QQuantum Hair on Colliding Black Holes
Lawrence Crowell , + and Christian Corda , ∗ July 3, 2020 AIAS, Budapest, Hu/Albuquerque, NM/ Jefferson, TX 75657 Department of Physics, Faculty of Science, Istanbul University, Istanbul34134, Turkey and International Institute for Applicable Mathematics and Infor-mation Sciences, B.M., Birla Science Centre, Adarshnagar, Hyderabad 500063,IndiaE-mails: + goldenfi[email protected] ; ∗ [email protected] } Abstract
Black hole collision produce gravitational radiation which is generallythought in a quantum limit to be gravitons. The stretched horizon ofa black hole contains quantum information, or a form of quantum hair,which in a coalescence of black holes participates in the generation ofgravitons. This may be facilitated with a Bohr-like approach to blackhole (BH) quantum physics with quasi-normal mode (QNM) approach toBH quantum mechanics. Quantum gravity and quantum hair on eventhorizons is excited to higher energy in BH coalescence. The near horizoncondition for two BHs right before collision is a deformed
AdS spacetime.These excited states of BH quantum hair then relax with the productionof gravitons. This is then argued to define RT entropy given by quantumhair on the horizons. These qubits of information from a BH coalescenceshould then appear in gravitational wave (GW) data. This is a form ofthe standard
AdS/CF T correspondence and the Ryu-Takayanagi (RT)formula.
Quantum gravitation suffers primarily from an experimental problem. It iscommon to read critiques that it has gone off into mathematical fantasies, butthe real problem is the scale at which such putative physics holds. It is not hardto see that an accelerator with current technology would be a ring encompassingthe Milky Way galaxy. Even if we were to use laser physics to accelerate particlesthe energy of the fields proportional to the frequency could potentially reducethis by a factor of about so a Planck mass accelerator would be far smaller;it would encompass the solar system including the Oort cloud out to at least light years. It is also easy to see that a proton-proton collision that producesa quantum BH of a few Planck masses would decay into around a mole of1 a r X i v : . [ phy s i c s . g e n - ph ] J un aughter particles. The detection and track finding work would be daunting.Such experiments are from a practical perspective nearly impossible. This isindependent of whether one is working with string theory or loop variables andrelated models.It is then best to let nature do the heavy lifting for us. Gravitation is afield with a coupling that scales with the square of mass-energy. Gravitation isonly a strong field when lots of mass-energy is concentrated in a small region,such as a BH. The area of the horizon is a measure of maximum entropy anyquantity of mass-energy may possess [2], and the change in horizon area withlower and upper bounds in BH thermodynamic a range for gravitational waveproduction. Gravitational waves produced in BH coalescence contains informa-tion concerning the BHs configuration, which is argued here to include quantumhair on the horizons. Quantum hair means the state of a black hole from a singlemicrostate in no-hair theorems. Strominger and Vafa [3] advanced the existenceof quantum hair using theory of D-branes and STU string duality. This infor-mation appears as gravitational memory, which is found when test masses arenot restored to their initial configuration [4]. This information may be used tofind data on quantum gravitation.There are three main systems in physics, quantum mechanics (QM), statisti-cal mechanics and general relativity (GR) along with gauge theory. These threesystems connect with each other in certain ways. There is quantum statisticalmechanics in the theory of phase transitions, BH thermodynamics connects GRwith statistical mechanics, and Hawking-Unruh radiation connects QM to GRas well. These are connections but are incomplete and there has yet to be anygeneral unification or reduction of degrees of freedom. Unification of QM withGR appeared to work well with holography, but now faces an obstruction calledthe firewall [5].Hawking proposed that black holes may lose mass through quantum tunnel-ing [6]. Hawking radiation is often thought of as positive and negative energyentangled states where positive energy escapes and negative energy enters theBH. The state which enters the BH effectively removes mass from the same BHand increases the entanglement entropy of the BH through its entanglementwith the escaping state. This continues but this entanglement entropy is lim-ited by the Bekenstein bound. In addition, later emitted bosons are entangledwith both the black hole and previously emitted bosons. This means a bipar-tite entanglement is transformed into a tripartite entangled state. This is nota unitary process. This will occur once the BH is at about half its mass at thePage time [7], and it appears the unitary principle (UP) is violated. In order toavoid a violation of UP the equivalence principle (EP) is assumed to be violatedwith the imposition of a firewall. The unification of QM and GR is still notcomplete. An elementary approach to unitarity of black holes prior to the Pagetime is with a Bohr-like approach to BH quantum physics [8–10], which will beshortly discussed in next section.Quantum gravity hair on BHs may be revealed in the collision of two BHs.This quantum gravity hair on horizons will present itself as gravitational mem-ory in a GW. This is presented according to the near horizon condition on2eissnor-Nordstrom BHs, which is AdS × S , which leads to conformal struc-tures and complementarity principle between GR and QM. At the