Bose-Einstein graviton condensate in a Schwarzschild black hole

Transcription

1 Bose-Einstein graviton condensate in a Schwarzschild black hole Jorge Alfaro, Domènec Espriu and Departament de Física Quàntica i Astrofísica & Institut de Ciències del Cosmos, Universitat de Barcelona ICCUB Winter Meeting (February 6 th, 2017)

2 Outline 1 Bose-Einstein Condensates as a picture for Black Holes Main characteristics of a BEC G. Dvali & C. Gómez Theory From Gross Pitaevskii to Einstein equations 2 Building up a condensate Regions of interest: outside/inside Exact solution 3

3 Bose-Einstein Condensates Main characteristics of a BEC G. Dvali & C. Gómez Theory From Gross Pitaevskii to Einstein equations High temperature Billiard balls thermal velocity v density d 3 Low temperature Wave packets λ db = h/p T 1/2 ( BEC: T=T matter wave ) c overlap Single collective quantum wave λ db d interatomic Pure Bose condensate: T = 0 Giant matter wave Quantum phase transition (indistinguishability) Macroscopic occupation of the fundamental state (coherent cloud) Macroscopic quantum phenomena become apparent (collective effects)

4 Main characteristics of a BEC G. Dvali & C. Gómez Theory From Gross Pitaevskii to Einstein equations Dvali & Gómez Theory [hep-th] Fortsch. Phys. 61 (2013) Within this framework BHs are understood as leaky bound-state in form of a self-sustained BEC (T=0) of N (>> 1) weakly-interacting gravitons maximally packed (long-wavelength λ / low interaction strength α) with λ r grav N L P length-scale for GR α L P 2 λ 2 1 N ratio of two lenght-scales: quantum / classical Not UV-physics but strong collective effects of IR-physics (Could quantum physics contribute to the information paradox?)

5 Main characteristics of a BEC G. Dvali & C. Gómez Theory From Gross Pitaevskii to Einstein equations Achievements: dynamics strongly affected by QG Geometric and semi-classical characteristics are emergent approximate notions in the limit of N >> 1. Hawking radiation is a thermal spectrum due to the quantum depletion of the BEC. Scattering from the collective potential Evaporation law dn dt = 1 NLP T H = NLP = λ Emergence of Bekenstein entropy as a natural degeneracy of N-graviton states; coincides with N. Quantum picture parameterized by a single characteristic: N (foundation for no-hair).

6 Main characteristics of a BEC G. Dvali & C. Gómez Theory From Gross Pitaevskii to Einstein equations From Gross Pitaevskii to Einstein equations Model for BEC: Gross-Pitaevskii equation ) ( 2 2m 2 + V (r) + g Ψ(r) 2 Ψ(r) = µψ(r) Hartree Fock approximation (MFT): Ψ(r 1,..., r N ) = ϕ 1 (r 1 )... ϕ N (r N ) Same quantum state = Same wave function dv Ψ 2 = N Non-linear Schrödinger equation G µν G (1) µν + G (2) µν + O(sup) Confining potential Schwarzschild background (curved geodesics) Chemical potential (conjugate to N) Missing Extend the formulation of perturbations around a classical BH to the grand canonical ensemble so as to incorporate the chemical potential to S HE.

7 Building up a condensate Building up a condensate Regions of interest: outside/inside Exact solution Pre-existence of a classical gravitational field that generates g (Schw) µν Graviton condensate tensor field h µν : bosonic perturbation Spherically symmetric perturbations (l = 0) of the classical metric ( ) 1 rs + h r tt g µν = g µν + h µν = 0 1 rs + h rr 0 0 r 0 0 r r 2 sin 2 θ e.o.m. = G µ ν (g µν ) G (1) µ ν (h µν ) + G (2) µ ν (h µν 2 ) + O(sup)? = µ h µν BEC is intimately related to the classical field that sustains it CF = BEC = (drive the CF as a mean field potential à la Hartree-Fock)

