gravitational wave emissions during binary inspirals. Stable orbits around these objects hold an imprint of their nature and in the asymptotically de Sitter case, there is also a qualitatively new feature with the emergence of a stable outer orbit. We consider binary inspirals of both magnetic and neutral, and magnetic and magnetic, black hole pairs. The electromagnetic emissions and the gravitational waveform evolution, along with interblack hole separation, display distinct features. Many of the astrophysical signatures may be observationally glaring—for instance, even in regions of parameter space where no electroweak corona forms, owing to magnetic fields that are still many orders of magnitude larger than even magnetars, their consequent electromagnetic emissions will be spectacular during binary inspirals. While adding new results, our discussions also complement works in similar contexts, that have appeared recently in the literature.
DOI: 10.1103/PhysRevD.103.023006
I. INTRODUCTION relatively long lived cosmologically and potential survivors from our universe’s early epochs. Furthermore, the paucity In our pursuit to understand the fundamental interactions of magnetic monopoles and magnetically charged matter that govern our universe, it is crucial to explore diverse contribute to their persistence, making them amenable to phenomena that may inform us. One such avenue is provided current observations, if they have survived from some by astrophysical objects with extreme properties. They are primordial cosmological era. potentially spectacular probes for fundamental physics and MBHs and some of their intriguing theoretical aspects beyond Standard Model phenomena, that are many times have been discussed in the past [8–10].Recently,various beyond the reach of terrestrial experiments. Rather than spectacular properties of MBHs have been highlighted [11], being merely relegated to fantasy, these avenues are coming drawingonresultsfrom[12–15]. As extensively discussed to fruition with the discovery of gravitational waves from and emphasized in [11], such objects may be windows to the black hole/neutron star mergers [1,2], observation of near nature of fundamental symmetries and other fundamental horizon features of black holes [3], discovery of evermore physics, hitherto inaccessible to other probes. The magnetic exotic astrophysical objects [4–6], and the advent of a fields near the horizon are generically large for these objects. multimessenger era in astronomy [7]. In certain regions of parameter space the magnetic fields may One such class of possible exotic astrophysical objects in fact even be large enough to restore electroweak symmetry are magnetically charged black holes (MBHs). Especially [11–15]. Binary pairs of such objects, with opposite mag- in the extremal or near-extremal limit, they may be netic charges and in the extremal limit, also allow us to speculate on many intriguing possibilities [16]. Thus, finding * astrophysical hints for such objects, and subsequent obser- [email protected] † [email protected] vations of them, may have great repercussions for our ‡ [email protected] understanding of the universe. There have been some phenomenological studies on Published by the American Physical Society under the terms of MBHs recently [17,18]. In the low-mass MBH region, the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to where an electroweak corona forms, a comprehensive the author(s) and the published article’s title, journal citation, phenomenological study has been performed in [17].The and DOI. Funded by SCOAP3. effective one-body motion of a dyonic black hole binary
2470-0010=2021=103(2)=023006(21) 023006-1 Published by the American Physical Society GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) system, along with some aspects of the corresponding considering both a vanishing and a positive cosmological electromagnetic and gravitational radiation, incorporating constant background. In Sec. IV we briefly review certain post-Newtonian corrections, was studied recently in [18]. quantum field theoretic aspects of extremal and near- Our study will be complementary to these, with a slightly extremal MBHs, with various QB. In this section, over different focus. There have also been studies in other this QB range, we also present bounds on MBH as a dark contexts in the literature—for instance, studying implica- mater candidate from galactic magnetic field and dark tions of a topologically induced black hole electric charge matter density measurements. In Sec. V and Sec. VI we [19], and interesting studies on MBH solutions arising in then consider electromagnetic and gravitational wave nonlinear electrodynamics, and their implications on black emission from binary inspirals where MBHs are involved. hole shadows [20]. In a broader context, with recent We summarize our results and conclude in Sec. VII. advances [1–3], it is also of great interest to probe and understand the exact nature of possible compact objects in II. ASTROPHYSICAL MAGNETICALLY our universe [21]. For instance, horizonless exotic compact CHARGED REISSNER-NORDSTROM objects may have a small reflectivity, with consequent BLACK HOLES observable manifestations in the postmerger ringdown and echoes [22]. Compact objects carrying magnetic charges In asymptotically flat spacetime, the standard Reissner- may introduce new elements to these considerations [23,24]. Nordstrom exterior solution of a black hole carrying Our aim in this work is to investigate and discuss various electric and magnetic charges is of the form potential astrophysical signatures for MBHs, while giving 2M ðQ2 þ Q2 Þ simple analytic results. As alluded to already, in the extremal ds2 ¼ − 1 − þ E B dt2 r r2 and near-extremal limit (or the analogous limit in a de Sitter background) MBHs are expected to be long-lived and 2M ðQ2 þ Q2 Þ −1 þ 1 − þ E B dr2 þ r2dΩ2: ð1Þ astrophysically viable. Moreover, as we shall see, across r r2 diverse MBH masses and magnetic charges, the magnetic fields near the horizon and near the innermost stable orbits We are using units where c ¼ 1;GN ¼ 1; 1=4πϵ0 ¼ can be extremely large. For instance, even for MBH masses μ0=4π ¼ 1. ðr; θ; ϕÞ are the coordinates, and M is the where no electroweak corona [11] forms, these electromag- mass of the black hole. QE and QB are the electric and netic fields are many orders of magnitude bigger than even magnetic charges of the black hole respectively. The some of the currently known largest astrophysical magnetic dimensionless magnetic charge, Q, as defined in [11],is fields—that of neutron stars such as Magnetars [4].These related to QB in our convention through Q ¼ QB=qB, unique characteristics, among others, may provide interest- 2πϵ ℏ 2 3 44 10−9 where qB ¼ 0 c =e ¼ . × A-m; it is the basic ing astrophysical signatures for these objects. unit of magnetic charge assuming Dirac quantization. We will discuss simple bounds on the electric and In most realistic astrophysical and cosmological settings, magnetic charges for black holes from astrophysical con- it is relatively easy for the electric charge (QE) to get siderations, features of astrophysically relevant stable neutralized by in-falling matter or be strongly limited by circular orbits around MBHs in asymptotically flat and other physical considerations. Consider for instance an de Sitter backgrounds, limits from galactic magnetic field electrically charged Reissner-Nordstrom black hole of mass observations on MBH abundance when they are considered M and electric charge þQE. For an ionized Hydrogen as a dark matter component, and characteristics of their nuclei to be absorbed, negating electrostatic repulsion, so as electromagnetic and gravitational radiation, along with to increase its intrinsic charge, one requires conservatively orbital evolution, during MBH binary inspirals. Our focus will not be just confined to parameter regions where an Q m E ≲ p ≃ 10−18: ð2Þ electroweak corona forms, but on relatively the full M jej parameter space of MBH masses; in the extremal and near extremal limits. Most of the results and estimates we Here, mp is the proton mass. This implies that in realistic present are new. Where there are minor overlaps, our situations the ratio will be very small in general. This results are consistent, within astrophysical uncertainties assumes that the ions are not being impinged on the black and modeling assumptions, with the extant studies. hole with a large kinetic energy. Imparting higher kinetic In Sec. II we briefly review the Reissner-Nordstrom energy will weaken the bound slightly, but the general solution, in asymptotically flat and de Sitter spacetimes. import of the result remains. Here, we also comment on various limits on the electro- Another theoretical consideration is that when the black magnetic charges from astrophysical considerations and hole is electrically charged, the near horizon electric fields discuss the location and relevant properties of the various may source Schwinger pair production of particle anti- horizons. In Sec. III we then investigate, in some detail, particle pairs [25], say of mass m and charge q. For the stable circular orbits in the vicinity of extremal MBHs, same charge, the nonperturbative rate will be dominated by
