Possible Astrophysical Observables of Quantum Gravity Effects near Black Holes
aa r X i v : . [ a s t r o - ph . H E ] J a n Possible Astrophysical Observables of Quantum Gravity Effects near Black Holes
Ue-Li Pen Canadian Institute for Theoretical Astrophysics, Toronto, Ontario, Canada (Dated: July 2, 2018)Recent implications of results from quantum information theory applied to black holes has led tothe confusing conclusions that requires either abandoning the equivalence principle (e.g. the firewallpicture), or the no-hair theorem (e.g. the fuzzball picture), or even more impalatable options.The recent discovery of a pulsar orbiting a black hole opens up new possibilities for tests of theoriesof gravity. We examine possible observational effects of semiclassical quantum gravity in the vicinityof black holes, as probed by pulsars and event horizon telescope imaging of flares. Pulsar radiationis observable at wavelengths only two orders of magnitude shorter than the Hawking radiation,so precision interferometry of lensed pulsar images may shed light on the quantum gravitationalprocesses and interaction of Hawking radiation with the spacetime near the black hole. This paperdiscusses the impact on the pulsar radiation interference pattern, which is observable through themodulation index in the foreseeable future, and discusses a possible classical limit of BHC.
PACS numbers: 97.60.Gb,04.70.Dy
Introduction –
The recent discovery of PSRJ1745-2900[1–3] orbiting the galactic center black hole opens upnew possibilities for precision tests of gravity. It allows usto investigate possible outcomes as its orbit is mapped,and possible quantum deviations from standard Einsteingravity.It has proven challenging to find experimental conse-quences of quantum gravitational effects. At the sametime, precision experimental probes of classical generalrelativity have a dearth of alternate theories to comparewith.In this letter, we explore possible semi-classical con-sequences of pulsar-black hole binaries and flares in thegalactic center black hole accretion flow. This is meantto stimulate concrete discussions of quantum mechanicsapplied to gravitational systems in scenarios that may betestable in the foreseeable future.
Motivation – The quantum mechanical nature of blackholes has provided a fruitful testbed for thought experi-ments and discussions. Hawking’s calculation led to thepossibility of black hole radiation and evaporation. Theradiation appears thermal, and appears not to dependon the interior of the black hole, or its formation history.This leads to the well known information loss problem[4].Historically the resolution of the problem included vi-olation of unitarity (i.e. causality), or the possibility ofremnants. String theory is a constructive example whichis unitary and contains the same black hole entropy andevaporation, and no remnants. In this context, the reso-lution of the paradox has to lie in the purity of Hawkingradiation vs the breakdown of the equivalence principlenear the horizon. As discussed in [5] (hereafter AMPS), amodification of Hawking radiation purity requires macro-scopic changes in space-time of order unity outside theScharzschild radius (see also [6] for a complementaryview). Different groups arrive at opposite aesthetic con-clusions from this line of reasoning: the firewall[5] pic- ture maintains radiation purity, and instead sacrifices theequivalence principle for infalling observers, who burn upat the horizon, thus preventing them from measuring vi-olations of quantum mechanics. The fuzzball picture ex-plores the opposite path[7]: the Hawking photons areemitted by a substantially non-Scharzschild geometry,but classical observers see a general relativistic spacetimeincluding the equivalence principle on both sides of thehorizon. This latter framework is consistent with prin-ciple of Black Hole Complementarity[8] (BHC, see alsothe Fuzzball interpretation of BHC[7]), that the classicaland quantum pictures depend on the nature of the mea-surement. Spefically, low energy probes, such as Hawk-ing radiation or grazing pulsar radiation, would be sub-ject to the quantum nature, while high energy probes,including protons, stars, and other likely matter see aclassical space-time. In this
Letter we follow this sce-nario to explore possible consequences for quantum mea-surement using pulsar radiation, which provide realisticprobes of BHC, using wavelengths comparable in energyto the Hawking radiation. This opens up the possibil-ity of testing physics using real experiments instead ofaesthetic considerations.Pulsars are highly compact, very bright light sourcesand exquisite clocks, enabling precision measurements ofspace-time. A pulsar orbiting a black hole provides ascenario which accentuates potential experimental out-comes. Very recently, the first candidate has been discov-ered [1–3], likely orbiting the galactic center black hole.While the orbital parameters are not known to the au-thor, there is a possibility that its orbit in projectionpasses close behind the black hole, such that a gravita-tionally lensed image becomes visible. Depending on or-bital parameters, such a conjunction could take decadesto occur during which time more pulsar-black hole bina-ries may be discovered. In addition to the galactic centresupermassive black-hole pulsar binary, ten double neu-tron star systems are known, and discovery of a pulsar-stellar mass black hole binary appears likely[9]. For pur-poses of this discussion, slow and fast pulsars are bothsuitable, with slow pulsars dominating the predicted pop-ulations. This is one of the goals of the planned SquareKilometer Array[22].