8 Building up a condensate Regions of interest: outside/inside Exact solution Einstein equations as a Gross-Pitaevskii equation S = S HE + S chem.pot. = d 4 x g R 1 dṽ 2 µ h αβ h αβ where h 2 = h αβ h αβ ρ gravitons because of N = dṽ Ψ 2 The equations of motion would change according to the choice of the volume element, dv or dṽ. As starting point dṽ for simplicity = G µ ν ( g µν + h µν ) = µh µ ν G t t ( g µν + h µν) = and G r r ; G θ θ = G φ φ (r rs)(r + rs) r 4 h rr + (r rs)2 (r + 2r s) h 2 rr = µ g tt h tt r 5 (r rs)2 h 2(r rs)3 r 3 rr h r 4 rr h rr

9 Building up a condensate Regions of interest: outside/inside Exact solution Outside the horizon: connection with the classical theory There is no other normalizable solution outside the BH horizon than the trivial solution for h µν. BEC disappears; µ becomes senseless. Therefore, for G µ ν (g µν = g µν + h µν ) = 0, g µν must be of g µν type. Birkhoff theorem any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat.! dimensionful quantity, r s : each δg µν = h µν has associated an amount of energy reflected into δr s. ds 2 = (1 δrs r )dt2 + 1 dr 2 + r 2 dω 2 1 δrs r

10 Inside the horizon Building up a condensate Regions of interest: outside/inside Exact solution Compatible asymptotic behaviour as boundary conditions. Near r = 0 : h tt = A (1 w)/w and h rr = B (1 w)w Near r = r s : h tt = A and h rr = A w/(1 w) 2 Maximum constrains over h µν once the interpolation from 0 and from r s are made to coincide: A = A = B. a free overall factor A in both components not fixed by the theory. (A = 0.8 in the graph)

11 Alternative description Building up a condensate Regions of interest: outside/inside Exact solution Mixed perturbation h µ ν = g νλ h µλ = constant Dimensionless chemical potential X = µr 2 = constant G t t : (1 h r r ) h r r + w 1 w 2 (1 2 h r r ) h r r = X h t t We find a simpler description it terms of constant magnitudes

12 Exact theory Building up a condensate Regions of interest: outside/inside Exact solution Change the focus: state the following ansatz h t t = ϕ t h r r = ϕ r h θ θ = h φ φ = 0 Therefore, the perturbed metric becomes g µν = diag [ (1 + ϕ t ) g tt, (1 + ϕ r ) g rr, g θθ, g φφ ] The non-perturbed Einstein equations simplify to G t t = ϕ r ϕ r r 2 = X r 2 ϕt Gr r = ϕ r ϕ r r 2 = X r 2 ϕr Gϕ ϕ = 0 = 0 Then ϕ t = ϕ r ϕ X = 1 + ϕ which agrees with the O(h µν 2 ) perturbative equations and retrieves the numerical solution if ϕ plays the role of A.

13 In what follows we will see how the relation between X and ϕ depends on how fluctuations are assumed to interact with the metric itself. In any case, there is a normalizable solution that can be interpreted as the collective wave function of a graviton condensate. The resulting e.o.m. is always one linearly independent equation obtained for a constant condensate wave function ϕ and we are able to consider it as a mean field-like Gross-Pitaevskii equation. Once we introduce a not-null µ, the relation between the constant wave function ϕ and the dimensionless chemical potential X is unique.

14 Collective many body interpretation µ r=rs 0 X = 1+ϕ The jump in X gives reasons to think that BHs are able to auto-sustain a graviton BEC. Outside, X 0, i.e. Schwarzschild. 2 branches of solutions: - X = 0 ϕ = 0 (Schwarzschild) but not the other way around. - When ϕ N is nonzero, X departs from -1 (not-schwarzschild). Consequence: if N 0 (ϕ 0) without disturbing geometry (r s = cte), i.e. ratio N/ ϕ fixed, X 1 independently of any hypothesis (Schw).

15 Gross-Pitaevskii picture From GP picture: N = dv Ψ 2 A good candidate for the role of probability density: Ψ 2 = ρĥ = 1 2ĥαβ 1 λĥαβ where λ is the graviton wavelength and coincides with r s. ĥ αβ = M P h αβ and h 2 = h α β h β α = 2ϕ 2. Where ϕ is entirely determined by the value of the dimensionless chemical potential X. The number of gravitons is N = 8π 3 M P r s 2 h 2

16 Implications of the theory: 3 NLP r s = 8π h Dvali & Gómez: λ NL P Semiclassical calculations lead us, using M M P N for one mode at c, to dm dt M 1 P 2 N dn dt = 1 dn 2r s dt = r 2 s dm dt = T 2 T 1 r s T = NLP