023006-2 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) the lightest charged particle in the spectrum. The oppositely monopoles might make this channel impossible, or even if charged particle falls into the black hole, reducing its they exist, the generic heaviness of viable magnetic monop- overall charge, while the same-charge particle is repelled oles would for all practical purposes stall the discharge. outwards. As the black hole discharges, the near horizon Next, consider an MBH of charge þQB and like-charged ∼1016 field subsequently decreases below the critical value and magnetic monopoles of charge qB and mass mB GeV pair production essentially stops. As the nonperturbative participating in the charging process, through accretion. 2 rate [26] goes as ΓE ∼ exp½−m =qE , it implies that when Repeating the arguments leading up to Eq. (2), one now obtains a much weaker bound 2 QE ≪ m 2 ; ð3Þ ð2MÞ q QB mB −4 ≲ ≃ 10 : ð7Þ M jqBj subsequent discharge would be suppressed. This then places a limit on the equilibrium black hole electric charge Here, we have assumed Dirac quantization. We therefore see that the bounds on the charge-to-mass Q e m 2 M E ≲ 10−5 : ð4Þ ratio for MBHs are many orders of magnitude weaker than M q me M⊙ that for electrically charged black holes. This then implies that MBHs may have larger charge-to-mass ratio relative to Again, one observes that generally the black hole charge their electrically charged counterparts. Nevertheless, this to mass ratio is expected to be small, and well below the also indicates that to achieve an even higher ratio (≳10−4), QE ¼ M extremal limit. the candidate MBH’s origin may have to be distinct from Even when an electrically charged Reissner-Nordstrom conventional astrophysical scenarios, like that of a low black hole potentially forms, it may also get quickly charge-to-mass ratio MBH just slowly capturing low- neutralized by accreting ionized plasma. Considering _ energy, like-charged magnetic monopoles. Of course, standard Eddington accretion rates [27] MEd: ¼ again, if sufficient kinetic energy may be imparted to the 4π σ ∼ 10−15 −1 cGMmp= Thom: ðM=M⊙ÞM⊙ s , for a black in-falling monopoles in some way, the bound may be hole charge QE, the neutralization timescale will be roughly weakened and some further charging is possible through this mode. A possibility is also that the only viable ∼ QE m ≃ 1015 QE m monopoles have small effective fractional charges [30–32], tneutr: _ s: ð5Þ q MEd: q M but are also heavy, which may weaken the bound. Another more plausible origin is that the MBH forms far from Even in the extremal case QE ¼ M, assuming neutraliza- extremality, but due to the enhanced Hawking radiation tion by accretion of ionized hydrogen, the charge neutrali- QB → 1 [11] it quickly tends to M . This scenario is also more zation is very rapid with pertinent, since primordial cosmological epochs may have furnished an amenable avenue for MBHs with large QE¼M ∼ 10−3 tneutr: s: ð6Þ charge-to-mass ratios to form [33,34]. We will adopt an agnostic viewpoint about the exact details of their cosmo- It has been commented that in certain very special sit- logical origin. uations a background magnetic field may be able to thwart In keeping with the main theme of the paper, and due neutralization by charge-selective accretion [25] and extend to the above points, we will assume QE ¼ 0 throughout. the effective timescale. Other accretion models and rates Let us consider therefore the Reissner-Nordstrom solution may change this estimate as well, but the timescales are purely with QB ≠ 0. The horizons for such a black hole are still expected to be relatively short. The broad message located at therefore seems to be that electrically charged black holes, even when they form, may not be very astrophysically qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − 2 long-lived. r ¼ M M QB ð8Þ Note that unlike the electrically charged counterpart, a classical magnetically charged black hole may persist for Note that when M ¼jQBj, the two horizons merge and much longer, owing to the paucity of magnetically charged one has an event horizon located at rH ¼jQBj. This case matter and magnetic monopoles, or if the monopoles are will be referred to as the extremal MBH. The r coordinate very heavy. Considerations from both Schwinger pair in this case is spacelike on either side of the horizon. production of magnetic monopoles and accretion of mag- The singularity at r ¼ 0 is timelike. We note for future netic monopoles lead to much weaker constraints. referenceqffiffiffiffiffiffiffiffiffi that in physical units, the extremal MBH has For instance, the rate of Schwinger pair production of μ0 M ¼ 4π QB. When M 023006-3 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) rffiffiffiffi The extremal solutions have many intriguing properties. Λ −23 M For instance, two well separated extremal black holes with ϵ ¼ 4M ¼ 3.6 × 10 : ð14Þ 3 M⊙ like charges will repel each other, exactly cancelling their gravitational attraction. In contrast, two extremal black For magnetically charged black holes in de Sitter space holes with unlike charges will attract each other, with a (MBHdS), and specifically for magnetic black holes with magnitude exactly matching their gravitational attraction. M ¼jQBj, the situation is much richer. When M ¼jQBj, These features will be particularly relevant to our later the horizons are now located at discussions. Another important observation is that extremal qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi magnetic black holes in asymptotically flat spacetime are pffiffiffipffiffiffiffi pffiffiffi 3 4 3 Λ Q − 3 TM¼jQBj 0 þ j Bj cold ( ¼ ), and may hence be long lived due to the r ¼ pffiffiffiffi ; suppression of Hawking radiation. This opens the possibil- i 2 Λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ity of such extremal MBHs surviving from earlier epochs of pffiffiffi pffiffiffipffiffiffiffi 3 − 3 − 4 3 Λ the universe to the present day. This makes them fascinat- jQBj r ¼ pffiffiffiffi ; ing candidates to constrain and search for in astronomical o 2 Λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi observations. pffiffiffi pffiffiffipffiffiffiffi Now consider the case when the cosmological constant 3 þ 3 − 4 3 ΛjQBj r ¼ pffiffiffiffi ; ð15Þ is nonvanishing and positive (Λ > 0). For a magnetic c 2 Λ Reissner-Nordstrom black hole embedded in asymptoti- cally de Sitter spacetime, the metric is given by Λ 3 16 2 when < =ð QBÞ. These correspond to the inner Cauchy horizon, outer event horizon and the de Sitter ds2 −Δ r dt2 Δ r −1dr2 r2dΩ2; 9 ¼ ð Þ þ ð Þ þ ð Þ cosmological horizon respectively; with ri 023006-4 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) μ ν state [35,36]. Then, such magnetic black holes embedded in 2 dx dx ϑ ≡ gμν an asymptotically de Sitter background may also ideally be ds ds relatively long lived and amenable to present-day astro- 2M dt 2 2M −1 dr 2 ¼ − 1 − þ 1 − physical observations. r ds r ds The horizons of an MBH provide a causal structure, dϕ 2 while also defining characteristic length scales for various þ r2 : ð19Þ astrophysical phenomena near the singularity. Another set ds of length scales are provided by stable orbits near the 2 compact object, which we explore next. In the next section ϑ is vanishing for massless particles, like photons, and we will then also compare the location of the stable orbits equal to −1 for massive particles and bodies. Utilizing the relative to the location of the MBH horizons. Killing symmetries for the Schwarschild spacetime, cor- responding to energy (E) and angular momentum (l), the above may be reduced to III. STABLE CIRCULAR ORBITS NEAR MBHs In general relativity, the innermost stable circular orbit dr 2 2M l2 ¼ E2 þ 1 − ϑ2 − : ð20Þ (ISCO) of a compact object, as the name suggests, is the ds r r2 smallest stable circular orbit for a massive or massless — test particle. The ISCOs are important in astrophysics From above, one may define an effective potential of for instance, in black hole accretion disks, where they the form potentially mark a characteristic inner edge of the disk. During binary inspirals they also mark the boundary after 1 2M l2 which the compact objects begin the merge phase and are ; ϑ2 − 1 − ϑ2 − VBH;effðr Þ¼ 2 2 : ð21Þ plunging into each other. The stable orbits provide r r information about the type and characteristics of the compact object. Their study is therefore pertinent and This may now be analyzed to deduce orbits and their even more prescient in the era where there are ongoing stabilities. 