arXiv:1603.02848v1 [gr-qc] 9 Mar 2016 † ∗ u fet,myalwfrteeeto fpr ftems of mass the of part of ejection quan- the to for due allow are d may which classically effects, singularity, tum is spacetime that the object by quantum scribed nak a the is to which close appear singularity, th could Also, that powers. pressures radiative negative high strong hole such black explain the ac- would electro and than emit the case, efficiently principle then more in much time, could radiation same short magnetic the but naked around topology limited a disk to the cretion for rise and be gives merger event may sin- hole an singularity, naked such black lived’ accompanies the ‘short that if change a fact, and of In merger, formation hole gularity. the of black in topology the results of change accompanies which the that to spacetime hol due the black occur may binary These from detected mergers. waves count gravitational electromagnetic to of existence parts the for explanation little ble if disk. burst the in ray explain present gamma possibly be should observed to matter the how of amount [4] of by duration suggested short collap- was standard the pro- It the an to The model. similar of very sar [3]. existence is system here the binary mechanism posed by the surrounding given disk is accretion electromagneti confirmed) the (if of explanation should counterpart possible waves One electromagnetic of allowed. emission be such no then uum electro- possible a of [2]. the detection counterpart by magnetic the claim of merger the binary collaboration by accompanied hole FERMI soon black was a [1] from (GW150914) waves gravitational of tion lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic ntepeetpprorproei osgetaohrpossi- another suggest to is purpose our paper present the In vac- in merge to were horizons hole black two if However, detec- the of collaboration LIGO the by claim recent The lcrmgei oneprst rvttoa ae from waves gravitational to counterparts Electromagnetic ASnmes 42.w 42.b 04.70.Bw 04.20.Jb, 04.20.Dw, numbers: PACS t during of presence counterpart the electromagnetic that an suggest of We luminosity c case. high the hole in black higher ‘pure’ much the scen be to the would from spectrum forms luminosity singularity emitted the naked lumin the emitted where the scenario then system the i hole matter, black s some the the case of In that surroundings singularity. topology naked in a change of the appearance app that the the out to point due We be merger. may the coalescence hole black binary in signals ecnie h usinhr hte h rpsdelectrom proposed the whether here question the consider We .INTRODUCTION I. aaIsiueo udmna eerh oiBah od C Road, Bhabha Homi Research, Fundamental of Institute Tata eateto hsc,Nzrae nvriy 100Asta 010000 University, Nazarbayev Physics, of Department ae singularities naked Dtd uut9 2021) 9, August (Dated: ail Malafarina Daniele akjS Joshi S. Pankaj er- ed e- e e c - ol hnb sda osbeosrainlts oconstr to test observational hypothesis. possible such a as spectr used luminosity be measured then could The ho times. black effi- all a at more which exists the in luminosity horizon situation higher around corresponding the much disk than a ciently accretion away transient give small a would of any same appearance and the or hand, singularity circular other naked stable the the innermost of On an (ISCO). presence of bit the existence by the and determined would horizon power [6], and accretion [3] in an considered have ones the as such merger, hole objects. original the crto ikbigmc oeefcierdao,adalso other. distingui each and observationally from principle radiator, in effective are more they therefore much singul being naked disk the a accretion objects, spectra emission these the for that different seen was significantly it There [11]. in singularities naked studied and holes properties black the around disks between accretion comparison the perspective, observa horizon, an tional a from behind Also, covered consequences. not observational have is may system st the the in when arising region th effects field during quantum time that limited possible a is for it a merger, forms occur is singularity would naked merger a change if hole topology Then such black which binary in situation A typical observati detectable implication. a and have principle signature in in occurring could effects horizon quantum vicinity a some topol its the then by in spacetime, change covered the to of leads not ogy that is process a that during singularity form naked singularity should a change a of topology if occurrence that the Therefore, changes [10] implies in topology spacetime shown a such was in of it result Furthermore a [9]. as occur to si linked generically that time and are ities long [8], manifold singularities a spacetime and/or for violations a known causality in been changes has topology quantum- It that on window [7]. a effects indicate gravitational may equations Einstein’s of emre uhmr easily. more much merger he h omo ml crto ik speeti the in present is disk, accretion small a of form the n st uigtemre ol lo odistinguish to allow would merger the during osity twsWelrwosgetdta iglrte nsolution in singularities that suggested who Wheeler was It black binary a surrounding disk accretion any that note We uhatasetnkdsnuaiywl xli the explain will singularity naked transient a such aac fa‘hr ie’nkdsnuaiyduring singularity naked lived’ ‘short a of earance roweetehrzneit taltms nfact In times. all at exists horizon the where ario aeieudrosdrn h egrcncause can merger the during undergoes pacetime s hr ae iglrt om sopposed as forms singularity naked a where ase † gei onepr ftegaiainlwave gravitational the of counterpart agnetic ∗ lb,Mma 005 India 400005, Mumbai olaba, a Kazakhstan na, lc oemresand mergers hole black ngular- shable rong arity onal was um ain of re le e s - - - . . 2

