Transcript ppt
Constraining quantum gravity phenomenology via analogue spacetimes Fourth Meeting on Constrained Dynamics and Quantum Gravity present ed at by Cala Gonone (Sardinia, Italy) September 12-16, 2005 Silke Weinfurtner Victoria University of Wellington, New Zealand Matt Visser Stefano Liberati Victoria University Wellington, New Zealand SISSA/INFN Trieste, Italy Analogue Models for gravity Extended Analogue Models for gravity in a coupled 2-component BEC Using the presented system for an Analogue Model for Quantum Gravity Phenomenology The first Analogue Models for Gravity Bill Unruh, “Experimental black hole evaporation?”, Phys. Rev. Lett., 46, (1981) 1351-1353. equations of motion for irrotational fluid flow linearizing equation about some solutions equations for ˜ interpretation: equation for massless scalar field in a geometry with metric space-time convergent fluid flow particle small excitations (sound waves) Bose-Einstein condensates as Analogue Models L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, Sonic Analog of Gravitational Black Holes in Bose-Einstein Condensates, Phys. Rev. Lett. 85, 4643–4647 (2000) Bose-Einstein condensate ~ in experiment gas of bosons, e. g. 87Rb (Eric Cornell) or 23Na (Wolfgang Ketterle) extremely low densities, 1015 atoms/cm3 very cold temperature, T1K Bose-Einstein condensate ~ in theory nearly all atoms occupy the ground state non condensed atoms are neglected microscopic system can be replaced by a classical mean-field, a macroscopic wave-function interpretation in terms of Analogue Models: The kinematics for sound waves in BEC is given by the Euler and continuity equation, in the so called hydrodynamic limit the BEC is a superfluid. Extending AM for massive scalar fields and use it for QGP application for Quantum Gravity Phenomenology: One expected Quantum Gravity Phenomena is the violation of spacetime symmetries, e.g. Lorentz violation: Universality and naturalness problem? We would need an analogue model for different interacting (naturalness problem?) particles (universality issue?)… It is possible to extend the Analogue Models to describe massive scalar fields: Matt Visser, Silke Weinfurtner, Massive Klein-Gordon equation from a BEC-based analogue spacetime, Phys.Rev. D72 (2005) 044020 Sound waves in a 2-component BEC Gross-Pitaevskii equations UBB UAA Macroscopic wave functions UAB Sound waves in a 2-component BEC From the GPE to a pair of coupled wave equations Physical interpretations: mass-density matrix background velocity this equation represents kinematics of sound waves in the 2-component BEC a small (in amplitude) perturbation in 2-component BEC results in pair of coupled sound waves coupling matrix this description holds for low and high energetic perturbations interaction matrix + quantum pressure term contains the modified interactions due to the external coupling Klein-Gordon equation for massive phonon modes. The two decoupled wave equations can be written as two scalar fields in curved space-times: in-phase mode anti-phase mode the in-phase mode represents a massless scalar field the anti-phase mode represents a massive scalar field the two effective metrics are different, due to different speeds of sound: Klein-Gordon equation for massive phonon modes. The fine tuning for the decoupling the wave equations: the densities and interactions within each condensate are equal The two speed of sounds are: the mono-metricity condition must be which requires the fine tuning the sign of can be positive or negative ( additional trapping frequency ), e.g it is Finepossible tuning of the interactions via the external coupling field : to make the modified XX or XY interactions zero: the external laser field modifies the interactions ~ ~ UBB UAA ~ ~ UBB UAA ~ UAB Dispersion relation for uniform condensate. Changing into momentum space leads to the dispersion relation: Note: The change to momentum space is only exact, if the densities are uniform and (c v ) v the background velocity is at rest ( Minkowski space-time ). v v 2 g ab interpretation in terms of Analogue Models: 2 0 0x 0y v 0 x 1 0 v 0 y 0 1 v 0 z 0 0 We recover perfect special relativity for the decoupled phonon modes in the hydrodynamic limit. 0z 0 0 1 The hydrodynamic limit Beyond the hydrodynamic limit the quantum potential has to be taken into account the quantum potential term (here in flat space-time) can be absorbed in the redefinition of the interaction matrix between the atoms (effective interaction matrix) this term gets relevant at wave length comparable to the healing length a change to momentum space shows the effective interaction is k-dependent For perturbations compareable to the healing length - high energy modes - the calculations have to be modified, by including the quantum potential. This will brake the Lorentz invariance in the dispersion relation! Dispersion relation for high energy phonon modes. Calculating the dispersion relation for the 2 coupled phonon-modes.. coupled wave equation for phonon modes in uniform condensate beyond the hydrodynamic limit: change into momentum-space perturbations have to fulfill the generalized Fresnel equation Dispersion relation for high energy phonon modes. mAmB Taylor expansion of k2 around zero Note that with H(k2) as a function of k2 only even parameters of k appear! Predictions of Quantum Gravity: UV LIV LIV purely UV physics (only QP due terms) this is equivalent to cancel on our LIV coefficients all the terms which do not depend on the quantum pressure potential; this requires the following constraints: the dimensionless coefficients - within that fine tuning - are: Note that for mA=mB it follows that 4,I= 4,II=1/2 ! Conclusions A coupled 2-component Bose-Einstein condensate can be used as an Analogue Model for a massive and massless scalar field in curved-spacetime At low energies one recovers perfect special relativity LI At high energies the theory has to be modified about the quantum potential LIV At both orders - k2 and k4 - deviations show up. Planck-suppressed! Order one! This is a typical situation studied in QG phenomenology with purely higher order LIV characterized by different coefficients of LIV are particle dependent (no universality) Thank you for your attention. Animation: Sound waves in a moving fluid Supersonic and subsonic region… c v0 0 2 c v0 0 2 2 v0 0 2 horizon fluid velocity fluid at rest