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Quantum Simulation of Rainbow Gravity by Nonlocal Nonlinearity

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Quantum Simulation of Rainbow Gravity by Nonlocal Nonlinearity Quantum simulation of rainbow gravity by nonlocal nonlinearity M. C. Braidotti1,2⋆ and C. Conti2,3 1 Department of Physical and Chemical Sciences, University of L’Aquila, Via Vetoio 10, I-67010 L’Aquila, Italy 2 Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome, Italy 3 Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy ∗Corresponding author: [email protected] Testing the unobserved quantum gravitational Many authors after Unruh reported emulations of black phenomena in different experimental frameworks holes in many research fields as acoustics[14], optics[10, is the challenge of analogue gravity. Laboratory 15, 16], Bose-Einstein condensates [17, 19, 20], 3He [21] emulation may validate theoretical models and and Fermi liquids[22], as reviewed in [12]. However, all give inspiration for further developments. The these studies mainly address to phenomenological as- simulations were limited to general relativity, in- pects, not providing frameworks to test the numerous cluding black holes, event horizons and superra- formulations of quantum gravity. To our knowledge, an diance. We report on the first analog of space- analogue of a quantum gravity scenario has never been time near a rotating black hole as in a recent proposed. quantum-gravity theory, called rainbow gravity. In this study, we report on classical and quantum analogs Nonlinear waves in nonlocal media, as those in of a recent theory of quantum gravity called “rainbow Bose-condensed gases and nonlinear optics, em- gravity”. This theory is a generalization of doubly spe- ulate the rainbow energy-dependent metric. A cial relativity[6] to incorporate curvature. Rainbow gra- fully quantized analysis is reported, showing that vity, proposed by Magueijo and Smolin in 2004[26, 27], the metric energy-dependence inhibits the exis- relies on a space-time geometry, which depends on the tence of an event horizon and superradiance. Our energy of the free-falling particle. This corresponds to results open the way to numerous fascinating ex- an energy-dependent metric with invariant line element: perimental tests of quantum gravity theories and 0 2 i 2 demonstrate that these theories can provide novel 2 µ ν (dx ) (dx ) ds = gµν (E)dx dx = 2 + 2 , (1) tools for open problems in nonlinear quantum −f (E/EP ) g (E/EP ) physics. with EP the Planck energy. The functions f and g One of the current major challenges in physics is uni- enclose all the metric energy-dependence and their fying general relativity and quantum mechanics. Nume- form affects the space-time properties, as inducing a rous attempts have been carried out, leading to the non-constant light speed c and/or an energy-dependent formulation of many quantum gravity theories[1, 2], gravitational constant. Some of the implications of such as string theory[3], loop quantum gravity[4], non- Eq.(1), as in black hole thermodynamics, are not known commutative geometry[5] and doubly special relativity[6, and inventing emulations may furnish many unexpected 7]. Despite all these efforts, however, it is still not pos- results and be an inspiration for further developments. sible to establish which theoretical proposal is the most In this paper, we study the behavior of excitations promising, due to the absence of experimental evidences. on a vortex background in a defocusing nonlinear arXiv:1708.02623v2 [gr-qc] 6 Jun 2018 As a result, a broad community of scientists is looking for nonlocal medium. We show that nonlocality allows analog systems to provide experimental confirmations of to mimic the rainbow gravity space-time of a rotating quantum-gravitational phenomena, as Hawking radiation black hole, where the degree of nonlocality σ deter- and superradiance[8–15, 18, 24, 25]. This research line mines the proximity to the Planck scale. Numerical has a two-fold advantage: on one hand, emulating these simulations of classical and quantum excitations give Planck-scale phenomena in the laboratory may suggest evidence of the fading of the black hole event horizon unexpected interpretations for quantum gravity, while, and the consequent weakening of superradiance when on the other hand, analog physics may furnish novel ex- increasing σ. Full quantum dynamics is analyzed in planations and unexplored regimes for classical and quan- the -representation by a pseudo-spectral stochastic tum nonlinear physics. Runge-KuttaP algorithm[28] and shows an enhancement The first analog dates back to 1981 [14] when Unruh pro- of the vanishing of the ergoregion. This furnishes a posed black-hole evaporation as a