aa r X i v : . [ h e p - t h ] A p r Quantum Harmonic Black Holes
Roberto Casadio a,b ∗ and Alessio Orlandi a,b † a Dipartimento di Fisica e Astronomia, Universit`a di Bolognavia Irnerio 46, 40126 Bologna, Italy c Istituto Nazionale di Fisica Nucleare, Sezione di Bolognavia Irnerio 46, 40126 Bologna, Italy
September 24, 2018
Abstract
Inspired by the recent conjecture that black holes are condensates of gravitons, we investigatea simple model for the black hole degrees of freedom that is consistent both from the pointof view of Quantum mechanics and of General Relativity. Since the two perspectives should“converge” into a unified picture for small, Planck size, objects, we expect our construction is auseful step for understanding the physics of microscopic, quantum black holes. In particular, weshow that a harmonically trapped condensate gives rise to two horizons, whereas the extremalcase (corresponding to a remnant with vanishing Hawking temperature) naturally falls out ofits spectrum.
One of the major mysteries in modern theoretical physics is to understand what are the internaldegrees of freedom of black holes. This issue becomes particularly relevant in any attempt todevelop a quantum theory which incorporates gravity along with the other forces of nature. Ofcourse, without experimental inputs, our best starting point is the classical description of blackholes provided by General Relativity [1], along with well established semiclassical results, such asthe predicted Hawking radiation [2].It was recently proposed by Dvali and Gomez that black holes are Bose-Einstein Condensates(BECs) of gravitons at a critical point, with Bogoliubov modes that become degenerate and nearlygapless representing the holographic quantum degrees of freedom responsible for the black holeentropy and the information storage [3]. In order to support this view, they consider a collectionof objects (gravitons) interacting via Newtonian gravity, V N ∼ − G N µr , (1.1) ∗ [email protected] † [email protected] µ is related to their characteristic quantum mechanical size via theCompton/de Broglie wavelength, ℓ ≃ ~ µ = ℓ p m p µ . (1.2)These bosons can superpose and form a “ball” of radius ℓ , and total energy M = N µ , where N isthe total number of constituents. Within the Newtonian approximation, there is then a value of N for which the whole system becomes a black hole. In details, given the coupling constant α = ℓ ℓ = µ m , (1.3)there exists an integer N such that no constituent can escape the gravitational well it contributedto create, and which can be approximately described by the potential U ( r ) ≃ V N ( ℓ ) ≃ − N α ~ ℓ Θ( ℓ − r ) , (1.4)where Θ is the Heaviside step function. This implies that components in the depleting region are“marginally bound”, E K + U ≃ , (1.5)where the kinetic energy is given by E K ≃ µ . This energy balance yields the “maximal packing” N α = 1 . (1.6)Consequently, the effective boson mass and total mass of the balk hole scale according to µ ≃ m p √ N and M = N µ ≃ √
N m p . (1.7)Note that one has here assumed the ball is of size ℓ (since bosons superpose) and, therefore, theconstituents will interact at a maximum distance of order r ∼ ℓ , with fixed ℓ . The Hawking radiationand the negative specific heat spontaneously result from quantum depletion of the condensate forthe states satisfying Eq. (1.5). This description is partly Quantum Mechanics and partly classicalNewtonian physics, but no General Relativity is involved, in that geometry does not appear in theargument.In this work, we will show how this picture, which draws from the conjectured UV-self-complete-ness of gravity [4], can be both improved within Quantum Mechanics and reconciled with the usualgeometric description of space-time in General Relativity. Some considerations about the possibleexistence of remnants will also follow. We shall use units with c = 1, ~ = ℓ p m p and the Newtonconstant G N = ℓ p /m p . To summarise, Ref. [3] assumes that a black hole is a BEC, trapped in a gravitational well de-scribed by the simple potential (1.4). We can improve on this description, by employing theQuantum Mechanical theory of the harmonic oscillator as a (better) mean field approximation for2he Newtonian gravitational interaction acting on each boson inside the BEC. The potential U inEq. (1.4) is therefore replaced by V = 12 µ ω ( r − d ) Θ( d − r ) ≡ V ( r ) Θ( d − r ) , (2.1)and we further set V (0) = U (0), so that12 µ ω d = N α ~ ℓ . (2.2)We also assume that the effective mass, length and frequency of a single graviton mode are relatedby µ = ~ ω = ~ /ℓ , which immediately leads to d = √ N α ℓ = √ N ℓ p . (2.3)If we neglect the finite size of the well, the Schr¨odinger equation in polar coordinates, ~ µ r (cid:20) ∂∂r (cid:18) r ∂∂r (cid:19) + 1sin θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) + 1sin θ ∂ ∂φ (cid:21) ψ = ( V − E ) ψ , (2.4)yields the well-known eigenfunctions ψ nlm ( r, θ, φ ) = N r l e − r ℓ F ( − n, l + 3 / , r /ℓ ) Y lm ( θ, φ ) , (2.5)where N is a normalization constant, F the Kummer confluent hypergeometric function of thefirst kind and Y lm ( θ, φ ) are the usual spherical harmonics. The corresponding energy eigenvaluesare given by E nl = ~ ω (cid:20) n + l + 32 − V (0) (cid:21) = ~ ω (cid:20) n + l + 12 (cid:18) − d ℓ (cid:19)(cid:21) , (2.6)where n is the radial quantum number and l the angular momentum (not to be confused with ℓ ).Following the idea in Ref. [3], we view the above spectrum as representing the effective QuantumMechanical dynamics of depleting modes, which can be described by the first (non-rotating) excitedstate ψ ( r ) = s ℓ √ π e − r ℓ (cid:0) r − ℓ (cid:1) . (2.7)The marginally binding condition (1.5), that is E ≃
0, then leads to the scaling laws ℓ = r N ℓ p and µ = r N m p , (2.8) This is nothing but Newton oscillator, which would correspond to a homogenous BEC distribution in the New-tonian approximation. Note we have already integrated out the angular coordinates.
3n perfect qualitative agreement with Eq. (1.7).We can now estimate the effect of the finite width of the potential well (2.1) by simply applyingfirst order perturbation theory and obtain∆ E = − Z ∞ d r d r ψ ( r ) V ( r ) ≃ − . √ N m p . (2.9)This can now be compared, for example, with the ground state energy E = − p /N m p ≃− . m p / √ N . Since | ∆ E | ≪ | E | , our approximation appears reasonable.We however remark that the ground state energy in this model has no physical meaning. Indeed,the Schr¨odinger equation (2.4) must be viewed as describing the effective dynamics of black holeconstituents, and the total energy of the “harmonic black hole” is still given by the sum of theindividual boson effective masses, M = N µ ≃ r N m p , (2.10)in agreement with the “maximal packing” of Eq. (1.7) and the expected mass spectrum of quantumblack holes (see, for example, Refs. [5, 6]). It is now reasonable to assume that the actual density profile of the BEC gravitational source isrelated to the ground state wave function in Eq. (2.5) according to ρ ( r ) ≃ M ψ ≃ m p e − r N ℓ √ π N ℓ . (3.1)Similar Gaussians profiles have been extensively studied in Refs. [7, 8], where it was proven thatsuch densities satisfy the Einstein field equations with a “de Sitter vacuum” equation of state, ρ = − p , where p is the pressure. Curiously, BECs can display this particular equation of state [9].This feature provides a connection between Quantum Mechanics and the geometrical description.Let us indeed take the static and normalised, energy density profile of Ref. [7], ρ ( r ) = M e − r θ √ π θ / , (3.2)where √ θ is viewed as a fundamental length related to space-time non-commutativity, and r is theradial coordinate such that the integral inside a sphere of area 4 π r , M ( r ) = Z r ρ (¯ r ) ¯ r d¯ r = M γ (3 / , r / θ )Γ(3 / , (3.3) The squared length θ should not be confused with one of the angular coordinates of the previous expressions.Also, note ρ has already been integrated over the angles. M of the object for r → ∞ . In the above,Γ(3 /
2) and γ (3 / , r / θ ) are the complete and upper incomplete Euler Gamma functions, respec-tively. This energy distribution then satisfies Einstein field equations together with the Schwarzschild-like metric d s = − f ( r ) d t + f − ( r ) d r + r dΩ , (3.4)where f ( r ) = 1 − G N M ( r ) r . (3.5)According to Ref. [7], one has a black hole only if the mass-to-characteristic length ratio is suffi-ciently large, namely for M & . √ θG N = 1 . m p √ θℓ p ≡ M ∗ . (3.6)If the above inequality is satisfied, the metric function f = f ( r ) has two zeros and there are twodistinct horizons. For M = M ∗ , f = f ( r ) has only one zero which corresponds to an “extremal”black hole, with two coinciding horizons (and vanishing Hawking temperature). The latter repre-sents the minimum mass black hole, and a candidate black hole remnant of the Hawking decay [10].Further, the classical Schwarzschild case is precisely recovered in the limit G N M/ √ θ → ∞ , so thatdepartures from the standard geometry become quickly negligible for very massive black holes.Going back to the BEC model, whose total ADM mass is given in Eq. (2.10), and comparingthe Gaussian profile (3.1) with Eq. (3.2), that is setting θ = N ℓ /