2 endeavors to observe near horizon features of black For photons, ϑ ¼ 0, and we obtain the effective poten- holes [3]. tial extremum by solving There have been extensive theoretical studies on stable orbits for test particles, in asymptotically flat l2 3l2M V0 ðr;0Þ¼− þ ¼ 0; ð22Þ Reisnner-Nordstrom spacetimes [37–43],aswellas BH;γ r3 r4 some studies in Reisnner-Nordstrom spacetimes with a nonzero cosmological constant [44,45]. There has also leading to the solution r ¼ 3M. From the sign of been extensive studies on photon surfaces in arbitrary space-times [46]. We would specifically like to inves- 3 2 12 2 00 l l M V γðr;0Þ¼ − ; ð23Þ tigate, and obtain simple analytic expressions for, stable BH; r4 r5 orbits for MBHs having M ¼jQBj, in asymptotically flat and asymptotically de Sitter backgrounds, with a focus for this solution, we conclude that on potential astrophysical observables. For instance, one of the intriguing results that we will obtain is the r γ ¼ 3M; ð24Þ observation that for MBHs in asymptotically de Sitter BH; backgrounds there is a new qualitative feature (absent both when Λ ¼ 0 or Λ < 0)—the emergence of outer is an unstable photon sphere. For massive particles, ϑ2 ¼ −1. Now, the extrema of the stable circular orbits, that depend on QB. We will also glean a simple analytic expression for this boundary effective potential are at when M ¼jQ j. Similar observations have been made B 2 2 for Schwarzschild de Sitter spacetimes [47,48] in the M l 3l M V0 ðr; −1Þ¼ − þ ¼ 0; ð25Þ past. We also relate the location of various length scales, BH;m r2 r3 r4 including those of the ISCOs, to comment on other observables in later sections. For finite and real-valued angular momenta (l), the above To clarify concepts and for comparisons, let us briefly equation has solutions for r in the range ð3M; ∞Þ.One review the status of ISCOs for a Schwarzschild black hole must analyze the stability of these orbits considering (BH) in an asymptotically flat background. Considering 2 2 θ ¼ const:, with a timelike or null affine parametrization, 2M 3l 12l M V00 ðr; −1Þ¼− þ − ; ð26Þ one has BH;m r3 r4 r5 023006-5 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) 2 to identify the ISCO. Substituting for l in above, from M ðQ2 þ l2Þ 3l2M 2l2Q2 00 0 ; −1 − B − B 0 Eq. (25), we identify that V r; −1 transitions from VMBH;mðr Þ¼ 2 3 þ 4 5 ¼ ; effð Þ r r r r positive values to zero at ð34Þ ISCO 6 rBH;m ¼ M: ð27Þ with their stabilities determined by the sign of This corresponds to the well-known ISCO for massive 2 2 2 2 2 particles in the Schwarschild spacetime background. Below 2M 3ðl þ Q Þ 12l M 10l Q V00 ðr; −1Þ¼− þ B − þ B : this radius, in the range ð3M; 6MÞ, the circular orbits are MBH;m r3 r4 r5 r6 unstable for massive bodies. We note from Eqs. (24) and ð35Þ (25) that below 3M, no circular orbits can exist for both photons and massive bodies. Again, in the interesting extremal limit M ¼jQBj, we are Let us now turn our attention to MBHs in asymptotically able to proceed analytically and investigate the status of flat spacetimes. Following similar arguments, as for the stable circular orbits. Schwarschild spacetime, one obtains using the Killing The viable extrema are now at symmetries 0 2 2 2 2 r ¼jQBj; dr 2 1 − M QB ϑ2 − l MBH;m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ E þ þ 2 2 ; ð28Þ ds r r r l2 l4 − 8l2Q2 r ¼ B : ð36Þ MBH;m 2jQ j with the corresponding effective potential identified as B 1 2M Q2 l2 Analysing Eq. (35), one notes that there is an apparently V ðr; ϑ2Þ¼− 1 − þ B ϑ2 − : ð29Þ stable circular orbit at r0 ∀ l2 ≥ 0. Again, due to the MBH;eff 2 r r2 r2 MBH;m presence of the event horizon at rH ¼jQBj, any perturba- 0 For massless particles like the photon, the extrema may tion of the massive test particle near rMBH;m may cause it to − ∀ 2 8 2 again be identified by solving fall in through the horizon. The orbit at rMBH;m l > QB is unstable. There are also stable circular orbits for 2 3 2 2 2 2 þ ∀ 2 ≥ 8 2 0 l l M l QB rMBH;m l QB. The smallest radii at which a V γðr;0Þ¼− þ − ¼ 0; ð30Þ MBH; r3 r4 r5 truly stable orbit exists, and therefore the location of the ISCO, is at and their stabilities analyzed considering the sign of 2 2 2 2 ISCO 4 3 12 10 rMBH;m ¼ jQBj: ð37Þ 00 l l M l QB V γðr;0Þ¼ − þ ; ð31Þ MBH; r4 r5 r6 Note also that stable orbits given by real-valued at the respective extrema. þ ∀ 2 ≥ 8 2 ∞ rMBH;m l QB extend all the way to þ . We will We focus on the extremal MBH with M ¼jQBj,inan note later that in a de Sitter background there is a qualitative asymptotically flat spacetime background. In this limit, difference. we may analytically investigate Eqs. (30) and (31) for the Let us now consider the case of MBHs in asymptotically possibility of stable circular orbits. We observe that there is de Sitter spacetimes with a positive cosmological constant an ostensibly stable circular orbit at the horizon (Λ > 0), and analyse the status of stable circular orbits 0 therein. rMBH;γ ¼jQBj; ð32Þ For comparisons, again consider first the Schwarzschild de Sitter spacetime (BHdS)—i.e., with. QB ¼ 0 and Λ > 0. and an unstable circular orbit at For massless particles there is again an unstable photon sphere at 1 2 rMBH;γ ¼ jQBj; ð33Þ r γ ¼ 3M; ð38Þ corresponding to the photon spheres. The event horizon is BHdS; located at rH ¼jQBj, and any perturbation of the test 0 and for massive particles an ISCO at around photon near rMBH;γ, toward the horizon, will cause it to plunge through it. Thus, it is stable only in one direction. 81 For massive neutral bodies, the analogous extrema are ISCO ≃ 6 1 ϵ2 O ϵ4 rBHdS;m M þ þ ð Þ : ð39Þ located at solutions to 4 023006-6 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) We have assumed again that M2Λ ≪ 1, consistent with the given by Eq. (18), and may therefore denote a thermody- observed value of the cosmological constant and mass namically stable situation—Eq. (43) implies that ranges we consider. Interestingly, as has been pointed out recently [47,48], r2 Λ r4 − Q r − Q 2 − ½3 Bð BÞ there is now an astrophysically intriguing qualitative differ- l ¼ − − 2 : ð45Þ ence in the outer orbits that is absent in asymptotically flat ðr QBÞðr QBÞ (Λ ¼ 0) or anti de Sitter (Λ < 0) backgrounds. Instead of the stable or quasi stable outer orbits extending all the way This may be substituted in Eq. (44) and analysed for to þ∞, there is now a boundary beyond which l2 is no vanishing points. 4 00 4 ; −1 0 longer positive and finite. This limit is denoted by an At r ¼ jQBj, VMBHdS;mð QB Þ < , and below this outer stable or quasistable circular orbit (OSCO). For radius it has a negative sign as well. Hence, we conclude Schwarzschild de Sitter spacetime this is located approx- that the ISCO must now lie slightly above this radius. For imately at [47,48] the observed positive cosmological constant, for which Q2 Λ ≪ 1 in the range of interest to us, the location of the 1 3 B 3M = ISCO may be deduced by Newton-Raphson iterations and rOSCO ≃ : ð40Þ BHdS;m 4Λ is found to be approximately at For some systems this boundary may potentially be of 64 astrophysical or cosmological significance. For example, rISCO ≃ 4 Q 1 ϵ2 O ϵ4 : MBHdS;m j Bj þ 9 þ ð Þ ð46Þ for typical galactic masses they are of the same scale as the intergalactic spacing, and for galaxy clusters they are of roughly the same size as the clusters themselves [48]. There is again a qualitatively new feature for the Let us now turn to the QB ≠ 0 case, assuming a back- magnetic Reissner Nordstrom black hole embedded in a ground with a positive cosmological constant (MBHdS). de Sitter background–the presence of an outer boundary for We now need to analyse the effective potential stable orbits—as earlier also observed in the Schwarschild de Sitter case. This feature may be deduced and inves- 1 2 2 Λ 2 2 2 M QB r 2 l tigated by analysing Eq. (45), cognisant of the requirement V ðr; ϑ Þ¼− 1 − þ − ϑ − : 2 MBHdS;eff 2 r r2 3 r2 that for physically allowed orbits l must be positive and finite. ð41Þ 2 As r → 2jQBj, l starts to blow up, and hence illustrates 2 the possible existence of an ISCO in the vicinity. With For massless particles (ϑ 0), when M Q , the ¼ ¼ B 2 Λ ≪ 1 photon sphere (stable in one direction) is at the same QB , we suspected from Eq. (37) that it must be close 4 radius as in the Λ ¼ 0 case, to jQBj. Iteratively solving, we already established the position of this ISCO in Eq. (46). 0 Focusing on the quartic in the numerator of Eq. (45), r γ ¼jQ j: ð42Þ MBHdS; B we deduce from Sturm’s theorem (see for instance [49]) 1 that there are exactly two real roots; also, since coefficients There is an unstable circular orbit at r ¼ 2jQ j, also RNdS;γ B of the quartic are real, the two imaginary roots come in as in the asymptotically flat case. complex conjugate pairs. From Descarte’s rule of signs For massive neutral bodies, the analogous extrema are [49], we also further note that there are zero negative roots located at and therefore exactly two positive roots. 