The paper is organized as follows. In section II we review during the merger event, a window would open on the ultra- the basic features of changes in topology of a spacetime in high density compact object located at the point of merger. connection with naked singularities and we discuss the im- For example, overspinning of a Kerr solution could provide plications of such changes for the coalescence of two black a mechanism for the formation of a naked singularity. Note holes. In section III we outline how the emission of radia- that this process would seem to violate the black hole area tion, when an accretion disk is present near a naked singularity theorem. This is consistent with fact that the black hole area in the system can help explain the proposed electromagnetic theorem assumes that the spacetime is weakly asymptotically counterpart to GW150914. Finally comments and remarks are simple and empty, which is cosmic censorship assumption, presented in section IV. that is violated at the time of the merger due to the topology change. In other words, for the black hole area theorem to hold one must assume that cosmic censorship is valid. On the II. TOPOLOGY CHANGE, NAKED SINGULARITIES AND other hand a violation of cosmic censorship, as in the case il- BLACK HOLE MERGERS lustrated here, allows for the area theorem not to hold, at least temporarily. It was Geroch who showed that any spacetime containing In this connection, a specific possibility that may occur is, two spacelike hypersurfaces with different topologies must when two Kerr black holes of spin very close to unity collide have either closed timelike curves or singularities [8]. The and merge, then at least momentarily a Kerr geometry with argument was refined by Tipler who showed that by impos- a> 1 could develop (where a is the adimensional spin param- ing the weak energy condition it is possible to ensure that eter given by a = J/M). This would then eventually settle to there cannot be any topology change without the presence a Kerr black hole by radiating away a sufficient amount of an- of a spacetime singularity [9]. The visibility or otherwise of gular momentum through a suitable mechanism. Such a sce- such singularities was then investigated in connection with the nario would be entirely consistent with the topology change Cosmic Censorship Hypothesis, and one of us showed that theorems mentioned above. naked singularities in terms of past incomplete nonspacelike Whether it is possible to create a Kerr solution with spin geodesics do arise as a consequence of topology change [10]. parameter greater than one is one of the key unanswered ques- There are several approaches that one can adopt in order tions of black hole physics today. Several astrophysical black to interpret such singularities. For example, topology change hole candidates are known with spin parameter close to unity. can be viewed as a purely classical phenomenon provided one Is it possible to add enough angular momentum to such ob- allows to extend general relativity to include degenerate met- jects in order to destroy the event horizon? In [14] it was sug- rics, and therefore the singularities associated with them can gested that infalling test particles carrying angular momen- be considered to be in some sense ‘mild’ [12]. On the other tum can indeed overspin a nearly extremal Kerr black hole hand, a very powerful and fruitful idea is to interpret the sin- and turn it into a naked singularity. On the other hand in [15] gularities that arise in classical GR as an indication of break- it was shown that the inclusion of the gravitational self-force down of the theory and therefore use them as milestones to- could prevent the above scenario from happening. All these wards a viable theory of quantum gravity. A fully quantum- examples involve test particles carrying angular momentum gravitational approach should take care of such singularities, and are restricted by simplifying assumptions on the initial removing the undesired infinities without breaking the causal set up. A completely different scenario is that of the merger structure of the manifold. Nevertheless, even without a full of two black holes of comparable size, both carrying angu- theory of quantum-gravity, one can use a semi-classical ap- lar momentum and with an arbitrary initial configuration. The proach to quantum corrections to show how these would af- mergerof two such black holes would have to carryaway most fect the strong field regime by using purely classical tools, i.e. of the excess angular momentum for the final configuration to by solving the usual Einstein field equations (see for example settle to a Kerr solution with a < 1. Simulations have shown [13]). that the merger of two black holes can carry away enough Then, if naked singularities can be viewed as a window on angular momentum to respect the Kerr bound [16]. However, quantum-gravity, the appearance of one in the merger of two there are several parameters that come into play when describ- black holes may possibly produce a powerful explosive phe- ing a merger in full and one cannot exclude at present that an nomenon(such as a GRB) whose nature would be intrinsically intermediate nakedly singular phase can form from some ini- quantum-gravitational. Such an event would be a cosmic lab- tial configurations(see Figure 1), as suggested by the topology oratory where one could observe and test quantum-gravity. change considerations. The coalescence of two black holes is a typical situation in We should also recall here that the ringdown phase in the which a topology change occurs. A binary black hole sys- detected GW150914 event should not be necessarily taken tem is characterized by an initial topology of two disjoint as a strong indication of the formation of the event horizon event horizon hypersurfaces, and the final topology (after the necessarily, as the same signal may be obtained by a small merger) is a single null hypersurface which is the resultant ringed compact object (see [17]). Therefore it is worth ask- event horizon of the merged black hole. In the simplest case ing the question whether a binary merger such as that asso- of two Schwarzschild black holes merging, the initial topol- ciated with GW150914 could produce a transient quantum- ogy is that of two disjoint circles evolving, and final topology gravitational phenomenon, in which the central object, classi- is given by S2 R. Then if a naked singularity were to form cally described by a naked singularity, is not covered by the × 3 horizon for a short finite time. Allowing for this possibility, magnetic signal could come from vicinity of such an object. we are now interested in investigating what kind of electro-