present time, there is a large agreement, among researchers in quantumgravity, that BHs should be highly excited states representing the fundamen-tal bricks of the yet unknown theory of quantum gravitation [8–10]. This isparallel to quantum mechanics of atoms. In the 1920s the founding fathersof quantum mechanics considered atoms as being the fundamental bricks oftheir new theory. The analogy permits one to argue that BHs could have adiscrete energy spectrum [8–10]. In fact, by assuming the BH should be thenucleus the “gravitational atom”, then, a quite natural question is: What arethe “electrons”? In a recent approach, which involves various papers (see [8–10]and references within), this important question obtained an intriguing answer.The BH quasi-normal modes (QNMs) (i.e. the horizon’s oscillations in a semi-classical approach) triggered by captures of external particles and by emissionsof Hawking quanta, represent the “electrons” of the BH which is seen as beinga gravitational hydrogen atom [8–10]. In [8–10] it has been indeed shown that,in the the semi-classical approximation, which means for large values of the BHprincipal quantum number n , the evaporating Schwarzschild BH can be con-sidered as the gravitational analogous of the historical, semi-classical hydrogenatom, introduced by Niels Bohr in 1913 [11, 12]. Thus, BH QNMs are inter-preted as the BH electron-like states, which can jump from a quantum level toanother one. One can also identify the energy shells of this gravitational hydro-gen atom as the absolute values of the quasi-normal frequencies [8–10]. Withinthe semi-classical approximation of this Bohr-like approach, unitarity holds inBH evaporation. This is because the time evolution of the Bohr-like BH is gov-erned by a time-dependent Schrodinger equation [9, 10]. In addition, subsequentemissions of Hawking quanta [6] are entangled with the QNMs (the BH elec-tron states) [9, 10]. Various results of BH quantum physics are consistent withthe results of [9, 10], starting from the famous result of Bekenstein on the areaquantization [13]. Recently, this Bohr-like approach to BH quantum physics hasbeen also generalized to the Large AdS BHs, [14]. For the sake of simplicity, inthis Section we will use Planck units ( G = c = k B = (cid:126) = π(cid:15) = 1 ). Assumingthat M is the initial BH mass and that E n is the total energy emitted by theBH when the same BH is excited at the level n in units of Planck mass (then M p = 1 ), one gets that a discrete amount of energy is radiated by the BH in aquantum jump in terms of energy difference between two quantum levels [8–10] ∆ E n → n ≡ E n − E n = M n − M n == (cid:112) M − n − (cid:112) M − n , (1)This equation governs the energy transition between two generic, allowed levels n and n > n and consists in the emission of a particle with a frequency ∆ E n → n [8–10]. The quantity M n in Eq. ( ), represents the residual mass of3he BH which is now excited at the level n . It is exactly the original BH massminus the total energy emitted when the BH is excited at the level n [8–10].Then, M n = M − E n , and one sees that the energy transition between thetwo generic allowed levels depends only on the two different values of the BHprincipal quantum number and on the initial BH mass [8–10]. An analogousequation works also in the case of an absorption, see [8–10] for details. In theanalysis of Bohr [11, 12], electrons can only lose and gain energy during quantumjumps among various allowed energy shells. In each jump, the hydrogen atomcan absorb or emit radiation and the energy difference between the two involvedquantum levels is given by the Planck relation (in standard units) E = hν . In theBH case, the BH QNMs can gain or lose energy by quantum jumps from oneallowed energy shell to another by absorbing or emitting radiation (Hawkingquanta). The following intriguing remark finalizes the analogy between thecurrent BH analysis and Bohr’s hydrogen atom. The interpretation of equation1 is the energy states of a particle, that is the electron of the gravitational atom,which is quantized on a circle of length [8–10] L = 4 π (cid:18) M + (cid:114) M − n (cid:19) . (2)Hence, one really finds the analogous of the electron traveling in circular orbitsaround the nucleus in Bohr’s hydrogen atom. One sees that it is also M n = (cid:114) M − n . (3)Thus the uncertainty in a clock measuring a time t becomes, with the Planckmass is equal to in Planck units, δtt = 12 M n = 1 (cid:112) M − n , (4)which means that the accuracy of the clock required to record physics at thehorizon depends on the BH excited state, which corresponds to the number ofPlanck masses it has. More in general, from the Bohr-like approach to BH quan-tum physics it emerges that BHs seem to be well defined quantum mechanicalsystems, having ordered, discrete quantum spectra. This issue appears consis-tent with the unitarity of the underlying quantum gravity theory and with theidea that information should come out in BH evaporation, in agreement with aknown result of Page [7]. For the sake of completeness and of correctness, westress that the topic of this Section, i.e. the Bohr-like treatment of BH quantumphysics, is not new. A similar approach was used by Bekenstein in 1997 [15]and by Chandrasekhar in 1998 [16]. 4 Near Horizon Spacetime and Collision of BlackHoles