2 2 2 2 2 2 Since again Q Λ ≪ 1, one of these roots must be Λr M ðQ þ l Þ 3l M 2l Q B V0 r;−1 − − B − B very near jQ j. We are interested in the other positive MBHdS;mð Þ¼ 3 þ 2 3 þ 4 5 B r r r r root, in the context of finding an outer stable or quasistable ¼ 0; ð43Þ circular orbit. A relatively sharp bound by Lagrange and Zassenhaus (see for instance [50,51]) for the positive roots with their stabilities determined by the sign of n n−1 of a polynomial pðxÞ¼anx þ an−1x …a1x þ a0 is given by Λ 2M 3ðl2 þ Q2 Þ V00 r; −1 − − B MBHdS;mð Þ¼ 3 3 þ 4 r r 1=2 1=n 2 2 2 an−1 an−2 a0 12l M 10l Q x ≤ 2 max ; …; : ð47Þ − þ B : ð44Þ a a a r5 r6 n n n For M ¼jQBj—in which case the outer event horizon Thus, the larger of the positive roots for the numerator of and de Sitter cosmological horizon have equal temperatures Eq. (45), and hence the position of the OSCO, is bound by 023006-7 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) bounds on MBH dark matter from galactic magnetic field 23 25 27 29 31 33 35 10 10 10 10 10 10 10 measurements. 1022 Let us consider the magnetic fields sourced by MBHs 1018 near the horizon and close to stable orbits in their vicinity. The magnetic field at the event horizon of an MBH is 1014 given by 1010 Q 106 μ 1 B 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM Bðr ¼ rHÞ¼ 2 2 4π 2 10 M 1 1 − QB þ ð M Þ 10–2 QB –6 M⊙ 10 2 34 1015 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM ¼ . × Tesla 2 –6 –4 –2 0 2 4 6 2 10 10 10 10 10 10 10 M 1 1 − QB þ ð M Þ M⊙ ≤ 2.34 × 1015 Tesla ; ð49Þ FIG. 1. Various length scales associated with an extremal M MBH. The ISCO furnishes a characteristic length scale for various astrophysical phenomena, such as suggestively marking where the equality holds for an extremal MBH, i.e., when the edge of accretion discs. During binary inspiral of two compact M ¼jQBj. In the extremal case, the event horizon is objects, they also characterise typical separations before which located at rH ¼jQBj. They fall inversely as the mass of the objects will begin to plunge into each other. In asymptotically the MBH. In Fig. 2, we show variation of the magnetic field de Sitter spacetime, an OSCO also appears which is displayed in the figure. For comparison, a few astrophysical length scales are at the horizon and at the ISCO as a function of the MBH overlaid—the radius of the Milky way galaxy, the indicative mass. Remarkably, even in mass ranges where no electro- horizon radius for the M87* BH as measured by the Event weak corona forms [11–15], or equivalently electroweak Horizon Telescope [3], and the inner edge of the accretion disc symmetry is restored, the magnetic fields near the horizon measured for the Aquila X-1 neutron star system [52]. and at the location of ISCO are immense. For a better perception of the magnitude of these MBH generated fields, across various masses, we may compare it 3Q 1=3 3Q2 1=4 rOSCO ≤ 2 B ; B : MBHdS;m max Λ Λ ð48Þ 1023 1025 1027 1029 1031 1033 1035 21 It is found numerically that the bound is relatively good 10 and gives an approximate analytic expression for the 1019 neighborhood of the OSCO. This supposition is also further 1017 strengthened by application of Sturm’s theorem in ever tighter intervals. Due to the smallness of Λ, or more 1015 2 Λ ≪ 1 13 precisely due to QB , the orbit corresponding to 10 the largest positive root will be stable or at least quasistable 1011 with a very long lifetime. In Fig. 1 we compare the various horizon scales and 109 stable orbits of an MBH. The ISCO lies very close to the 107 event horizon, and as we shall discuss in the next section, will have a tremendous magnetic field in its neighborhood. 10–6 10–4 10–2 100 102 104 106 With an increase in the MBH mass, the OSCO and ISCO separation gradually reduces. A few astrophysical length scales are also shown for comparisons. FIG. 2. Magnetic field at the horizon and at the ISCO of an extremal MBH. The horizontal grey band corresponds to the typical magnetic field of a magnetar [4]. It is observed that for a IV. QUANTUM FIELD THEORETIC AND large range of MBH masses, the magnetic fields are very large PHENOMENOLOGICAL ASPECTS OF MBH near the horizon and at the location of the ISCO; even rivalling the currently know strongest cosmic magnets—magnetars [4]. In this section we briefly review few relevant quantum This opens the possibility that the MBH neighborhoods may be field theoretic aspects of MBH and then some consequent very promising as probes of strong field quantum field theoretic constraints. We will focus on aspects of the near horizon phenomena [53,54]. As we will also discuss, in Sec. V, these large magnetic fields, leading to symmetry restoration, Hawking fields could also potentially lead to striking electromagnetic radiation from extremal and near-extremal MBHs, and emissions. 023006-8 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) to other large fields currently observed in the universe. Magnetars are a subset of neutron stars with extremely large magnetic fields— ≳ 109 BMagnetar Tesla: ð50Þ They are among the strongest cosmic magnets that are currently known [4]. We therefore note that the MBH fields near the event horizon, near the ISCOs, and even farther out, are generically much larger than typical Magnetar external fields. Hence, if MBHs exist in our universe, their neighborhoods, where presumably accretion phenomena FIG. 3. Different regions of MBH mass with qualitatively −4 may be operational or trapped particles may exits, would be different features as discussed in the text. For M ≳ 10 M⊙, exquisitely suitable for testing strong field quantum field the magnetic field at the horizon is not large enough to render ≤ 10−4 theoretic phenomena (see for instance [53,54]). Apart from electroweak symmetry restoration. For M M⊙, the this, these immense fields, also make MBH interactions electroweak symmetry can in principle be restored near the horizon depending on the magnetic charge QB. Furthermore, for with neighboring plasma or other compact objects, sources −17 M ≤ 10 M⊙ the Hawking radiation rate is enhanced due to the for striking electromagnetic phenomena. In Sec. V, we will emergence of massless 1 þ 1 dimensional fermionic modes with discuss a few aspects in this context. large degeneracy which can escape to a large distance., see text – Also, as discussed in [13 15], in the presence of a large for more details. 2 magnetic field (larger than mh), the electroweak symmetry 1 may be restored and only the Uð ÞY component of the magnetic field survives; here Y denotes the hypercharge. In Horizon Telescope may offer an opportunity to probe such this case, as emphasized in [11], the near horizon region of potential near-horizon phenomena in the future [3]. the MBH will be in an electroweak symmetry unbroken The electroweak corona and electroweak hair surround- phase, with intriguing properties. In order for the magnetic ing such low-mass MBHs may have interesting features, ≥ 2 ≡2 25 1019 such as being nonspherically symmetric [10,11]. These field to satisfy Bðr ¼ rHÞ mhð . × TeslaÞ,a −4 different regions of MBH mass are pictorially shown in necessary condition is then to have M ≤ 1.04×10 M⊙. Fig. 3. A comprehensive exploration of the phenomenology For extremal MBHs, this would also correspond to of MBHs, with electroweak coronas, in this low-mass Q ≤ 5 3 1024 M B . × A-m; for a given , the maximum region is presented in [17]. magnetic field is obtained when M ¼jQBj. As one moves Let us now briefly discuss a few relevant aspects of MBH radially away from the horizon, the magnetic field keeps thermodynamics. The temperature of a MBH is given by decreasing. The electroweak symmetry will remain unbro- 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ken as long as B r r ≥ m . In the region where ð ¼ HÞ h Q 2 ≤ 2 ≡0 93 1019 1 4 1 − ð BÞ Bðr ¼ rHÞ mWð . × TeslaÞ, the electroweak qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM Tðr ¼ rHÞ¼ 2 symmetry is subsequently broken. 