t 4 t 4

t t 3 3 t 2 t 2 t 1

t t 1 r

FIG. 1: Qualitative depiction of the merger of two black holes with formation of a transient naked singularity. On both panels, at the time t = t1 the black holes are separated. Thick lines represent the horizons and dotted lines the central singularities. At the time t = t2 the two horizons merged into a single horizon, the singularities are still separated. At the time t = t3 the singularities merge forming a temporary overspinning Kerr spacetime. There is no horizon. After t = t3 the excess angular momentum is radiated away and the system settles to a Kerr black hole, as at the time t = t4.

III. ACCRETION ONTO A NAKED SINGULARITY as

2 acc = ǫmc˙ , (1) L In the observed GW150914 the merger carried away an es- timate of three solar masses in gravitational waves. In the where m˙ is the accretion rate of the mass in the disk and case where the singularities of the colliding black holes are ǫ = 1 EISCO is the efficiency, which is related to the bind- − hidden behind the horizon at all times this amount of energy ing energy per unit mass of circular orbits evaluated at the cannot come from the irreducible mass of the black holes that ISCO (EISCO) (see for example [18]). For a Schwarzschild is possibly concentrated in the singularity. The energy radi- black hole the ISCO is located at 6M, a is equal to zero and ated away in gravitational waves must come from the bind- ǫ = 0.057. In the limiting case of a Kerr black hole with a ing energy of the system, loss of angular momentum and/or going to one the ISCO shrinks to rISCO = M and the effi- from the matter in the accretion disk. Accretion disks around ciency reaches the maximum allowed value for a black hole black hole candidates are known for being not very massive, of ǫ = 0.42. As we can see, the efficiency depends on the typically of the order of 0.1M⊙, and therefore the amount of location of the ISCO. For a naked singularity the stable cir- energy that they could provide is not enough. Further to this, cular orbits could in principle extend all the way to the center for a binary system like the one that originatedGW150914 the and the efficiency can be as high as one hundred percent. If surrounding medium is expected to be almost entirely devoid we interpret the singularity as a measure of our ignorance on of matter. On the other hand, as we shall see below, a naked quantum gravity, we can assume that it must be replaced by singularity is a much more efficient engine to convertthe grav- a finite sized exotic compact object of Planck size. Then the itational energy of the infalling particles into radiation. The ISCO would not be exactly zero and thus the emitted power repulsive forces that may occur near the singularity as a result would be lower than a ‘pure’ naked singularity but still much of quantum effects could also possibly strip some mass away higher than that of a black hole. Similarly, the time interval from the central object to eject it in the surrounding medium, during which the source is not coveredby the horizon depends thus providing a source of matter for the accretion disk. upon the specific nature of the central object. It may reduce to an instant in the limit of a naked singularity in the collapse In [11] we studied the properties of accretion disks around frame but would appear longer and finite to faraway observer. naked singularities. For steady