This paper proposes how the quantum basis of black holes may be detected ingravitational radiation. Signatures of quantum modes may exist in gravitationalradiation. Gravitational memory or BMS symmetries are one way in whichquantum hair associated with a black hole may be detected [17]. Conservationof quantum information suggests that quantum states on the horizon may beemitted or entangled with gravitational radiation and its quantum numbersand information. In what follows a toy model is presented where a black holecoalescence excites quantum hair on the stretched horizon in the events leadingup to the merger of the two horizons. The model is the Poincare disk for spatialsurface in time. To motivate this we look at the near horizon condition for anear extremal black hole.The Reissnor-Nordstrom (RN) metric is ds = − (cid:18) − mr + Q r (cid:19) dt + (cid:18) − mr + Q r (cid:19) − dr + r d Ω . (5)Here Q is an electric or Yang-Mills charge and m is the BH mass. In previoussection, considering the Schwarzschild BH, we labeled the BH mass as M in-stead. The accelerated observer near the horizon has a constant radial distance.For the sake of completeness, we recall that the Bohr-like approach to BH quan-tum physics has been also partially developed for the Reissnor-Nordstrom blackhole (RNBH) in [14]. In that case, the expression of the energy levels of theRNBH is a bit more complicated than the expression of the energy levels of theSchwarzschild BH, being given by (in Planck units and for small values of Q )[14] E n (cid:39) m − (cid:114) m + q − Qq − n , (6)where q is the total charge that has been loss by the BH excited at the level n .Now consider ρ = ˆ rr + dr √ g rr = ˆ rr + dr (cid:112) − m/r + Q /r , (7)with lower integration limit r + is some small distance from the horizon and theupper limit r removed from the black hole. The result is ρ = m log [ (cid:112) r − mr + Q + r − m ] + (cid:112) r − mr + Q (cid:12)(cid:12)(cid:12) rr + . (8)With a change of variables ρ = ρ ( r ) the metric is ds = (cid:16) ρm (cid:17) dt − (cid:18) mρ (cid:19) dρ − m d Ω , (9)5here on the horizon ρ → r . This is the metric for AdS × S for AdS in the ( t, ρ ) variables tensored with a two-sphere S of constant radius = m in theangular variables at every point of AdS . This metric was derived by Carroll,Johnson and Randall [18].In Section 4 it is shown this hyperbolic dynamics for fields on the horizon ofcoalescing BHs is excited. This by the Einstein field equation will generate grav-itational waves, or gravitons in some quantum limit not completely understood.This GW information produced by BH collisions will reach the outside worldhighly red shifted by the tortoise coordinate r ∗ = r (cid:48) − r − m ln | − m/r | .For a solar mass BH, which is mass of some of the BHs which produce grav-itational waves detected by LIGO, the wavelength of this ripple, as measuredfrom the horizon to δr ∼ λδr (cid:48) = λ − m ln (cid:18) λ m (cid:19) (cid:39) × m. (10)A ripple in spacetime originating an atomic distance − m from the hori-zon gives a ν = 150 Hz signal, detectable by LIGO [19]. Similarly, a ripple − to − cm from the horizon will give a − Hz signal detectable bythe eLISA interferometer system[20]. Thus, quantum hair associated with QCDand electroweak interactions that produce GWs could be detected. More exactcalculations are obviously required.Following [21], one can use Hawking’s periodicity argument from the RNmetric in order to obtain an “effective” RN metric which takes into account theBH dynamical geometry due to the subsequent emissions of Hawking quanta.Hawking radiation is generated by a tunneling of quantum hair to the exterior, orequivalently by the reduction in the number of quantum modes of the BH. Thisprocess should then be associated with the generation of a gravitational wave.This would be a more complete dynamical description of the response spacetimehas to Hawking radiation, just as with what follows with the converse absorptionof mass or black hole coalescence. This will be discussed in a subsequent paper.These weak gravitons produced by BH hair would manifest themselves ingravitational memory. The Bondi-Metzner-Sachs (BMS) symmetry of gravita-tional radiation results in the displacement of test masses [22]. This displace-ment requires an interferometer with free floating mirrors, such as what willbe available with the eLISA system. The BMS symmetry is a record of YMcharges or potentials on the horizon converted into gravitational information.The BMS metric provide phenomenology for YM gauge fields, entanglements ofstates on horizons and gravitational radiation. The physics is correspondencebetween YM gauge fields and gravitation. The BHs coalescence is a processwhich converts qubits on the BHs horizons into gravitons.Two BHs close to coalescence define a region between their horizons witha vacuum similar to that in a Casimir experiment. The two horizons havequantum hair that forms a type of holographic "charge" that performs work onspacetime as the region contracts. The quantum hair on the stretched horizonis raised into excited states. The ansatz is made that
AdS × S for two nearlymerged BHs is mapped into a deformed AdS for a small region of space between6wo event horizons of nearly merged BHs. The deformation is because theconformal hyperbolic disk is mapped into a strip. In one dimension lower, thespatial region is a two dimensional hyperbolic strip mapped from a Poincare diskwith the same SL (2 , R ) symmetry. The manifold with genus g for charges hasEuler characteristic χ = 2 g − and with the dimensions of SL (2 , R ) this isthe index g − for Teichmuller space [22]. The SL (2 , R ) is the symmetry ofthe spatial region with local charges modeled as a U (1) field theory on an AdS .The Poincare disk is then transformed into H p that is a strip. The H p ⊂ AdS is simply a Poincare disk in complex variables then mapped into a strip withtwo boundaries that define the region between the two event horizons. AdS geometry in BH Coalescence