8πM Q 2 1 þ 1 − ð BÞ In the intermediate region, dubbed the electroweak M 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi corona [11], where m ≤ Bðr ¼ r Þ ≤ m , there is 2 W H h 4 1 − ðQBÞ W-boson condensation and the Higgs vacuum expectation 6 15 10−8 M⊙ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM ¼ . × K 2 – 2 value is smaller than 246 GeV [13 15]. The characteristic M 1 1 − QB þ ð M Þ radius of the electroweak corona is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1 − QB M⊙ ð Þ 5 3 10−18 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM QB ¼ . × MeV 2 ∼ M Q 2 rcorona 2 : ð51Þ 1 þ 1 − ð BÞ qBmWmH M 1 ≤ : ð52Þ In the relevant low-mass range of Figs. 1 and 2 and, 8πM where an electroweak corona may form, one has r ∼ pffiffiffiffiffi corona Oð1Þ mm and decreasing as M for the smaller mass The temperature is zero for an extremal MBH. Due to ranges. For the higher mass-ranges in this region of the existence of the near-horizon electroweak symmetric parameter space, one therefore has a macroscopic region region, and thus massless fermions, the Hawking radiation extending outside the horizon. This region where electro- from a MBH can be modified. It was shown in [55] that a weak symmetry is restored may affect processes and 3 þ 1 dimensional massless chiral fermion with charge q emissions close to the horizon, during accretion, just before (in the current context, q would be the hypercharge) the in-falling matter enters the event horizon. The Event gives rise to qQB=qB (where again as before we define 023006-9 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) 2πϵ ℏ 2 3 44 10−9 1 1 qB ¼ 0 c =e ¼ . × A-m) massless þ This is the speed a monopole, starting from rest, gains in a dimensional chiral fermions. The vanishing mass may be constant magnetic field after traversing unit coherence understood by noting that the lowest Landau levels for length lc. lc is the characteristic length of the galactic fermions in the presence of magnetic field have zero domain over which the magnetic field may be assumed energy. This means that the radiated power will be propor- to be roughly constant. In order for the MBHs to 2 tional to ðqQB=qBÞT , which is the same temperature constitute the DM of our galaxy, their average velocity −3 dependence as for a blackbody radiation in 1 þ 1 dimen- near the solar system should be v ∼ 10 c (assuming ρlocal ≈ 0 4 3 ≫ sion. This is bigger by a factor qQB than the power radiated DM . GeV=cm ) and v vmag. This is because, if from a Schwarzschild black hole of the same horizon size. we are assuming that the MBHs constitute the DM of the Thus, there is increased radiation as compared to a neutral galaxy, their velocity must be smaller than the galactic black hole. virial velocity to remain bound to the galaxy. Also, if For example, consider that the MBH temperature sat- v ∼ 10−3c ≪ v , then the monopole will quickly accel- −17 mag T>m M ≲1 03 10 M⊙ isfies e, which corresponds to . × . erate to vmag and eventually leave the galaxy. This clearly is The Hawking radiation will be enhanced by the above not the case we are interested in. degeneracy factor. Now, even for T 023006-10 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) Here Δt ¼ lc=v, and we are assuming that the MBH is A near-extremal MBH, as a DM candidate would be subject deflected only slightly. Hence, by looking at the second to a weaker bound. For instance, if there is say a 10% order contribution, it is clear that the kinetic energy deviation from extremality, increases on the average, although the first order contri- bution is vanishing. M=Kg ¼ 1.1 × 38.7; ð65Þ Assuming that the MBHs account for a faction fDM of QB=A-m the observed local DM density, their flux may be approxi- mated as the bound would be approximately 20% weaker −16 ≲ 2 1 10−3 2.13 × 10 Kg v fDM . × : ð66Þ F ¼ f MBH m2 sec : DM M 10−3c These indicative bounds show that, unlike the uncharged ρ × DM : ð58Þ primordial BHs (in certain mass ranges), it is very unlikely 0.4 GeV=cm3 that the extremal or near-extremal MBHs constitute a very significant fraction of DM in our universe. The Parker bound [57–59] is obtained by requiring Note that the above bounds on the DM fraction are that the average energy gained by the MBHs, during the derived assuming parameter values corresponding to 10 regeneration time treg of the galactic magnetic field, be Andromeda galaxy (M31), for which lc ¼ kpc, treg ¼ 10 ρ 0 4 3 10−3 – smaller than the energy stored in the magnetic field. From Gyr, DM ¼ . GeV=cm and v ¼ c [60 63]. The this line of argument, one gets possibility to strengthen the Parker bound by using M31 magnetic field measurements was first pointed out in [17]. 1 4 Improvements in galactic magnetic field measurements Δ 4π 2 ≤ 2 π 3 Ek FMBH ð lcÞ treg 2μ B 3 lc ; ð59Þ opens up the possibility that this bound may be further 0 strengthened in the future [64]. Before we conclude this section, we would like to which implies that the flux is bound by very briefly comment on the possibility of nucleon decay catalysis by the MBHs, similar to the Rubakov-Callan Mv2 – F ≤ : effect for magnetic monopoles [65 70]. We like to note MBH 3μ 2 ð60Þ 0QBlctreg here that magnetic monopoles need not always catalyse proton decay. It depends on whether the physics at the core Using Eq. (60) now in Eq. (58), we get a limit on the of the monopole violates baryon number or not. It should MBH DM fraction however be noted that already the Standard Model weak interaction do violate baryon number, by the triangle 2 2 anomaly, and can in principle lead to unsuppressed proton −6 M A-m v f ≤ 1.5 × 10 decay catalysis [71,72]. In the original proposal by DM Kg Q 10−3c B Rubakov and Callan, it was the baryon number violating 3 0.4 GeV=cm 10 kpc 10 Gyr gauge field configuration inside the monopole core which × ρ : ð61Þ DM lc treg was responsible for the baryon number violation. In fact, soon after Rubakov and Callan’s work, examples were Assuming typical astrophysical parameter values, the constructed where monopoles indeed did not catalyse nucleon decay [73,74]. Moreover, a Rubakov-Callan like fraction of DM, fDM, that can be made of the MBHs, is therefore constrained to be around analysis in the background of a MBH may have more subtleties [11] due to the fact that the magnetically charged 2 object now is a black hole, with an event horizon. Thus, to ≲ 1 5 10−6 M=Kg fDM . × : ð62Þ our understanding, no robust and model independent bound QB=A-m can be derived at present on the MBH abundance based on Rubakov-Callan like effect. For an extremal MBH, which has V. ELECTROMAGNETIC EMISSIONS M=Kg FROM MBH BINARY INSPIRALS ¼ 38.7 ð63Þ Q =A-m B If an MBH is involved in a binary inspiral, due to the magnetic charge, one expects electromagnetic emissions one gets from the system. We would like to estimate the power radiated as well as understand some characteristics of ≲ 1 7 10−3 fDM . × : ð64Þ this electromagnetic emission, for typical extremal MBH 023006-11 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) parameters. We will consider two cases. where only one of the black holes in the binary is an extremal MBH, and the case where both black holes in the binary are extremal MBHs. Let us consider the case of a neutral black hole and an extremal MBH undergoing binary inspiral (MBH-BH case). At sufficiently large separations (relative to event horizon radii), during an epoch long before the actual merger of the black holes, we may neglect corrections to Maxwell’s equations due to spacetime curvature, as a first approximation. If ω_ =ω2 ≪ 1, where ω is the orbital angular velocity, the orbits may be further approximated as an adiabatic sequence of quasicircular ones. This is well FIG. 4. While for β ≪ 1, most of the electromagnetic Poynt- β → 1 satisfied in the earlier phase of the binary inspiral. The ing flux is directed perpendicular to the orbital plane, as the radiation profile qualitatively changes. Most of the emitted electromagnetic and gravitational radiation reactions also electromagnetic power for β → 1 is in the orbital plane, and tend to make the orbits circular [75], thus making the points in the direction of the MBH’s instantaneous velocity. If assumption of quasi-circular orbits well motivated. All an observer is in the line of sight of this beam, they would see a these then imply that the instantaneous acceleration and quasi-periodic emission that gradually increases in frequency. velocity are also well approximated to be perpendicular to Also, as we will quantify, if MBHs exist and undergo capture each other. Finally, to focus just on the essentials, we will by another compact object to form an MBH-BH binary pair or also take QB;1 ¼ QB;2 ≡ QB for the extremal black holes. something similar, the total electromagnetic power output is These well motivated assumptions, as we shall see, will generically very large—rivalling even some of the most enable us to gain a clearer analytic understanding of the energetic gamma ray bursts known to date (see discussions electromagnetic emissions from the MBH-BH system. in the text and Fig. 5). With the above assumptions, we may use the Li´enard generalization of the Larmor formula to estimate the emitted in the direction of the MBH’s instantaneous differential power radiated per solid angle for MBH-BH velocity. inspiral (see Appendix B). To estimate the radiation pattern and the net power emitted it suffices to consider MBH-BH 1 dP ∼ 0 0 3 : ð69Þ the situation at a particular instant. The instantaneous dΩ β→1 ð1 − sin θ sin ϕ Þ velocity and acceleration of the MBH are perpendicular to each other and the radiated power per solid angle for the The radiation profile is dominant in the forward direction MBH-BH may be computed as (Appendix B) to the instantaneous velocity and symmetric about it. The electromagnetic emission sweeps forward like a headlight, MBH-BH 2 2 4 dP μ0Q a ω ¼ B as the MBH orbits (see Fig. 4). dΩ 64π2c3 This is of course just analogous to synchrotron radiation ½ð1 − β sin θ0 sin ϕ0Þ2 − ð1 − β2Þsin2θ0cos2ϕ0 from an electrically charged particle moving in a circle. · : ð1 − β sin θ0 sin ϕ0Þ5 Thus, if earth is in the path of this intense beam at some point in the orbit, we may be able to detect this large and