Strong Gravitational Lensing –
We consider the dy-namics of a pulsar orbiting a black hole. As a pulsarpasses behind a black hole, multiple images of the pulsarappear. In the weak field limit, one sees two images. Thisphenomenon is called “strong lensing”. In the strong fieldregime, an infinite number of exponentially fainter im-ages appear [10, 11]. In this section, we will confine ourdiscussion to the two images under “weak field” stronggravitational lensing.We will consider the regime where the pulsar is manySchwarzschild radii behind the black hole, and the weakfield limit applies, with only small perturbations. Thepulsar radiation is lensed by the black hole’s gravitationalfield, which is well described by geometric optics. Wefirst review the geometric optics, and then estimate theinterference pattern of this double-slit experiment.Generally, the brighter image is further away from theblack hole, and less affected by post Einsteinian effects.This two image scenario is much like a quantum doubleslit experiment. The wavelength of the photons is notdrastically different from that of the thermally emittedHawking radiation photons, and is expected to probe thelow energy limit.A background source always has two images: one insidethe Einstein radius, which we call the interior image, andone outside the Einstein radius, the exterior image. TheEinstein radius is defined in [12]: θ E = √ r s D ds D d , (1)where the Scharzschild radius r s = 2 GM/c , D ds is thedistance from the black hole to the pulsar, and D d isthe distance to us. We are considering the limit where r s ≪ D ds ≪ D d . From here onward, we will use unitswhere the speed of light c = 1. The apparent imagepositions for a source at angular separation β are at θ ± = 12 (cid:18) β ± q β + 4 θ E (cid:19) (2)with magnifications µ ± = u + 22 u √ u + 4 ±
12 (3)where u = β/θ E [13]. For large separations, the interiorimage gets faint as 1 /u , while the exterior image goesto its unlensed brightness. The time delay between thetwo images is∆ t = r s " u p u + 4 + ln √ u + 4 + u √ u + 4 − u (4) We now consider potential fuzzball corrections to thisstandard picture. In Scharzschild coordinates, the metricis ds = − [1 − ψ ( r )] dt + [1 − ψ ( r )] − dr + r d Ω (5)with ψ ( r ) = r s /r . For a light source many r s away,the lensing equation depends on the projected potential ϕ ( θ ) = 2 D ds D d R ψ ( p ( θD d ) + z ) dz . We consider a gen-eral multipole expansion of the potential, which in pro-jection becomes ϕ ( θ, φ ) = θ E ln( θ/θ E ) + θ E X m a m θ ms cos[ m ( φ − φ m )] θ m , (6)with the apparent Schwarzschild radius θ s = r s /D d . Wedefine the ratio b ≡ θ E /θ s , which is the impact parameterof the lensed image in units of Schwarzschild radii. Thegeneric orbit has b ≫
1, and our data probes θ ∼ θ E . Atthese large radii, the low m harmonics dominate. Sincethe deviations to the space-time are of order unity nearthe horizon, one expects a m to be order unity, and byisotropy the φ m are uniformly distributed. In a firewallpicture, all coefficients a m = 0.The lowest order perturbation is a dipole, m = 1. Thisis analogous to a displacement of the black hole positionby a Schwarzschild radius. This results in a variation oftime delay δ ∆ t = α r s b (7)for a order unity proportionality constant α .In a fuzzball scenario, the delayed pulse will appearbroadened: its pulse profile will appear wider, convolvedby the distribution of delays from the different multi-pole perturbations. While the actual separation of im-ages may be challenging to resolve in angle, the delaysare readily observable. As in interstellar plasma lensing,the multiple images interfere constructively and destruc-tively when observed in a single dish telescope[14]. Fortime estimates, we will scale to a solar mass black hole.When applied to the galactic center supermassive blackhole, we will explicitly state that. Typical pulsars havethe most sensitive detections at ∼ GHz. In a classicalblack hole, this results in a very precise measurement ofdelay and dopplershift: with a 1 GHz bandwidth, de-lays are measurable to a nanosecond. The characteristicdelay ∆ t for a solar mass black hole is several microsec-onds, and expected fuzzball fluctuations smaller by b .For a given set of orbital parameters, the fringe patternis fully determined, and can be tracked. After shiftingthe self-interference pattern by the geometric delay, oneis sensitive to the stochastic quantum fluctuations in timedelay (7), which induce phase differences of