The near horizon condition for a near extremal black hole approximates
AdS × S . In [18] the extremal black hole replaces the spacelike region in ( r + , r − ) with AdS × S . For two black holes in near coalescence there are two horizons,that geodesics terminate on. The region between the horizons is a form ofKasner spacetime with an anisotropy in dynamics between the radial directionand on a plane normal to the radial direction. In the appendix it is shownthis is for a short time period approximately an AdS spacetime. The spatialsurface is a three-dimensional Poincare strip, or a three-dimensional region withhyperbolic arcs. This may be mapped into a hyperbolic space H . This is afurther correlation between anti-de Sitter spacetimes and black holes, such asseen in AdS/BH correspondences [23].The region between two event horizons is argued to be approximately
AdS by first considering the two BHs separated by some distance. There is an ex-pansion of the area of the S that is then employed with the AdS × S . Wethen make some estimates on the near horizon condition for black holes veryclose to merging. To start consider the case of two equal mass black holes in acircular orbit around a central point. We consider the metric near the centerof mass r − and the distance between the two black holes d >> m . Indoing this we may get suggestions om how to model the small region betweentwo black holes about to coalesce. An approximate metric for two distant blackholes is of the form ds = (cid:16) − m | r + d | − m | r + d | (cid:17) dt − (cid:16) − m | r + d | − m | r + d | (cid:17) − dr − r ( dθ + sin θd Φ ) , (11)where d Φ = dφ + ωdt , for ω the angular velocity of the two black holes around r = 0 . With the approximation for a moderate Keplerian orbit we may thenwrite this metric as This metric is approximated with the binomial expansionto O ( r ) and O ( ω ) as 7 s = (cid:16) − md (cid:16) r d (cid:17)(cid:17) dt − (cid:16) − md (cid:16) r d (cid:17)(cid:17) − dr − r ω sin θdφdt − r ( dθ + sin θdφ ) . (12) g tt is similar to the AdS g tt metric term plus constant terms and and similarly g rr . It is important to note this approximate metric has expanded the mea-sure of the angular portion of the metric. This means the -sphere with theseangle measures has more “area” than before from the contribution of angularmomentum. The Ricci curvatures are R tt = R rr (cid:39) − md , R tφ (cid:39) (cid:104) (cid:0) md (cid:1) + mr d (cid:105) ωsin θ,R φφ = g tφ g tt R tφ (cid:39) − r ω sin θ + O (cid:16) ω d (cid:17) , R θθ = 0 , (13)where O ( d − ) terms and higher are dropped. The R r r and R φφ Ricci curvatureare negative and R tφ positive. The -surface in r, φ coordinates has hyperbolicproperties. This means we have at least the embedding of a deformed versionof AdS in this spacetime. This exercise expands the boundary of the disk D ,in a -spacial subsurface, with boundary around each radial distance so there isan excess angle or “wedge” that gives hyperbolic geometry. The ( t, φ ) curvaturecomponents comes from the Riemannian curvature R rφtr = − ωα − and itscontribution to the geodesic deviation equation along the radial direction is d rds + R rφtr U t U φ r = 0 (14)or that for U t (cid:39) and U φ (cid:39) ωd rdt (cid:39) ω r. (15)This has a hyperbolic solution r = r cosh ( √ ωt ) . The U φ will have higherorder terms that may be computed in the dynamics for φ Similarly the geodesicdeviation equation for φ is d φds + R φrtr U t U r r = 0 (16)or cryptically d φdt (cid:39) Riem
A cosh ( αt ) sinh ( αt ) , (17)for Riem → R φ rtr . This has an approximately linear form for small t thatturns around into exponential or hyperbolic forms for larger time. The spatialmanifold in the ( r, φ ) variables then have some hyperbolic structure. It is8orth a comment on the existence of Ricci curvatures for this spacetime. TheSchwarzschild metric has no Ricci curvature as a vacuum solution. This -black hole solution however is not exactly integrable and so mass-energy is notlocalizable. This means there is an effective source of curvature due to thenonlocalizable nature of mass-energy for this metric. This argument is made inorder to justify the ansatz the spacetime between two close event horizons priorto coalescence is AdS . Since most of the analysis of quantum field is in onedimension lower it is evident there is a subspace AdS . This is however followedup by looking at geometry just prior to coalescence where the S has more areathan it can bound in a volume. This leads to hyperbolic geometry. Abovewe argue there is an expansion of a disk boundary ∂ D , and thus hyperbolicgeometry. It is then assume this carries to one additional dimension as well.Now move to examine two black holes with their horizons very close. Con-sider a modification of the AdS × S metric with the inclusion of more “area”in the S portion. The addition of area to S is then included in the metric. Inthis fashion the influence of the second horizon is approximated by a change inthe metric of S . The metric is then a modified form of the near horizon metricfor a single black hole, ds = (cid:16) rR (cid:17) dt − (cid:18) Rr (cid:19) dr − ( r + ρ ) d Ω . (18)The term ρ means there is additional area to the S making it hyperbolic. TheRiemann curvatures for this metric are: R trtr = − r − ρ r ( r + ρ ) , R rθrθ = − ρ r + ρ , R rφrφ = − ρ r + ρ sin θ, R θφθφ = ρ sin θ (19)From these the Ricci curvatures are R rr = − r − ρ r + ρ , R θθ = R φφ = − (cid:18) R r (cid:19) ρ r + ρ , (20)that are negative for small values of r . For r → all Ricci curvatures diverge Ric → −∞ . The R rr diverges more rapidly, which gives this spacetime regionsome properties similar to a Kasner metric. However, R rr − R θθ is finite for r → ∞ . This metric then has properties of a deformed AdS . With thetreatment of quantum fields between two close horizons before coalescence thehyperbolic space H is considered as the spatial surface in a highly deformed AdS . A Poincare disk is mapped into a hyperbolic strip.The remaining discussion will now center around the spatial hyperbolic spa-tial surface. In particular the spatial dimensions are reduced by one. This isthen a BTZ-like analysis of the near horizon condition. The dimensional spa-tial surface will exhibit hyperbolic dynamics for particle fields and this is thena model for the near horizon hair that occurs with the two black holes in thisregion. 