14, one finds that the conditionin Eq. (3.6) reads 1 . √ N & . √ N , (3.7)and is always satisfied (for N ≥ We have shown that the scenario of Ref. [3], in which black hole inner degrees of freedom (aswell as the Hawking radiation) correspond to depleting states in a BEC, can be understood andrecovered in the context of General Relativity by viewing a black hole as made of the superpositionof N constituents, with a Gaussian density profile, whose characteristic length is given by theconstituents’ effective Compton wavelength. From the point of view of Quantum Mechanics, suchstates straightforwardly arise from a binding harmonic oscillator potential. Moreover, requiring theexistence of (at least) a horizon showed that the extremal case, corresponding to a remnant withvanishing Hawking temperature, is not realised in the harmonic spectrum (2.10). Such states willtherefore have to be described by a different model.5t the threshold of black hole formation (see, for example, Ref. [12] and References therein),for a total ADM mass M ≃ m p (thus N ≃ . However, we canalready anticipate that quantum black holes with spin should be relatively easy to accommodatein our description, by simply considering states in Eq. (2.5) with l >
0. This should allow us toconsider more realistic quantum black hole formation from particle collisions, since particles mostlikely scatter with non-zero impact parameter.Many questions are still left open. First of all, the discretisation of the mass has an importantconsequence in the classical limit. For example, let us look again at Eq. (2.10), and consider twonon-rotating black holes with mass M = q N m p and M = q N m p , where N and N arepositive integers, which slowly merge in a head-on collision (with zero impact parameter). Theresulting black hole should have a mass M which is also given by Eq. (2.10). However, there is ingeneral no integer N such that √ N = √ N + √ N . It therefore appears that either the mass shouldnot be conserved, M = M + M , or the mass spectrum described by Eq. (2.10) is not complete.This problem, which is manifestly more significant for small black hole masses (or, equivalently,integers N ), is shared by all those models in which the the black hole mass does not scale exactlylike an integer. If we wish to keep Eq. (2.10), or any equivalent mass spectrum, we might thenargue that a suitable amount of energy (of order M + M − M ) should be expelled during themerging, in order to accommodate the overall mass into an allowed part of the spectrum. In thiscase, one may also wonder if this emission can be thought of as some sort of Hawking radiation ,or if it is completely different in nature.Another issue regards the assumption in Eq. (3.1), i.e. the idea that the classical density profilecorresponds to the square modulus of the (normalised) wavefunction. At the semiclassical level, thisseems reasonable and intuitive, but necessarily removes the concept of “point-like test particle” fromGeneral Relativity, thus forcing us to reconsider the idea of geodesics only in terms of propagation ofextended wave packets, which might show unexpected features or remove others from the classicaltheory. Also, elementary particles would not differ from extended massive objects and thereforeshould have an equation of state (see, for instance, the old shell model in Refs. [14]). Would thisequation of state be an observable and enter the description of the particle on the same level asany other quantum number? Do different particles have different equations of state?Last but not least, there is the question of describing the formation of a BEC during a stel-lar collapse. Condensation is usually achieved at extremely low temperature, when the thermalde Broglie wavelength becomes comparable to the inter-particle spacing. Whereas one has no doubtthat particles inside a black hole are extremely packed, it is not clear how such a dramatic dropof temperature could occur. One might find a reason for this in some modification of the laws ofthermodynamics inside the event horizon. For example, one might adapt the construction yielding the effective potential acting on collapsing nested shellsobtained in Refs. [13]. Note that for vanishing impact parameter, one does not expect any emission of classical gravitational waves. cknowledgements This work is supported in part by the European Cooperation in Science and Technology (COST)action MP0905 “Black Holes in a Violent Universe”.
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