17 Indices are raised and lowered with the full metric g µν The wave-function does not propagate over a fixed background but is the background. Ansatz: h t t = ϕ h r r = ϕ h θ θ = h φ φ = 0 [ ] 1 g µν = diag 1 ϕ g 1 tt, 1 ϕ g rr, g θθ, g φφ and dv = d 4 x g = d 4 x r 2 1 ϕ The non-perturbed Einstein tensor is modified by the variation of the volume element. The e.o.m. (valid to every order in the expansion) is the mean field-like GP equation with G t t = G r r = ϕ r 2 = X r 2 ϕ 3 2 X r 2 ϕ2 X = ϕ ϕ +...

18 The functional form of the metric perturbation is the same as the original background; so, is it possible to derive the latter from the former in some sort of self-consistent derivation? Would it be possible for the BEC to be sustained by other quanta? Why only gravitons? Similarities/differences? Extend the theory to other BHs: - Rotating Kerr. Do rotations contribute to quantum depletion? Is it possible to drive some sort of Bogoliubov mode? - Try to find condensation in BTZ. Do particles confine in lower dimensions? THANK YOU.

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20 Comments on the reasons of the form of the µ term The µ must be introduced in the grand canonical ensemble. When particles have no charge there are no conserved currents. The number of gravitons N is not a conserved current; gravitons can be created and annihilated inside the BH. While there is no conserved current, E total of the system is a conserved magnitude. Analogously to a massless neutral scalar field E = 1 2 dv φ 2 φ = E = 1 2 dv ε 2 ĥ 2 = dv ερ grav. Each graviton has constant energy ε = 1/λ = 1/r s, so N = E total /E grav.

21 General covariance A two rank tensor under an infinitesimal displacement D in the coordinates, δ D [ξ] x µ = ξ µ, transforms as g αβ = g µν x µ x α x ν x β = ξ µ,ν + ξ ν,µ = L ξ g µν The same happens for the background metric; under an infinitesimal diffeomorphism it changes as δ g µν = L ξ g µν From the definition of the perturbation, general coordinates transformations are read as δh µν = δ(g µν g µν ) = ξ ρ h µν,ρ + ξ ρ,νh µρ + ξ ρ,µh ρν = L ξ h µν The facts that h µν transforms covariantly and that µ is a scalar, ensure the general covariance as leaves the action invariant.

22 Numerical solution outside Useful boundary at r : h µν must vanish. Decaying law ansatz for h µν such as power law ( 1/r α ), exponential vanishment ( e αr ) or even Gaussian decay ( e αr 2 ) force µ as well as h µν to be null. Numerical integration coincides with this result. A magnitude associated with the total number of gravitons seems to vanish in the region when decreasing the boundary at. h µν = 0 µ = 0

23 Event horizon boundary Exclude borders: r (0; r s ) Our {G t t, G r r, G ϕ ϕ } {µ, h tt, h rr } system is algebraic for µ. In the limit r r s the 2 2 system is solved by (w = r/r s ) Volume-preserving fluctuation at O(h) (non-normalizable), µ = 0 h tt = A h rr = A Square-integrable solution (finite norm) w (1 w) 2 µ = 1 + A r 2 h tt = A 1 w w h rr = A w 1 w At a linear level, µ = 1/r 2. Quadratic equations give O(A) corrections to µ. Good candidate for being boundary condition at the event horizon.

24 Origin boundary condition Exclude borders: r (0; r s ) In the limit r 0 the 2 2 system is solved by (w = r/r s ) h tt = A 1 w w h rr = B (1 w)w Another good candidate for being a boundary condition. Procedure: perform numerical integrations starting from the event horizon and from the origin. Then, make them match.

25 Covariant conservation There is one more ingredient that gives consistency to our solution. Bianchi identities dictates that This is translated to our µ-term ν G ρ ν = 0 ν ( µhρ ν ) = 0 From these 4 equations, ρ = r is the only nontrivial: [ r (µh r r ) + µ Γ t ( tr t r ht h ) ] r + Γ θ θr h r r + Γ φ φr h r r = 0 Using h t t = h r r = ϕ: r µ + 2µ r = 0 = µ = µ 0 r 2 with µ 0 = 1 + ϕ