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, T1K
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