ð67Þ quasiperiodic Poynting flux as it repeatedly sweeps by. Fast radio bursts [5,6] are mysterious, radio transients that last β ω 2 ¼ð aÞ= c is the boost and a is the separation for Oð1Þ ms, whose origins are not currently understood. It between the MBH and BH at that instant. The prime is intriguing to speculate if, at least in some cases, Fast denotes that we are adopting a coordinate frame of radio bursts may be related to this sweeping of the reference attached to the MBH and that the angles are electromagnetic emission, as an MBH undergoes orbital with respect to it. motion. Even when β ≪ 1, the binary system would be When β ≪ 1, during the early inspiral phase, most of the driving a very energetic Poynting flux with ever increasing radiation is emitted normal to the orbital plane. power. For electrically charged black holes some specula- tion in this general direction exists in the literature [76]. MBH-BH dP 2 2 There has also been interesting ideas proposed related to ∝ 1 − sin θ0cos ϕ0: ð68Þ dΩ β≪1 the collapse of magnetospheres in Kerr-Newman black holes [77]. The beaming effect in binaries involving MBHs, As the orbital speeds increase and β → 1, later in the as β → 1, nevertheless has not been speculated on before. inspiral evolution, the radiation profile changes qualita- Compared to electrically charged compact objects, for tively. One notes that the radiation is now dominantly MBHs, the considerations may also be markedly 023006-12 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) different—due to their better astrophysical viability, poten- tially larger charges, as well as corresponding changes in 1023 1025 1027 1029 1031 1033 1035 the electromagnetic emissions and orbital evolutions. The total power radiated may be computed from above 1050 to be 1047 μ 2 2ω4 MBH-BH 0QBa P ¼ 3 : ð70Þ 24πc 1044 To understand the above emitted power as a function of 1041 the MBH-BH separation or orbital angular velocity, one may use the Keplerian relation to express the orbital 1038 angular velocity also as a function of a, or vice versa. 10–6 10–4 10–2 100 102 104 106 The total electromagnetic power emitted varies as a−4 or 8=3 equivalentlyqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as ω . Now, the Lorentz factor defined as ω2a2 γ 1 1 − 1023 1025 1027 1029 1031 1033 1035 ¼ = 4c2 gradually increases as the binary inspiral β → 1 evolves and closer to the merger phase one expects . 1046 The emitted electromagnetic power in that regime is 4 expected to be further enhanced by a factor ∼γ . This 1036 can lead to even larger electromagnetic emissions as the black holes inspiral closer to each other. The other effect is 1026 that the frequency of the emitted radiation will increase as the black hole separations decreases, evolving roughly as 1016 ω ∼ −3=2 EM a , starting from the far radio band and pro- gressing to gamma ray frequencies. In the late phase of the 106 binary inspiral, the curvature effects will start becoming important and will also have to be gradually accounted for 10–4 while treating electromagnetic emissions. Some aspects of 10–6 10–4 10–2 100 102 104 106 these curvature corrections, for dyonic black hole inspirals, have been discussed in the literature [18]. In Fig. 5 we compare the total electromagnetic power FIG. 5. The figure displays the total electromagnetic power emitted assuming various separations. When the separation emitted, during binary inspiral of a blackhole system where one of is quantified as a fixed multiple of the horizon radius, one the objects is an extremal MBH. For the MBH-MBH case, the total power emitted is just a factor of four larger. The emitted power is sees that the emitted power is independent of QB (or M) due to the assumption of extremality. This is because, when the displayed as a function of the common black hole masses and the different curves are for different black hole separations. When the Keplerian relation is used along with the expression for r , H separation is quantified as a fixed multiple of the horizon length, the dependence on QB drops out. The power emitted by a the dependence on mass drops-out, due to the assumption of few astrophysical sources are also displayed in the figure. extremality. For comparison, we display typical power emitted by Remarkably, one notes that the total power emitted in Type Ia supernovae, and note that the inspiral electromagnetic binary inspirals, involving an MBH, may be much larger power expected from MBHs may be many orders of magnitude than those from Type Ia supernovae and very energetic larger. Intriguingly, the emitted power may even rival some of the gamma ray bursts. most powerful gamma ray bursts observed in our universe—for Let us now consider the second case of interest— instance, GRB 080916C [78] as overlaid on the figure. the inspiral of two oppositely charged extremal MBHs (the MBH-MBH case). For simplicity and to focus on the Now, under the assumptions we are working with, the salient aspects, we again take the magnitude of the charges retarded potentials and electromagnetic fields due to the to be the same for both the magnetic black holes. We also two revolving MBHs may be approximated as that due to follow all the simplifying assumptions that we had adopted the superposition of two oscillating magnetic dipoles. The for the MBH-BH case. Now, as the two MBHs revolve two dipoles just differ by a π=2 phase difference, under the around each other, they will again generate magnetic dipole assumption of quasicircular orbits. The electromagnetic radiation. The emitted Poynting flux is now expected to be fields and the corresponding Poynting vector may be larger than the MBH-BH case, owing to the fact that both computed readily (see Appendix B), giving the instanta- objects are now magnetically charged. neous power radiated per unit area as 023006-13 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) 2 2 4 μ0Q a ω be extended to cases where M1 ≠ M2 and Q 1 ≠ Q 2,as S⃗MBH-MBH ¼ B ½1 − fsin θ cos ðωt − ϕÞg2 r:ˆ B; B; 16π2c3r2 r well as for motions with nonzero eccentricities (see for ð71Þ instance [79]). We do not expect the main points to drastically change from the present analysis nevertheless. Here, r is the distance from the binary system to the These electromagnetic emissions and the associated radiation reactions will contribute to the gradual evolution observer and tr ¼ t − r=c is the retarded time. When averaged over a complete cycle, this gives the mean of the orbit. Thus, to analyse the orbital and gravitational directional intensity of the emitted electromagnetic radia- wave frequency evolution, unlike conventional binary tion in the MBH-MBH case, inspirals with just neutral black holes, we will need to include these in addition to the power radiated by gravi- 2 2 4 tational waves. We will investigate these aspects in the next μ0Q a ω 1 MBH-MBH ˆ ⃗MBH-MBH B 1 − 2θ ˆ section. There, we will analyse the modification to the IEW r ¼hS i¼ 2 3 2 sin r: 16π c r 2 orbital frequency evolution, and consequently the evolution ð72Þ of emitted gravitational frequencies, along with changes to the inter black hole separation. The maximum intensity is again directed normal to the 1 2 VI. GRAVITATIONAL WAVES FROM BINARY orbital plane, for β ≪ 1. It falls off as 1 − 2 sin θ with increasing polar angle, with moderate emissions in the INSPIRALS OF MBH orbital plane. The generated radiation is found to be in The dynamics of the MBH-BH or MBH-MBH binary general elliptically polarized, but normal to the orbital systems will be modified from conventional expectations, plane, along the direction of maximum intensity (zˆ), it will due to the large additional electromagnetic Poynting flux. be circularly polarized. Apart from a possible electromagnetic counterpart, this Within our approximations, the total power radiated may must also manifest in the evolution of the orbits and the now be estimated by integrating over a sphere surrounding gravitational waveforms. We would like to investigate this the binary system. This gives in some detail, and we adopt the same assumptions as those made in the previous section, to obtain simple analytic 2 2 4 μ0Q a ω results. PMBH-MBH ¼ B : ð73Þ EW 6πc3 For two binary black holes in an elliptic orbit, with eccentricity e, the total radiated power in gravitational This is twice the power radiated by a single oscillating waves is given by [80] magnetic dipole. This is because, though quadratic in the 8G 1 73 37 fields, the cross terms with fields from each of the super- bin: N 2 4ω6 1 2 4 PGW ¼ 5 M a 2 7 2 þ e þ e : posing dipoles average to zero, being out of phase by π=2. 