bySilke Weinfurtner

Victoria University of Wellington, New Zealand

Stefano Liberati SISSA/INFN

Trieste, Italy

Constraining quantum gravity phenomenology via analogue spacetimes

Fourth Meeting onConstrained Dynamics and Quantum Gravity

Cala Gonone (Sardinia, Italy)September 12-16, 2005

presented at

Matt Visser Victoria University

Wellington, New Zealand

Using the presented system for an Analogue Model for Quantum Gravity Phenomenology

Extended Analogue Models for gravityin a coupled 2-component BEC

Analogue Models for gravity

The first Analogue Models for Gravity

Bill Unruh, “Experimental black hole evaporation?”, Phys. Rev. Lett., 46, (1981) 1351-1353.

space-time convergent fluid flow particle small excitations (sound waves)

equations of motion for irrotational fluid flow

linearizing equation about some solutions

equations for

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˜ Ψ

interpretation: equation for massless scalar field in a geometry with metric

Bose-Einstein condensates as Analogue Models

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

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

Bose-Einstein condensate ~ in theory

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)

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

Matt Visser, Silke Weinfurtner, Massive Klein-Gordon equation from a BEC-based analogue spacetime, Phys.Rev. D72 (2005) 044020

It is possible to extend the Analogue Models to describe massive scalar fields:

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?)…

Gross-Pitaevskii equations

Macroscopic wave functions

Sound waves in a 2-component BEC

UAA

UBB

UAB

From the GPE to a pair of coupled wave equations

Physical interpretations:

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

mass-density matrix background velocity

Sound waves in a 2-component BEC

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 two speed of sounds are:

the mono-metricity condition must be which requires the fine tuning

the densities and interactions within each condensate are equal

Fine tuning of the interactions via the external coupling field :

the external laser field modifies the interactions

UAAUBB

UAB

~

~

~

the sign of can be positive or negative ( additional trapping frequency ), e.g it is possible to make the modified XX or XY interactions zero:

UAAUBB

~ ~

Dispersion relation for uniform condensate.

Changing into momentum space leads to the dispersion relation:

interpretation in terms of Analogue Models:

We recover perfect special relativity for the decoupled phonon modes in the hydrodynamic limit.

Note: The change to momentum space is only exact, if the densities are uniform and the background velocity is at rest ( Minkowski space-time ).

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gab ∝

−(c 2 − v02) −v0x −v0y −v0z

−v0x 1 0 0

−v0y 0 1 0

−v0z 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

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!

The hydrodynamic limit

Dispersion relation for high energy phonon modes.

change into momentum-space

coupled wave equation for phonon modes in uniform condensatebeyond the hydrodynamic limit:

perturbations have to fulfill the generalized Fresnel equation

Calculating the dispersion relation for the 2 coupled phonon-modes..

Dispersion relation for high energy phonon modes.

Taylor expansion of k2 around zero

Note that with H(k2) as a function of k2 only even parameters of k appear!

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mA mB

the dimensionless coefficients - within that fine tuning - are:

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:

Predictions of Quantum Gravity: UV LIV

LIV purely UV physics (only QP due terms)

Note that for mA=mB it follows that 4,I= 4,II=1/2 !

A coupled 2-component Bose-Einstein condensate can be used as an Analogue Model for a massive and massless scalar field in curved-spacetime

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)

Conclusions

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!

Thank you for your attention.

Supersonic and subsonic region…

ho

rizon

fluid velocity fluid at rest

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c 2 − v02 < 0

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c 2 − v02 > 0

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rv 0 =

r 0

Animation: Sound waves in a moving fluid