9or the sake of simplicity now reduce the dimensions and consider AdS in plus spacetime. The near horizon condition for a near extremal black holein 4 dimensions is considered for the BTZ black hole. This AdS spacetime isthen a foliations of hyperbolic spatial surfaces H in time. These surfaces underconformal mapping are a Poincare disk. The motion of a particle on this diskare arcs that reach the conformal boundary as t → ∞ . This is then the spatialregion we consider the dynamics of a quantum particle. This particle we startout treating as a Dirac particle, but the spinor field we then largely ignore bytaking the square of the Dirac equation to get a Klein-Gordon wave.Define the z and ¯ z of the Poincare disk with the metric ds p − disk = R g z ¯ z dzd ¯ z = R dzd ¯ z − z ¯ z (21)with constant negative Gaussian curvature R = − /R . This metric g z ¯ x = R / (1 − ¯ zz ) is invariant under the SL (2 , R ) ∼ SU (1 , group action, which, for g ∈ SU (1 , ,takes the form z → gz = az + b ¯ bz + ¯ a , g = (cid:18) a b ¯ b ¯ a (cid:19) . (22)The Dirac equation iγ µ D µ ψ + mψ = 0 , D µ = ∂ µ + iA µ on the Poincaredisk has the Hamiltonian matrix H = (cid:18) m H w H ∗ w − m (cid:19) (23)for the Weyl Hamiltonians H w = √ g z ¯ z α z (cid:0) D z + ∂ z ( ln g z ¯ z ) (cid:1) ,H ∗ w = √ g z ¯ z α ¯ z (cid:0) D ¯ z + ∂ ¯ z ( ln g z ¯ z ) (cid:1) , (24)with D z = ∂ z + iA z and D ¯ z = ∂ ¯ z + iA ¯ z . here α z and ¯ α z are the × Weyl matrices.Now consider gauge fields, in this case magnetic fields, in the disk. Thesemagnetic fields are topological in the sense of the Dirac monopole with vanishingAhranov-Bohm phase. The vector potential for this field is A φ = − i φ (cid:18) dzz − d ¯ z ¯ z (cid:19) . (25)the magnetic field is evaluated as a line integral around the solenoid opening,which is zero, but the Stokes’ rule indicates this field will be φ (¯ z − z ) /r , for r = ¯ zz . A constant magnetic field dependent upon the volume V = dz ∧ d ¯ z in the space with constant Gaussian curvature R = − /R A v = i BR (cid:18) zd ¯ z − ¯ zdz − ¯ zz (cid:19) . (26)10he Weyl Hamiltonians are then H w = − r R e − iθ (cid:18) α z (cid:16) ∂ r − ir ∂ θ − √ (cid:96) ( (cid:96) + 1) + φr + i kr − r (cid:17)(cid:19) H ∗ w = − r R e iθ (cid:18) α ¯ z (cid:16) ∂ r − ir ∂ θ + √ (cid:96) ( (cid:96) + 1) + φr + i kr − r (cid:17)(cid:19) , (27)for k = BR / . With the approximation that r << or small orbits theproduct gives the Klein-Gordon equation ∂ t ψ = R − (cid:18) ∂ r + (cid:96) ( (cid:96) + 1) + φ r + k r + ( (cid:96) ( (cid:96) + 1) + φ ) k (cid:19) ψ. (28)For (cid:96) ( (cid:96) + 1) + φ = 0 this gives the Weber equation with parabolic cylinderfunctions for solutions. The last term ( (cid:96) ( (cid:96) + 1) + φ ) k can be absorbed intothe constant phase ψ ( r, t ) = ψ ( r ) e − it √ E + (cid:96) ( (cid:96) + 1) + φ (29)This dynamics for a particle in a Poincare disk is used to model the samedynamics for a particle in a region bounded by the event horizons of a black hole.With AdS black hole correspondence the field content of the
AdS boundary isthe same as the horizon of a black hole. An elementary way to accomplishthis is to map the Poincare disk into a strip. The boundaries of the strip thenplay the role of the event horizons. The fields of interest between the horizonsare assumed to have orbits or dynamics not close to the horizons. The map is z = tanh ( ξ ) . The Klein-Gordon equation is then ∂ t ψ = R − (cid:18) (1 + 2 ξ ) ∂ ξ ∂ ¯ ξ + (cid:96) ( (cid:96) + 1) + φ | ξ | − k | ξ | (cid:19) ψ, (30)where the ξ is set to zero under this approximation. The Klein-Gordon equa-tion is identical to the above. The solution to this differential equation for Φ = (cid:96) ( (cid:96) + 1) + φ is ψ = (2 ξ ) / √ −
4Φ + 1) e − kξ × [ c U (cid:16) (cid:16) E R k + √ −
4Φ + 1 (cid:17) , ( √ −
4Φ + 1) , kξ (cid:17) + c L √ − E R k + √ − ( kξ )] . (31)The first of these is the confluent hypergeometric function of the second kind.For Φ = 0 this reduces to the parabolic cylinder function. The secondterm is the associated Laguerre polynomial. The wave determined by the11arabolic cylinder function and the radial hydrogen-like function have eigen-modes of the form in the diagram above. The parabolic cylinder function D n = 2 n/ e − x / H n ( x/ √ with integer n gives the Hermite polynomial. Therecursion formula then gives the modes for the quantum harmonic oscillator.The generalized Laguerre polynomial L (cid:96) +1 n − (cid:96) − ( r ) of degree n − (cid:96) − givesthe radial solutions to the hydrogen atom. The associated Laguerre polynomialwith general non-integer indices has degree associated with angular momentumand the magnetic fields. This means a part of this function is similar to thequantum harmonic oscillator and the hydrogen atom. The two parts in a gen-eral solution have amplitudes c and c and quantum states in between the closehorizons of coalescing black holes are then in some superposition of these typesof quantum