5c ð1 − e Þ = 24 96 This is also four times greater than the total power emitted ð74Þ in the MBH-BH case. Again, as they spiral inward and the orbital speeds The binary system radiates both energy and angular- become relativistic, the total power radiated will be momentum through gravitational waves, and through 4 enhanced by a factor γ . Thus in the later epochs of electromagnetic emissions when charged. A careful study the inspiral, the electromagnetic power radiated can be of the orbital evolution suggests (see for instance [75]) that further enhanced. Also, for β → 1,asintheMBH-BH in general the eccentricities decrease quite rapidly due to case, most of the emitted power is expected to be in the the radiation back reaction. Thus a fair approximation is to orbital plane. The emitted electromagnetic frequency also consider e → 0 in Eq. (74), and focus on quasicircular increases from the far radio band to gamma ray bands, as orbits in the early stages of the inspiral. the inspiral evolves, going again as a−3=2. One point to In Fig. 6 we show the total gravitational wave power, as note though is that the Keplerian relations are slightly well as the ratio of the electromagnetic to gravitational modified in the MBH-MBH case—owingtothefactthat power emitted. Note that due to the slightly different for extremal MBH pairs, the electromagnetic attraction is Keplerian relations, when expressed solely as a function exactly equal to the gravitational attraction. Hence, the of the inter black hole separation, the total gravitational ω actual numerical factors and coefficients relating EM power radiated is different for the MBH-BH and MBH- to orbital separation will be slightly different from the MBH cases. The latter is about eight times larger than the MBH-BH case. former, for a fixed separation. For providing some intuition, For simplicity and to focus on the most salient aspects, the peak gravitational wave power observed in the first we have assumed equal masses and equal magnitudes for LIGO detection event GW150914 [1] is also shown. The the magnetic charges everywhere, but the expressions may GW170817 neutron star merger event [2] had a very weak 023006-14 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) 23 25 27 29 31 33 35 10 10 10 10 10 10 10 1023 1025 1027 1029 1031 1033 1035 50 1050 10 1049 30 1048 10 1047 1010 1046 1045 10–10 1044 10–6 10–4 10–2 100 102 104 106 10–6 10–4 10–2 100 102 104 106 23 25 27 29 31 33 35 10 10 10 10 10 10 10 1023 1025 1027 1029 1031 1033 1035 10 1014 5 1011 108 1 105 0.5 100 10–6 10–4 10–2 100 102 104 106 10–6 10–4 10–2 100 102 104 106 FIG. 6. The upper panel shows the gravitational wave power emitted by a binary black hole system where one of them is an extremal MBH. For the MBH-MBH case, the total power radiated in gravitational waves is eight times larger than for the MBH-BH case. For comparison, the peak power emitted in gravitational waves during LIGO first-detection event GW150914 [1] is overlaid. The lower panel shows the ratio of electromagnetic power (Fig. 5) to the gravitational power. A guidepost may be provided by the LIGO event GW170817 [2]. This neutron star merger event also had a relatively weak short gamma ray burst (≲2 s) counterpart—GRB170817A [7,81]. For this case, a very crude estimate of the ratio of the isotropic equivalent electromagnetic power [7,81,82] to gravitational power [2] gives a value ≲10−4. 2 electromagnetic counterpart [7,81], and a rough estimate dE MBH-BH 8G μ0Q ¼ − N M2a4ω6 − B a2ω4: ð75Þ of the electromagnetic to gravitational power ratio in that dt 5c5 24πc3 case gives around 10−4 or less. Thus, in the early stages of the inspiral the electromagnetic power generally exceeds Here, the Keplerian relations give the gravitational power output. As the binary system radiates electromagnetic and 2 gravitational waves, the radial separation between the black GNM E ¼ − ; holes decrease. As long as ω_ =ω2 ≪ 1, this evolution is 2a well-approximated by an adiabatic sequence of quasi- 2G M ω2 ¼ N ; ð76Þ circular orbits. In the regime we are making our approx- a3 imations, this is satisfied to a very good extent. qffiffiffiffiffiffiffiffiffi As in the last section, let us first consider the case of a μ0 with the association QB ¼ 4π M, for the extremal case neutral black hole and an MBH undergoing inspiral (MBH- GN BH case). Assuming a very early epoch in their binary we are interested in. inspiral, when they are well separated, we may write by Equation (75) then leads to an evolution equation for the energy conservation binary system’s orbital frequency 023006-15 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) ffiffiffi 2 p3 2 ω_ MBH-BH 96 μ0 Q ω_ MBH-MBH 96 4 μ M 5=3ω5=3 B ω 5=3 5=3 0 QB 2 ¼ 5 ðGN chÞ þ 3 : ¼ ðG M Þ ω þ ω: ω 5c 4πc M ω2 5c5 N ch πc3 M ð77Þ ð82Þ M Equation (82), similar to before, may be translated into The chirp mass as usual is defined by ch ¼ 3=5 2=5 an evolution equation for the emitted gravitational wave μ ðM1 þ M2Þ , where μ ¼ M1M2=ðM1 þ M2Þ is the reduced mass of the binary system. In our case, with the frequency simplified assumptionffiffiffi of equal masses, we therefore have pffiffiffi p5 3 8=3 M ¼ M= 2. dν MBH-MBH 96 4π 11 3 ch GW ¼ ðG M Þ5=3ν = Since the gravitational waves from a binary black dt 5c5 N ch GW hole system is predominantly emitted at a frequency πμ 2 ω 2ω 0 QB ν3 GW ¼ [75], Eq. (77) may be translated into an þ 3 GW: ð83Þ evolution for the emitted gravitational wave frequency c M 8=3 Equations (81) and (82) now give the adiabatic evolution dν MBH-BH 96π 11 3 GW ¼ ðG M Þ5=3ν = of the inter black hole separation as dt 5c5 N ch GW πμ 2 pffiffiffi 0 QB ν3 da MBH-MBH 512 58 þ 3 GW: ð78Þ − M 3 −3 4c M ¼ 5 ðGN chÞ a dt 5c 8μ − 0GN 2 −2 The approximate adiabatic time evolution of the radial 3 Q a : ð84Þ 3πc B separation may also be deduced from Eqs. (76) and (82) as pffiffiffi The complete gravitational waveform has the form MBH-BH 128 5 8 μ da − M 3 −3 − 0GN 2 −2 ¼ 5 ðGN chÞ a 3 QBa : dt 5c 3πc G Ma2ω2 h ðtÞ¼ N ð1 þ cos2 θÞ cosð2ωtÞ; ð79Þ þ c4r 2 2ω2 GNMa θ 2ω h×ðtÞ¼ 4 cos sinð tÞ: ð85Þ Consider now the case of two oppositely charged MBHs c r in early stages of their inspiral. From Eqs. (73) and (74) the evolution is therefore approximately given by Here, θ and r are the orientation and distance to the observer, with θ ¼ π=2 corresponding to the orbital plane 2 of the inspiralling black holes. The waves are circularly dE MBH-BH 8G μ0Q − N M2a4ω6 − B a2ω4: 80 polarized along the direction perpendicular to the orbital ¼ 5 5 6π 3 ð Þ dt c c plane and linearly polarized along the orbital plane direc- tion. We may plot the time evolution of the gravitational Here, we now have wave frequencies and inter black hole separation for the two cases of interest, and contrast it with the case of 2 two neutral black holes undergoing inspiral (BH-BH case). G M E ¼ − N ; This is illustrated in Fig. 7. The onset of chirping, of the a gravitational wave frequencies, in the three cases display 4 ω2 GNM distinct characteristics. ¼ 3 : ð81Þ a As an aside, we must mention that for precise detection qffiffiffiffiffiffiffiffiffi of gravitational wave signals, via current matched filtering μ0 techniques, in late stages of the compact binary inspiral For the extremal case of interest we have QB ¼ 4π M GN it is crucial to include post-Newtonian (PN) corrections again. The additional factors of two in Eq. (81), relative to [83–88]. Defining Eq. (76), are due to the fact that in the extremal case, the Coulombic attraction is exactly equal to the gravitational δ 2 ω 3 2=3 ∼ O 2 2 attraction and modifies the Keplerian relations in that ¼ð GNM =c Þ ðv =c Þ; ð86Þ manner. The orbital frequency of the binary system therefore this would modify the first terms on the right-hand side of evolves now as Eqs. (77) and (82) by factors of the form 023006-16 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) 100 5 107 50 2 107 20 1 107 10 5 106 10–8 10–7 10–6 10–5 10–4 10–8 10–7 10–6 10–5 100 500 50 200 20 100 10 50 10–8 10–6 10–4 0.01 1 10–8 10–6 10–4 0.01 1 100 0.005 50 0.002 20 0.001 –4 10 5. 10 10–6 0.001 1 1000 106 10–8 10–5 0.01 10 104 FIG. 7. Evolution of the inter black hole separations and frequencies for binary black hole systems in the three cases discussed in the text. Three prototypical mass values are taken from the range of interest to us, to illustrate features and variations. One observes that the gravitational wave frequency and inter black hole separations evolve differently in the three cases of interest. As expected, the onset of chirping of the gravitational wave frequency occurs first for the MBH-MBH case. XN contributions, that cause the binary orbits to display δ PN δn=2 gð Þ¼ cn=2 : ð87Þ complex trajectories. n¼0 VII. SUMMARY AND CONCLUSIONS PN 7 The coefficients cn=2 have been computed to n ¼ Magnetically charged black holes [8–11],iftheyexist, [83–88]. Some aspects of the PN corrections in the context provide a hitherto unexplored avenue to probe findamen- of dyonic black holes were discussed in [18]. Dyonic black tal physics and exotic quantum field theoretic phenomena hole inspirals have richer features compared to purely in the Standard Model of particle physics, and beyond. magnetic (or electric) black holes. For instance, as pointed Unlike electrically charged black holes, magnetically out and analyzed in [18], in the PN limit, there may be charged black holes may be relatively long lived and noncentral, angular momentum dependent forces, and other hence are promising candidates for future observations. 