state accretion, the luminosity From these considerations we see that knowing the mass for an accretion disk around a compact source can be written of the final configuration and assuming a scenario such as the 4 one presented here we could use the measured luminosity to depending on the specific features of the accreting fluid. Nev- constraint the size and ‘lifespan’ of the exotic central object. ertheless it is generally believed that the Eddington limit may Given a theoretical model for a compact source one can not be surpassed by much. In [4] it was pointed out how the evaluate Lacc in order to compare the observed spectrum with proposed electromagnetic counterpart to GW150914 detected the predicted one. For circular accretion disk the net luminos- by FERMI, that lasted approximately one second, had a lumi- ity for the observer at infinity is given by the integral over ln r nosity that exceeds by ten orders of magnitude than that ex- of pected for a 60M⊙ black hole. Therefore, while keeping the assumption that the two detections were caused by the same d L =4πr√ gE , (2) event, to make sense of this high luminosity one can either in- d ln r − F voke an extremely high accretion rate on a black hole or one can postulate a lower and more realistic accretion rate on an where r is the radial coordinate in the equatorial plane, √ g exotic compact object. is the determinant of the three dimensional metric outside− the Note that a transient naked singularity forming from the compact source (black hole or not) in the equatorial plane of merger would require an accretion rate of the order of the disk, E is the energy of the test particle of the disk at a 10−5M /sec to produce a GRB of luminosity distance r from the center and is the radiative flux emitted ⊙ F by the disk at a distance r (see [11] and references therein). erg 1049 , (9) The flux is in turn given by L≃ sec r m˙ ω,r which is therefore much smaller than that of black hole. This = , (E ωL)L dr (3) 2 ,r would allow or require for a much less massive accretion disk F −4π√ g (E ωL) ZrISCO − − − to be present in the surrounding medium at the time of the where ω and L are the angular velocity and angular momen- event to accomplish the electromagnetic signal. A naked sin- tum of the test particle in the disk. For a general stationary gularity event of the duration of one second would allow the and axially symmetric metric of the form compact source to emit the electromagnetic burst before set- tling down to a final black hole configuration. The duration of 2 2 2 2 2 ds = gttdt +2gtφdtdφ + grrdr + gθθdθ + gφφdφ , (4) the GRB would then be related to the characteristic lifetime of the naked singularity. This in turn would be given by the test particles in the accretion disk are confined to the equato- time required by the horizon to collapse plus the time neces- rial plane θ π/2 and their energy, angular momentum and ≃ sary to shed away the excess angular momentum. The time angular velocity can be written as scale of the latter would then be related to the natural scale of the underlying quantum-gravity theory. Then the measure- gtt gtφω E = − 2 , (5) ment of the duration of the burst could provide information − gtt 2gtφω gφφω − − − on the nature of the exotic compact object at the center of the gtφ gφφω process. L = − 2 , (6) gtt 2gtφω gφφω − − − 2 gtφ + (gtφ,r) gtt,rgφφ,r ω = − − . (7) IV. DISCUSSION p gφφ,r