model to study sound true quantum-simulation of second-quantized fields in a waves in moving fluids, showing that it is possible to find curved space-time with energy dependent metric. Hawking radiation in a non-gravitational system. This remarkable finding opened the way to a wide investiga- We start considering the classical regime and the way tion of analogue gravitational phenomena. an energy dependent metric occurs in the hydrodynami- 2 cal approximation of the normalized nonlocal nonlinear by the background ρ0. Schr¨odinger equation [r = (x, y)] [29–31] The geometrical description of the optical system fails when the background density varies on a scale smaller 1 ı∂ ψ + 2 ψ P R(r) ψ 2ψ =0. (2) ˜ t 2∇xy − ∗ | | than the healing length ξ =1/(2 P Rρ0), i.e. in the high energy limit where E K2. Inq terms of the quantum- Equation (2) describes the field evolution in many phy- gravity analog, the scale∝ of background density variation sical systems as nonlinear optics with thermal[29, 32] or exceeds the analog Planck length ξ, violating the Lorentz re-orientational nonlinearity, Bose-Einstein condensates invariance. Hence, the analogue is self-consistent only (BEC)[33, 34] and plasma-physics[35, 36]. In (2), de- ∗ in the low energy and low-momentum regime for which notes a convolution integral. The form of the kernel K < 1/ξ. R(x, y) depends on the specific physical system and its Following general relativity, a metric determines the in- ˜ Fourier transform is R(Kx,Ky). The field ψ is norma- variant square of a line element, ds2, given by lized such that ψ 2dr = 1, and P measures the | | 2 µ ν strength of the nonlinearity. ds =g ˜µν dx dx = Thanks to the hydrodynamicalR approach, commonly used R˜ for dispersive shock waves [32] and also analog gravity 2 ˜ 2 2 = 2 csR v0 dt + (6) cs !(− − [37], we study the behavior of small excitations on top of the metric induced by the fluid of light. By writing ıφ 2 2 the field ψ as ψ = ρe with v = φ, Eq. (2) re- + dr + (rdθ) 2vrdrdt 2vθrdθdt . √ − − duces to the continuity equation and∇ the Euler equa- " # ) tion, with bulk pressure [ρ] = ρP R(r) ρ (see Sup- P ∗ Equation (6) is written in polar coordinates (r, θ), with plementary Information). Small excitations in the pho- 2 2 2 1 v0 = vr + v , vr = ∂rφ0 and vθ = ∂θφ0. The local case ton fluid are described by letting ρ = ρ + ǫρ + O(ǫ2) θ r 0 1 [8] is found letting R˜ = 1 in Eq. (5). and φ = φ + ǫφ + O(ǫ2). In a slowly varying back- 0 1 Within the geometrical description validity limit region ground ρ and in the eikonal approximation, we have 0 (K < 1/ξ), we can mimic a black hole through vortex ρ =ρ ¯ eı(Kxx+Kyy Et) and φ = φ¯ eı(Kxx+Ky y Et), with 1 1 − 1 1 − solutions of Eq. (2). This, in addition with the presence E the angular frequency for an inertial observer at infi- of nonlocality, enables to simulate rainbow gravity. nity. The generalized dispersion relation is (see Supple- Vortexes are formed by a dark hole with circular sym- mentary Information) [38–41]: metry and a helical wave front. Their wavefunctions, iφ0(θ) 4 in polar coordinates, is ψ0 = ρ0(r)e , with phase K v 2 ˜ 2 K (E 0) = ρ0P R(K)K + . (3) φ = mθ. The integer m is the winding number, or vor- − · 4 0 p tex charge. The region in proximity of r = 0, where In Eq. (3) v0 = φ0 is the background velocity field. ρ(0) = 0, is called the vortex core. v ∇ When 0 = 0, the high energy limit, i.e., short wave- The form of the metricg ˜µ,ν determines the properties of 2 2 1/2 lengths and large momentum K = Kx + Ky , corre- the analog space-time, as the presence of an event hori- sponds to E K2 as for free particles. Whereas, in the zon and ergoregion. ∝ long wavelength limit (i.e. for small K) E csK, where The analog system presents an event horizon surface if 2 ∂≃ [ρ] the velocity of sound equals the radial velocity of the cs = Pρ0 is the local speed of sound cs P∂ρ . ≡ ρ=ρ0 fluid, i.e. cs = vr. Low energy modes cannot escape In the hydrodynamical regime, we obtain the following from this region. Furthermore, by geometrical consider- equation for the Fourier transformed massless scalar field ations, one can find thatg ˜ 0, and hence c2 < v2, ˜ tt ≥ s 0 φ1 in a 2 + 1 dimensional curved space (see Supplemen- determines the ergoregion, which is the region of space- tary Information): time where low energy modes are dragged in the moving light flow. 1 µν ] ∆φ1 = ∂µ( g˜g˜ ∂ν φ1), (4) In the case of local response function R˜ = 1, this geo- √ g˜ − metry has an ergosurface placed at c = v . In order to − p s θ µν g µν 1 mimic a black hole, it is necessary to introduce a radial whereg ˜ is the covariant metric (˜g )− =g ˜µν , inward velocity vr, by letting φ0(r, θ)=2π r/r0 + mθ, ˜ 2 2 T which induces an event horizon [8, 42, 43].
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