023006-17 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) They generically have large magnetic fields in the near inter black hole separation, the total gravitational wave horizon neighborhoods—in many cases, approaching power emitted by the MBH-MBH system is larger than fields many of orders of magnitude larger than neutron the MBH-BH system, owing to the different factors in the stars like magnetars [4]. In some regions, the field is even Keplerian relations. We also find that the electromagnetic strong enough to provide electroweak symmetry emitted power dominates the gravitational wave emitted restoration [11]. power in the early inspiral phase. The gravitational We explored various possible astrophysical signatures waveform for MBH-BH and MBH-MBH display inter- and limits on such objects in this work, in the extremal and esting time evolutions, and were observed to have near-extremal limits, over the full range of astrophysically distinct evolution histories compared to BH-BH binary interesting mass ranges. We have added new results and inspirals. The onset of the frequency chirping occurs estimates in this work, while also complementing recent much before for the MBH-MBH and MBH-BH cases. studies [17,18] that have appeared. The binary merger time-scale, as inferred from the We pointed out that horizons and innermost stable evolution of the inter black hole separation, is quicker circular orbits in the vicinity of extremal MBHs, which for these cases, as expected. set a characteristic distance for various astrophysical A deeper exploration of some of these and other phenomena such as accretion and the beginning of plunge astrophysical pointers is left for future work. As already phase during binary inspirals, show a distinct imprint of mentioned earlier, an interesting facet to ponder on is their intrinsic nature. In the case of a positive cosmological how a magnetic charge affects the ringdown and quasi- constant, there is also an intriguing qualitative feature normal modes in the postmerger phase. An understanding embodied by the appearance of an outer stable circu- of these aspects [22] is especially pertinent in the broader lar orbit. context of understanding the nature of dark compact Current measurements of galactic magnetic fields and objects [21]. In a similar vein, further exploration of dark matter density measurements were found to furnish potential near-horizon phenomena and effects on black interesting bounds on these objects across the full mass hole shadows [20] may be interesting future directions to range of astrophysical relevance. In the lower mass ranges, pursue. It may be hoped that some of the astrophysical this include cases where an electroweak corona forms near signatures, whose characteristics we have broached in the horizon, and electroweak symmetry is restored in the present work, may pave the way to the discovery of macroscopically large regions. Considering galactic mag- these exotic compact objects, if they indeed exist in our netic field regeneration times, we were able to place strong universe. bounds on magnetic black holes as dark matter candidates, with the conclusion that they are unlikely to constitute a ACKNOWLEDGMENTS significant component of dark matter in our universe. Within astrophysical and modeling uncertainties, our We thank Y. Bai, J. Berger, S. Jain, M. Korwar, R. results are consistent with the seminal result found in [17]. Loganayagam, and J. Maldacena for discussions, and thank Even for other masses, we pointed out that the fields D. Sachdeva for help with figures. D. Ghosh acknowledges are still generally enormous—enough to furnish spec- support through the Ramanujan Fellowship and the tacular electromagnetic astrophysical signatures, as well MATRICS grant of the Department of Science and as affect gravitational waveforms during binary inspirals. Technology, Government of India. A. Thalapillil would Considering binary inspirals of MBH-BH, as well as like to acknowledge support from an Early Career Research MBH-MBH, we noted that the electromagnetic fluxes are award from the Department of Science and Technology, extremely large with very characteristic angular profiles, Government of India. with respect to the orbital plane. The total electromag- netic power emitted we found may even overwhelm some APPENDIX A: HAWKING TEMPERATURE of the most energetic gamma ray bursts known to date OF MBH HORIZONS IN ASYMPTOTICALLY [78]. In this context, we speculated on connections that DE SITTER SPACETIME MBHs may have to mysterious phenomena in the uni- verse, like fast radio bursts [5,6]. Interestingly, in the The temperature of the Black hole is given by extremal MBH case, for a fixed separation quantified as a κ multiple of the horizon length, the electromagnetic power T ¼ 2π ; ðA1Þ radiated is independent of the MBH mass. We analysed some aspects of the gravitational wave where κ is the surface gravity given by emissionsfrombinarysystems,inboththeMBH-BHand MBH-MBH cases, and contrasted it with the conven- 2 1 κ ¼ − DaξbD ξ : ðA2Þ tional case when both black holes are neutral. For a fixed 2 a b 023006-18 ASTROPHYSICAL HINTS FOR MAGNETIC BLACK HOLES PHYS. REV. D 103, 023006 (2021) Here, ξ is the timelike killing vector, Thus, the Hawking temperature at these two horizon radii are equal, with ξa 1 0 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ð ; ; ; Þ; ðA3Þ rffiffiffiffi 1 Λ Λ 1 − 4 Tðr0Þ¼TðrcÞ¼2π 3 M 3 : ðA14Þ ξa ¼ð−ΔðrÞ; 0; 0; 0Þ; ðA4Þ with APPENDIX B: ELECTROMAGNETIC 2 2 2M Q Λr RADIATION INVOLVING MBHs ΔðrÞ¼ 1 − þ B − : ðA5Þ r 2 3 r Many of the results we require may be computed readily, utilizing the superposition principle and electric-magnetic Using the above, we thus get, duality transformations (see for instance [89]). Under the duality transformations, solutions for electric charges and 1 2 ac 0 currents may be converted to their magnetic charge and κ ¼ − g Dcξ Daξ0; ðA6Þ 2 current analogues, through the mapping 1 rr 0 0 0 0 B → E=c; E → −Bc; Q → −Q =c: ðB1Þ − g ∂ ξ Γ ξ ∂ ξ0 − Γ ξ0 E B ¼ 2 ð r þ r0 Þð r r0 Þ: ðA7Þ Li´enard’s generalization of the Larmor formula for Γ0 1 00∂ 1Δ0ðrÞ rr Δ ∂ ξ0 0 Using r0 ¼ 2g rg00 ¼ 2 ΔðrÞ , g ¼ ðrÞ, and r ¼ , electric charges gives 0 ∂ ξ0 ¼ −Δ ðrÞ (rest of the terms are zero either because the r 2 rˆ v⃗ ⃗ 2 Christoffel symbols are zero or because derivative of a dP QE j × ð × aÞj ¼ 2 5 ; ðB2Þ component of the metric vanishes), we further get dΩ 16π ϵ0 ðrˆ · v⃗Þ which on integrating over the full solid angle gives the total Δ0ðrÞ2 κ2 electromagnetic power emitted as ¼ 4 ; ðA8Þ μ 2 γ6 ⃗ ⃗ 2 0QE 2 v × a jΔ0ðrÞj P ¼ ja⃗j − : ðB3Þ ⇒ κ 6πc c ¼ 2 : ðA9Þ ⃗ Hence, the temperature of the black hole at coordinate Here, for the particle trajectory sðtrÞ, tr denoting the ⃗ distance “r” is given by retarded time, and observation point r, we have defined _ _ ⃗ ⃗ ˆ v⃗¼ s⃗ðtrÞ, a⃗¼ v⃗ðtrÞ, r ¼ r⃗− s⃗ðtrÞ and v ¼ cr − v⃗. γ as jΔ0ðrÞj usual is the Lorentz factor. TðrÞ¼ 4π ; ðA10Þ To compute the power radiated in the MBH-BH case easily, we may choose a coordinate system (S0) whose with origin is at the position of the MBH. The MBH instanta- neous velocity and acceleration are perpendicular to each other. Assume that at some instant it has velocity along 2M 2Q2 2Λr Δ0 B − the yˆ direction and acceleration along −xˆ. Then, using the ðrÞ¼ 2 þ 3 : ðA11Þ r r 3 electric-magnetic duality transformation of Eq. (B1) and the generalized Larmor formula of Eq. (B2), one gets Using the expressions for r0 and rc, from the text, we get for M ¼jQBj MBH-BH μ 2 2ω4 dP 0QBa Ω ¼ 64π2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi d c Λ Λ 1 − β θ0 ϕ0 2 − 1 − β2 2θ0 2ϕ0 0 ½ð sin sin Þ ð Þsin cos Δ ðr0Þ¼2 1 − 4M ; ðA12Þ · : 3 3 ð1 − β sin θ0 sin ϕ0Þ5 B4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ rffiffiffiffi Λ Λ Δ0 −2 1 − 4 This on integrating over the full solid angle then gives the ðrcÞ¼ 3 M 3 : ðA13Þ total emitted power as, 023006-19 GHOSH, THALAPILLIL, and ULLAH PHYS. REV. D 103, 023006 (2021) 2 2 4 μ0Q a ω electromagnetic fields generated during the MBH-MBH PMBH-BH ¼ B : ðB5Þ 24πc3 binary inspiral as For the MBH-MBH binary system, considering μ ω2 ⃗MBH-MBH 0QBa quasicircular orbits, the fields may be computed by super- B ¼ ½fcos ðωt Þ cos ϕ þ sin ðωt Þ sinϕg 4πc2r r r posing the fields from two oscillating magnetic dipoles θθˆ ω ϕ − ω ϕ ϕˆ offset by π=2. For concreteness, let us assume that the cos þfsin ð trÞ cos cos ð trÞ sin g ; binary is in the X-Y plane of a suitably chosen coordinate E⃗MBH-MBH ¼ cðB⃗MRNbin: × rˆÞ: ðB7Þ frame whose origin is at the center of mass of the system. Then, one of the oscillating magnetic dipoles is in the xˆ The usual implicit assumptions have been made—that the direction and the other along yˆ, albeit with the π=2 phase observation is made in the radiation zone (r ≫ a) and that lag. 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