states, see Figure 1.The Hamiltonian H = 12 | π | − gr , π = − i∂ r , (32)which contains the monopole field, describes the motion of a gauge particle inthe hyperbolic space. In addition, there is a contribution from the constantmagnetic field U = − kr / . Now convert this theory to a scalar field theorywith r → φ and π = − i∂ r φ . Finally introduce the dilaton operator D andthe scalar theory consists of the operators H = 12 | π | − gφ , U = − k φ , D = 14 ( φπ + πφ ) , (33)where H + U is the field theoretic form of the potential in equation 9. Thesepotentials then lead to the algebra [ H , U ] = − iD, [ H , D ] = − iH , [ U, D ] = iM. (34)This may be written in a more compact form with L = 2 − / ( H + U ) ,which is the total Hamiltonian, and L ± = 2 − / ( U − H ± iD ) . This leaves12he SL (2 , R ) algebra [ L , L ± ] = ± iL , [ L + , L − ] = L . (35)This is the standard algebra ∼ su (2) . Given the presence of the dilaton operatorthis indicates conformal structure. The space and time scale as ( t, x ) → λ ( t, x ) and the field transforms as φ → λ ∆ φ . The measure of the integral d x √ g isinvariant, where λ = ∂x (cid:48) /∂x gives the Jacobian J = det | ∂x (cid:48) ∂x | that cancels the √ g and the measure is independent of scale. In doing this, we are anticipatingthis theory in four dimensions. We then simply have the scaling φ → λ − φ and π → π . For the potential term − g/ φ invariance of the action requires g → λ − g and for U = − k φ clearly k → λ k . This means we can considerthis theory for space plus time and its gauge-like group SL (2 , R ) as onepart of an SL (2 , C ) ∼ SL (2 , R ) .The differential equation number 28 is a modified form of the Weber equation ψ xx − ( x + c ) ψ = = 0 . The solution in Abramowit and Stegun are paraboliccylinder functions D − a − / ( x ) , written according to hypergeometric functions.The ξ − part of the differential equation contributes to the Laguerre polynomialsolution. If we let ξ = e x/ and expand to quadratic powers we then have thepotential in the variable x V ( x ) = − ( g + k ) + 12 ( k − g )( x + x ) , (36)for g and k the constants in H and U . The Schrodinger equation for thispotential with a stationary phase in time has the parabolic cylinder functionsolution ψ ( x ) = c D β − α +2 √ α / √ α / (cid:18) β (1 + 4 x ) √ α ) / (cid:19) + c D − β − α − √ α / √ α / (cid:18) iβ (1 + 4 x ) √ α ) / (cid:19) , where α = g + k and β = k − g . The parabolic cylinder function describes atheory with criticality, which in this case has with a Ginsburg-Landau potential.The field theory form also has parabolic cylinder function solutions. The fieldtheory with the field expanded as φ = e χ is expanded around unity so φ (cid:39) χ + χ . A constant C such that χ → Cχ is unitless is assumed orimplied to exist. The Lagrangian for this theory is L = 12 ∂ µ χ∂ µ χ + α + 12 µ χ + 2 βµχ. (37)The constant µ , standing for mass and absorbing α , is written for dimensionalpurposes. We then consider the path integral Z = D [ χ ] e − iS − iχJ . Considerthe functional differentials acting on the path integral (cid:18) ( p + m ) δδJ − iβ (cid:19) Z = − i (cid:28) δSδχ (cid:29) , (38)where ∂ µ χ = p µ χ . The Dyson-Schwinger theorem tells us that (cid:68) δSδχ (cid:69) = (cid:104) J (cid:105) mean we have a polynomial expression (cid:104) ( p + m ) χ − iβ − J (cid:105) = 0 ,13here we can trivially let J − iβ → J . This does not lead to paraboliccylinder functions. There has been a disconnect between the ordinary quantummechanical theory and the QFT. We may however, continue the expansion toquartic terms. This will also mean there is a cubic term, we may impose thatonly the real functional variation terms contribute and so only even power ofthe field define the Lagrangian L → ∂ µ χ∂ µ χ + α + 12 µ χ + 14 λχ , (39)where α → λ . The functional derivatives are then (cid:18) ( p + m ) δδJ + λ δ δJ (cid:19) Z = − i (cid:28) δSδχ (cid:29) , (40)This cubic form has three parabolic cylinder solutions. We may think of this as ap + bp = J and is a cubic equation for the source J that is annulled at threepoints. The correspond to distinct solutions with distinct paths. These three so-lutions correspond to three contours and define three distinct vacua. The overallaction is a quartic function, which will have three distinct vacua, where one ofthese is the low energy physical vacua. It is worth noting this transformationof the problem has converted it into a system similar to the Higgs field. Thissystem with both harmonic oscillator and a Coulomb potentials is conformaland it maps into a system with parabolic cylinder functions solutions. In effectthere is a transformation harmonic oscillator states ↔ hydrogen − like states .The three solutions would correspond to the continuance of conformal symme-try, but where the low energy vacuum for one of these may not appear to beconformally invariant.This scale transformation above is easily seen to be the conformal transfor-mation with λ = Ω . The scalar tensor theory of gravity for coupling constant κ = 16 πG S [ g, φ ] = ´ d x √ g (cid:0) κ R + ∂ µ φ∂ µ φ + V ( φ ) (cid:1) ,S [ g, φ ] = ´ d x √ g (cid:0) κ R + ∂ µ φ∂ µ φ + V ( φ ) (cid:1) . (41)This then has the conformal transformations g (cid:48) µν = Ω g µν , φ (cid:48) = Ω − φ, Ω = 1 + κφ , (42)with the transformed action S [ g (cid:48) , φ (cid:48) ] = ˆ d x (cid:112) g (cid:48) (cid:18) κ R (cid:48) + 12 g (cid:48) µν ∂ µ φ (cid:48) ∂ ν φ (cid:48) + V ( φ (cid:48) ) + 112 Rφ (cid:48) (cid:19) . (43)There is then a hidden SO (3 , (cid:39) SL (2 , C ) symmetry. Given an inter-nal index on the scalar field φ i there is a linear SO ( n ) transformation δφ i = C ijk φ j δτ k for τ k a parameter. There