Then the location of the ISCO can easily be evaluated from The detection of the gravitational wave signal GW150914 the second derivative of the effective potential for particles in from the LIGO observatory has opened a new era in astro- the disk physics. The possibility of the existence of an electromagnetic counterpart to the signal as detected by the FERMI observa- 2 2 E gφφ +2ELgtφ + L gtt tory has been suggested. If the existence of electromagnetic Veff = 2 1. (8) gtφ gttgφφ − counterparts to gravitational waves during binary black hole − merger was to be confirmed that would raise a lot of questions Note how the above quantities depend on the specific na- about the nature of the sources of gravitational waves. This ture of the spacetime metric coefficients. Therefore the ISCO would be especially so if the measured luminosity of the event is determined by the metric. By looking at equation (3) it is would prove to be much higher than the Eddington luminosity not difficult to see that for rISCO going to zero diverges, for the binary black hole system. therefore suggesting that in a realistic scenario theF singular- In the present paper we asked the question whether a tran- ity must be replaced by a finite Planck sized exotic compact sient naked singularity appearing during the merger could object. be used as a source for the observed electromagnetic signal. Every object in space has a maximum luminosity Edd, Given our ignorance on the quantum nature of the exotic com- called the Eddington luminosity, beyond which radiationL pres- pact object that is classically described by the singularity we sure overcomes gravity. Once the accretion flow reaches Edd cannot construct a full model describing the process. For ex- the material outside the object stops falling inwards andL gets ample we do not know if the matter in the accretion disk is pushed away from it. It is possible for an accreting compact coming from the medium outside the original black holes or object to have a luminosity that exceeds its Eddington limit if the repulsive pressures originating close to the singularity 5 can eject part of the mass of the original object and add it to If hypothesis such as the one above were to be proven cor- the accretion disk. On the other hand we have seen how the rect the study of these events would then providea measurable measurement of the luminosity of these events could provide cosmic laboratory where to test and observe quantum-gravity observational constraints on the nature of the compact object at work. that forms during the merger. Therefore, while more obser- Acknowledgement: The authors thank Prashant Kocher- vations are needed in order to better understand the nature of lakota, Jun Qi Guo, Andrzej Krolak, Abraham Loeb, K. I. the phenomenon, at present it is worth asking whether it is Nakao, and Ramesh Narayan, for useful discussions on these possible to use such observations to confirm or rule out the themes. DM would like to thank TIFR for support and hospi- possibility of the creation of a transient naked singularity dur- tality while this work was done. This article was supported by ing the coalescence. the Nazarbayev University Social Policy grant.

[1] B. P. Abbot et al. Phys Rev Lett, 116, 061102 (2016). [12] G. T. Horowitz, Class. Quant. Grav. 8, 587 (1991). [2] V. Connaughton et al. arXiv:1602.03920. [13] C. Bambi, D. Malafarina and L. Modesto, Phys Rev D 88, [3] A. Loeb, arXiv:1602.04735. 044009 (2013). [4] M. Lyutikov, arXiv:1602.07352. [14] T. Jacobson and T. P. Sotiriou, Phys. Rev. Lett. 103, 141101 [5] X. Li et al. arXiv:1602.04460. (2009). [6] K. Murase, K. Kashyama, P. Meszaros, I. Shoemaker and N. [15] E. Barausse, V. Cardoso, and G. Khanna, Phys. Rev. Lett. 105, Senno, arXiv:1602.06938. 261102 (2010). [7] J. A. Wheeler, Ann. Phys. 2, 604, (1957). [16] B. Giacomazzo, L. Rezzolla and N. Stergioulas, Phys. Rev. D [8] R. Geroch, J. Math. Phys. 8, 782 (1967). 84, 024022 (2011). [9] F. Tipler, Ann. Phys. 108, 1 (1977). [17] V. Cardoso, E. Franzin and P. Pani, arXiv:1602.07309. [10] P. S. Joshi and R. V. Saraykar, Phys. Lett. A 120, 111 (1987). [18] K. S. Thorne, Astrophys. Journ. 191, 507 (1974). [11] P. S. Joshi, D. Malafarina and R. Narayan, Class. Quantum Grav. 31 015002 (2014).