order unityfor b . ⊙ BH, at 4 × r S . The orbital FIG. 1: Pulsar modulation index. The aesthetic choice of“no-hair” is thought to result in a firewall, with classical mod-ulation index shown by the solid line. The alternative choiceembracing a equivalence/complementarity principle, and thusrejecting the firewall, could lead to the dashed line, where themodulation index is reduced due to exterior quantum effects. period is 2 hours, and gravitational radiation decay time ∼ yr, comparable to the life time of a slow pulsar(the most common type[15]). We use an inclination ofa quarter degree, which leads to conjunction at one Ein-stein radius. The modulation index h [16] resulting fromthe interference pattern depends on the orbital phase,and is shown in Figure 1. We model the phase changein (7) as a Gaussian random field, which changes everydynamical time. For illustration convenience, we tookthe prefactor α ∼ .
4. In a future data set, one could fitfor α from the data. The first principles computation of α is beyond the scope of this Letter, which we leave asa future exercise. We used an observing wavelength of λ = 20m, the low end of the LOFAR[23] telescope. If theequivalence principle is obeyed, the modulation index isdecreased, as is apparent in this picture. The fractionaleffect is smallest at superior conjunction, when the innerimage is the furthest from the black hole, and thus hasthe least quantum effects.PSRJ1745-2900 is currently about 1 millionScharzschild radii away from the galactic centerblack hole, in projection. If its orbit is inclined within0.1% to the line of sight, gravitational lensing effectsbecome order unity. In this particular system, plasmascattering decoheres the radiation by δt s ∼ (GHz /ν ) sec, and at frequencies of ∼ THz the scattering becomesnegligible. It is not known how bright the pulsar is atTHz frequencies. The ALMA[24] telescope might detect the pulsar at these frequencies. The gravitationallylensed images will form an interference pattern if theemission size is less than ∼ λθ E /θ s , or about 1m. Somepulsar emission, for example Crab giant pulses, arethought to come from regions sufficiently compact toscintillate. Current plasma lensing limits the emissionsize to be less than about a kilometer [17, 18]. Thereis about a 0.1% chance that this pulsar will have afavourable orientation to test quantum gravity. It wouldseem prudent to search for more black-hole pulsarcandidates, encouraged by the discovery of this first one.One might worry that plasma effects confuse quantumgravity. While the presence of plasma lensing could cer-tainly overwhelm gravitational lensing in some systems,this should not lead to a misidentification: plasma dis-persion depends on wavelength squared, while gravita-tional lensing is achromatic. Should one find a strongwavelength dependence of the effect, one would needto find a new and cleaner system with less plasma. Itseems unlikely to cause a gravitational misinterpretation.Holography of propagation effects can be used to removeplasma distortions, or even be used constructively to re-solve the gravitationally lensed images, enabling a directmeasure of phase coherence[17, 19].To summarize the observational test of pulsar lens-ing phase coherence: According to firewall supporters,the BHC picture predicts that the interior image under-goes substantial phase changes, of order r s /b . For wave-length longer than that, the interference pattern shouldbe obervable, and for shorter wavelengths, it shouldweaken and disappear. In the course of the orbit, thiswill change, with the fractional BHC impact minimizedduring conjunction (assuming conjunction is outside theeinstein radius). Finite emission size could also lead to anabsence of an interference pattern. This has a differentdependence on lensing geometry, so again seems unlikelyto mimic the BHC signal. Accretion Flow Flares
The Event Horizon Telescope(EHT)[25] could image radiation emitted by the plasmanear the horizon of the galactic center black hole Thisflow is known to be unsteady, with flares and other phe-nomena, that are strongly lensed[20]. As for the pulsarcase, the accretion flow and flare are classical objects, andexpected to follow the classical dynamics. The low energyphotons emitted from the flare are low energy probes, andsubject to interaction in BHC. Here, the effects would belarge. In a fuzzball picture, the interior lensed imagewould appear extended, i.e. fuzzy. Normally, this fainterimage, the one closer to the black hole, is smaller due toconservation of surface brightness. The opposite couldhappen in a fuzzball picture: any lensed image wouldmaintain a constant minimum size θ s ∼ µ ” due to thefuzziness. For a flare smaller than this size, deviationswill be observable.EHT experiments are unlikely to show interference be-tween lensed images, since accretion disk emission is gen-erally extended. Here, the test would rely on localizedflares, where multiple strongly lensed images would bevisible, and time profiles are observables. Discussion–