is also a nonlinear transformation from14quation 12 as δφ i = (1 + κφ ) / κδχ i for χ i a parameterization. In theprimed coordinates the scalar field and metric transform as δφ i = δτ i − κφ (cid:48) i φ j δχ j δg µν = g (cid:48) µν κφ (cid:48) i δχ i − κφ (cid:48) . (44)The gauge-like dynamics have been buried into the scalar field. With this semi-classical model the scalar field adds some renormalizability. Further this modelis conformal. The conformal transformation mixes the scalar field, which is by it-self renormalizable, with the spacetime metric. Quantum gravitation is howeverdifficult to renormalize. Yet we see the linear group theoretic transformation ofthe scalar field in SO ( n ) is nonlinear in SO ( n, .Conformal symmetry is manifested in sourceless spacetime, or spatial regionswithout matter or fields. The two dimensional spatial surface in AdS is thePoincare disk that with complexified coordinates has metric with SL (2 , R ) algebraic structure. This may of course be easily extended into SL (2 , C ) as SL (2 , R ) × SL (2 , R ) . In this conformal setting quantum states share featuressimilar to the emission of photons by a harmonic oscillator or an atom. Theorbits of these paths are contained in regions bounded by hyperbolic surfaces, orarcs for the two dimensional Poincare disk. The entropy associated with thesearcs is a measure of the area contained within these curves. This is in a nutshellthe Mirzakhani result on entropy for hyperbolic curves.This development is meant to illustrate how radiation from black holes is pro-duced by quantum mechanical means not that different from bosons producedby a harmonic oscillator or atom. Hawking radiation in principle is detectedwith a wavelength not different from the size of the black hole. The wavelengthapproximately equal to the Schwarzschild radius has energy E = hν corre-sponding to a unit mass emitted. The mass of the black hole is n of these unitsand it is easy to find m p = (cid:112) (cid:126) c/G . These modes emitted are Planck units ofmass-energy that reach I ∞ . In the case of gravitons, these carry gravitationalmemory. For the coalescence of black holes gravitational waves are ultimatelygravitons. For Hawking radiation there is the metric back reaction, which in aquantum mechanical setting is an adjustment of the black hole with the emis-sion of gravitons. The emission of Hawking radiation might then be comparedto a black hole quantum emitting a Planck unit of black hole that then decaysinto bosons. The quantum induced change in the metric is a mechanism forproducing gravitons.In the coalescence of black holes the quantum hair on the stretched horizonssets up a type of Casimir effect with the vacuum that generates quanta. Ingeneral these are gravitons. We might see this as not that different from ascattering experiment with two Planck mass black holes. These will coalesce,form a larger black hole, produce gravitons, and then quantum states excitedby this process will decay. The production of gravitons by this mechanism isaffiliated with normal modes in the production of gravitons, which in principleis not different from the production of photons and other particles by other15uantum mechanical processes. I fact quantum mechanical processes underlyingblack hole coalescence might well be compared to nuclear fusion.The 2 LIGOs plus now the VIRGO detector are recording and triangulatingthe positions of distant black hole collisions almost weekly. This informationmay contain quantum mechanical information associated with quantum grav-itation. This information is argued below to contain BMS symmetries or in-formation. This will be most easily detected with a space based system suchas eLISA, where the shift in metric positions of test masses is most readily de-tectable. However, preliminary data with the gross displacement of the LIGOmass may give preliminary information as well. The coalescence of two black holes is a form of scattering. We may think of blackholes as an excited state of the quantum gravity field and a sort of elementaryparticle. The scattering of two black holes results in a larger black hole plusgravitational radiation. This black hole will then emit Hawking radiation. Thusin general the formation of black holes, their coalescence and ultimate quantumevaporation can be seen as intermediate processes in a general scattering theory.Quantum hair is a set of quantum fields that build up quantum gravitation,in the manner of gauge-gravity duality and BMS symmetry. This is holography,with the fields on the horizons of two BHs that determine the graviton/GWcontent of the BH coalescence. A detailed analysis of this may reveal BMScharges that reach I + are entangled with Hawking radiation by a form of en-tanglement swap. In this way Hawking radiation may not be entangled with theblack hole and thus not with previously emitted Hawking radiation. This willbe addressed later, but a preliminary to this idea is seen in [24], for disentan-glement between Hawking radiation and a black hole. The authors are workingon current calculations where this is an entanglement swap with gravitons. Theblack hole production of gravitons in general is then a manifestation of quantumhair entanglement.It is illustrative for physical understanding to consider a linearized form ofgravitational memory. Gravitational memory from a physical perspective is thechange in the spatial metric of a surface according to [4] ∆ h + . × = lim t →∞ h + , × ( t ) − lim t →−∞ h + , × ( t ) . (45)Here + and × refer to the two polarization directions of the GW. See \cite{key-20} for more on this. Quantum hair on two black holes just before coalescenceare highly excited and contribute to spacetime curvature, or in a full contextof quantum gravitation the generation of gravitons. As yet there is no com-plete theory of quantum gravity, but it is reasonable to think of gravitationalradiation as a classical wave built from many gravitons. Gravitons have twopolarizations and a state | Ψ + , × (cid:105) the density matrix ρ + , × = | Ψ + , × (cid:105)(cid:104) Ψ + , × | thendefines entropy S = ρ + , × log ( ρ + , × ) that with this near horizon condition of16 dS with a black hole is a form of Mirzakhani entropy measure in hyperbolicspace. The gravitons emitted are generated by quantum hair on the collidingblack holes. These will contribute to gravitational waves, and in general withBMS translations that bear quantum information from quantum hair.This theory connects to fundamental research, The entanglement entropy of CF T entropy with AdS lattice spacing a is S (cid:39) R G ln ( | γ | ) = R G ln (cid:20) (cid:96)L + e ρ c sin (cid:18) π(cid:96)L (cid:19)(cid:21) , (46)where the small lattice cut off avoids the singular condition for (cid:96) = 0 or L for ρ c = 0 . For the metric in the form ds = ( R/r ) ( − dt + dr + dz ) thegeodesic line determines the entropy as the Ryu-Takayanagi (RT) result [1] S = R G ´ π/ (cid:96)/L dssin s = − R G ln [ cot ( s ) + csc ( s )] (cid:12)(cid:12)(cid:12) π/ (cid:96)/L (cid:39) R G ln (cid:0) (cid:96)L (cid:1) , (47)which is the small (cid:96) limit of the above entropy. The RT result specifies entropy,which is connected to action S a ↔ S e [25]. Complexity, a form of Kolmogoroffentropy [26], is S a /π (cid:126) which can also assume the form of the entropy of asystem S ∼ k log ( dim {H} ) for H the Hilbert space and the dimension overthe number of states occupied in the Hilbert space. There is also complexityas the volume of the Einstein-Rosen bridge [27] vol/GR ads or equivalently theRT area ∼ vol/R AdS . There is an equivalency between entropy or complexityaccording to the geodesic paths in hyperbolic H by geometric means [22]. Thisshould generalize to H ⊂ AdS .The generation of gravitational waves should have an underlying quantummechanical basis. It is sometimes argued that spacetime physics may not beat all quantum mechanical. This is probably a good approximation for energysufficient orders of magnitude lower than the Planck scale. However, if wehave a scalar field that define the metric g (cid:48) = g (cid:48) ( g, φ ) with action S [ g, φ ] then a quantum field φ and a purely classical g means the transformation of g by this field has no quantum physics. In particular for a conformal theory Ω = 1 + κφ a φ a , here a an internal index, the conformal transformation g (cid:48) µν = Ω g µν has no quantum content. This is an apparent inconsistency. Forthe inflationary universe the line element ds (cid:48) = g (cid:48) µν dx µ dx ν = Ω ( du − d Σ (3) ) (48)with dt/du = Ω gives an FLRW or de Sitter-like line element that expandsspace with Ω = e t √ Λ / . The current slow accelerated universe we observe isapproximately of this nature. The inflaton scalars are then fields that stretchspace as a time dependent conformal transformation and are quantum mechan-ical.The generation of gravitational waves is ultimately the generation of gravi-tons. Signatures of these quantum effects in black hole coalescence will entail17he measurement of quantum information. Gravitons carry BMS charges andthese may be detected with a gravitational wave interferometer capable of mea-suring the net displacement of a test mass. The black hole hair on the stretchedhorizon is excited by the merger and these results in the generation of gravi-tons. The Weyl Hamiltonians in equation 9 depend on the curvature as ∝ √R .For the curvature extreme during the merging of black holes this means manymodes are excited. The two black holes are pumped with energy by the colli-sion, this generates or excites more modes on the horizons, where this results ina black hole with a net larger horizon area. This results in a metric response,or equivalently the generation of gravitons.Quantum normal modes are given by independent eigen-states, such as withquantum harmonic oscillator states. The harmonic oscillator states are wellknown to be given by the Hermite polynomials, which are a special case ofparabolic cylinder functions. Rydberg states are also a form of normal modes.The quantum states for the hyperbolic geometry of black hole mergers are ageneralization of these forms of states. The excitation of quantum hair in sucha merger and the production of gravitons is a converse situation for the emis-sion of Hawking radiation. In both cases there is a dynamical response of themetric, which is associated with gravitons. Currently a “by hand” correctioncalled back reaction is used in models. A more explicit discussion on the pro-duction of gravitons is beyond the scope here. However, the parabolic cylinderfunctions and the Laguerre functions clearly play a role in quantum productionof gravitons in BH coalescence. This means quantum gravitation should havesignatures of much the same physics as atomic physics or the role of electronsand phonons in solids.The major import of this expository is to propose quantum gravitationalsignatures in the coalescence of black holes. While there is plenty of furtherdevelopment needed to compute more firm predictions, the generic result is thatgravitational waves from colliding black holes have some quantum gravitationalsignatures. These signatures are to be found in gravitational memory. Further,this long-term adjustment of spacetime metric deviates form a purely classicalexpected result. 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