The lensing framework gives a simplifiedpicture to discuss Hawking radiation entanglement. In-stead of the typical exp(10 ∼ ) microstates, we can focuson the low order multipoles, reducing the variables to a , θ . To clarify the situtation further, we consider afurther restricted subspace: φ = 0, with the axis cho-sen such that the classical interior lensed image is at φ = 0. We will further restrict a ∈ {− , } , i.e. a 2-statesystem | + i , |−i , and consider a pure state black hole.Any incoming photon state becomes entangled with theperturbed lensing eigenstates: | i i = α | + , ↑i + β |− , ↓i ,where | α | + | β | = 1. It may be perturbed outward( ↑ ) or inward( ↓ ), resulting in a different phase delay asdiscussed above. An ensemble of photons, as expectedfrom pulsars or flares, is projected one photon at a time.This has some analogies to Stern-Gerlach space quantiza-tion, with the distinction that here the deflection field isquantum mechanical, perhaps like SQUID quantum su-perposition experiments[21]. The photon becomes effec-tively entangled with the fuzzball. For a static fuzzball,whose eigenstates are not spherically symmetric, the in-terference pattern is affected. When we then add thedegrees of freedom that the perturbations not discrete,we see the persistence of quantum perturbations due tonon-commutation of propagation operators and fuzzballeigenstates.In any matter flow, for example an orbiting neutronstar or accretion flow, one needs to compare the rate atwhich particles exchange energy with each other, com-pared to the differential energy shift from the fuzzballentanglement. For a star the outcome is clear: its largeinternal degrees of freedom lead to decoherence, and theyprobe an average space-time, which is Schwarzschild. Forthe flare under consideration, the fact that a lensing mea-surement requires the flare to be small compared to its or-bital radius, requires its self-interaction to be larger thanthe differential tidal effect, and the flare is also treatedclassically. This interpretation differs slightly from thecommonly stated BHC picture about energy: a high en-ergy photon may still be seeing a quantum space time,while an infalling observer with sufficient complexity tobe considered classical, will see the classical space-time.An interesting intermediate question would be the trajec-tory of a complex composite particle, such as a proton.Its constituent strongly interacting gluons and quarksmight each probe a different space-time, and the full pro-ton would see an averaged effect, closer to classical.This analysis combines uncertain speculation from op-posite opinion camps on the nature of black hole evapo-ration, namely the firewall group and the fuzzball group.We caution that the scenario is by no means inevitable,even in a fuzzball picture it is not clear that the O (1)variation of the space-time coefficients This was argued by AMPS as a weakness of the fuzzball picture. Summary. – We have explored semiclassical quantumeffects of black holes. A pulsar black hole binary pro-vides a concrete setup where such effects might becomeobservable. This proposed experiment distinguishes be-tween classical measurements, such as a star orbiting ablack hole, and quantum measurements, such as the in-terference of two light paths bent by the gravitationalfield of the black hole. The interference pattern could bechanged by the quantum nature of the black hole if theresolution of the Hawking paradox lies in the non-purityof Hawking radiation. The outcome is speculative in na-ture, and is hoped to stimulate further investigation ofnon-Einsteinian outcomes of strong lensing experiments.We have argued that the recent pulsar-black hole sys-tem, and likely more future discoveries, allow us to probenew aspects of quantum gravity. At least, until the idealpulsar-black hole system is discovered it provides a newsandbox to test ideas, to answer questions about comple-mentarity and the interaction of Hawking radiation withspace-time.
Acknowledgements.
I am grateful to Jason Gallichio,Avery Broderick, Aephraim Steinberg, Daniel James andAggie Branczyk for stimulating discussions. I thank theTsinghua University for Advanced Studies where thiswork was completed. [1] R. M. Shannon and S. Johnston, MNRAS , L29(2013), 1305.3036.[2] N. Rea, P. Esposito, J. A. Pons, R. Turolla, D. F. Torres,G. L. Israel, A. Possenti, M. Burgay, D. Vigan`o, A. Pa-pitto, et al., ApJ , L34 (2013), 1307.6331.[3] R. P. Eatough, H. Falcke, R. Karuppusamy, K. J. Lee,D. J. Champion, E. F. Keane, G. Desvignes, D. H. F. M.Schnitzeler, L. G. Spitler, M. Kramer, et al., Nature(London) , 391 (2013), 1308.3147.[4] S. W. Hawking, Phys. Rev. D , 2460 (1976).[5] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Jour-nal of High Energy Physics , 62 (2013), 1207.3123.[6] S. D. Mathur, Classical and Quantum Gravity , 224001(2009), 0909.1038.[7] S. G. Avery, B. D. Chowdhury, and A. Puhm, Journalof High Energy Physics , 12 (2013), 1210.6996.[8] L. Susskind, L. Thorlacius, and J. Uglum, Phys. Rev. D , 3743 (1993), hep-th/9306069.[9] K. Belczynski, V. Kalogera, and T. Bulik, Astrophys. J. , 407 (2002), astro-ph/0111452.[10] L. Boyle and M. Russo, ArXiv e-prints (2011), 1110.2789.[11] R. R. Rafikov and D. Lai, Phys. Rev. D , 063003(2006), arXiv:astro-ph/0512417.[12] P. Schneider, J. Ehlers, and E. E. Falco, Gravita-tional Lenses (Springer-Verlag Berlin Heidelberg NewYork. Also Astronomy and Astrophysics Library, 1992).[13] S. Mao, ApJ , L41 (1992).[14] U.-L. Pen and L. King, MNRAS , L132 (2012),1111.6806.[15] E. Pfahl, P. Podsiadlowski, and S. Rappaport, Astrophys. J. , 343 (2005), astro-ph/0502122.[16] D. R. Lorimer and M. Kramer, Handbook of Pulsar As-tronomy (2012).[17] U.-L. Pen, J. P. Macquart, A. Deller, and W. Brisken,ArXiv e-prints (2013), 1301.7505.[18] M. D. Johnson, C. R. Gwinn, and P. Demorest, Astro-phys. J. , 8 (2012), 1208.5485.[19] U.-L. Pen and Y. Levin, ArXiv e-prints (2013),1302.1897. [20] A. E. Broderick and A. Loeb, MNRAS , 353 (2005),astro-ph/0506433.[21] J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, andJ. E. Lukens, Nature406