Comment on 'Accelerated observers emerging from a Bose-Einstein condensate through analogue gravity'
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Comment on ”Accelerated observers emergingfrom a Bose-Einstein condensate throughanalogue gravity”
Hristu Culetu,Ovidius University, Dept.of Physics,Mamaia Avenue 124, 900527 Constanta, Romania,e-mail : [email protected] 11, 2019
Abstract
Few comments upon Gonzalez-Fernandez and Camacho paper (arXiv:1904.02299) are pointed out. We bring evidences that the analogue gravityrecipe does not work for a BEC in an anisotropic harmonic-oscillator trap.The analogy with an accelerated observer does not seem to be realistic.Some loopholes related to the physical units are emphasized.
Most textbooks on Statistical Mechanics treat the phenomenon of Bose-Einstein condensate (BEC) in an uniform, non-interacting gas of bosons. In thesemi-classical approximation, the energy spectrum is considered as a continuum.In the fully-condensed state, all bosons are in the same single-particle state [1].Gonzalez-Fernandez and Camacho [2] use the Analogue Gravity recipe in astudy on moving fluids and an analog spacetime based on the acoustics of thefluid. The BEC is trapped in an anisotropic 3-dimensional harmonic-oscillatorpotential. In the acoustic representation, the authors’ effective metric g µν ( t, ~r )(their Eq.4) contains the prefactor n c /mc s , where c s is the phonons speed inthe medium, m is the mass of the particles of the BEC and n c is the densitydistribution.From authors’ Gross-Pitaevskii equation (1) (or from their Eq.2) we noticethat the coupling constant k ( a ), characterising the effective interaction betweenthe particles, has units of energy × volume while n c is 1/(volume). Therefore,the prefactor n c /mc s from their Eq.4 has the dimension T M − L − ( T- time,M - mass, L - length) (see also Eq.13). But g µν - the metric tensor in GeneralRelativity (GR) should be dimensionless or a length squared if we take thecoordinates to be dimensionless. In either case, it turns out that the units areerronous, even though the authors use geometrical units ( C = 1, where C is thevelocity of light in vacuo). Having dimensions, we cannot absorb the prefactor in1he velocities c s and v . In addition, the spacetime (4) has no an event horizonbecause there is no any value of v = v i v i to satisfy the equation [3, 4] g tt − X g ti g ii = 0 , i = 1 , , − c s , that isnonzero).Let us observe that K from Eq.17 has the correct dimensions - a velocitysquared. However, when x << b, y << c and z << d , the exponential factortends to unity and K = c s , using (10). Therefore, the metric is not Minkowskiansince the constant c s is not Lorentz-invariant, like the speed of light ( ds willnot be invariant under the Lorentz transformation). In other words, we maynot define the fourth coordinate as x = c s t . In our view, there is no way to getthe Minkowski geometry from the metric (17) because K cannot become thevelocity of light in any approximation, even in the case the BEC is removed.We express serious doubts on the authors’ physical interpretation of the line-element (17). The metric (18) is nothing but the Rindler metric which is flatbut covers only a part of Minkowski’s spacetime. As the authors use geometricalunits - see below Eq.27 - their g in (18) is, in fact, g = − − gξ /C . Same isvalid for Eq. (20) and (21). g is the constant acceleration of an observer locatedat the origin ξ = 0 of the accelerated reference system or the surface gravity κ = p a b a b √− g | ξ = − /g = g gξ (1 + gξ ) = g, (0.2)with a b = (0 , g/ (1+ gξ ) , ,
0) the acceleration of a static observer and ξ = − /g is the location of the Rindler horizon.As far as the generalized form (21) of the approximated line-element (20) isconcerned, it is worth noting that the spatial components of the acceleration arenot g i , as the authors of [2] claim, but g i / (1 + 2 g i ξ i ). Moreover, we stress thatthe line-elements (20) and (21) are curved, though the starting Rindler metric(18) was flat. In other words, the authors generated a stress tensor by meansof an approximation ( gξ /C << t ′ from dt ′ = √ Kdt is not a time, but alength ( √ K is a velocity). The points inside the harmonic-oscillator trap obey,of course, ( x i /b i ) < x i /b i ) <<
1, asthe authors have claimed in Eq.23 ( from, say, x i /b i = 0 . < x i /b i = 0 . << g i / (1 + 2 g i ξ i ), which have a very different behavior compared to − x i / b i .How do the authors ensure the same physical units in their Eq.25 ? The l.h.s.is an acceleration but the r.h.s has dimension 1 / length. Because they used geo-metrical units, the l.h.s. should be g i /C . But there is no any C in the metriccoefficients of (24), since dt ′ = Kdt . It turns out there is a contradiction here. At p.6, the authors introduced a g µν with the right dimensions (Eq.16) by simply gettingrid of a (non dimensionless) conformal factor. r << b in (32) (a case of amore interest), we find that T sµν = ( C / πG ) G sµν is of the order of C / πGb ,where the width of the wave function b = p ~ /mω (see [1], Eq.2.34), with ω - the trap frequency. If we consider m = 1 amu and ω = 10 Hz, one obtains b = 1 µm , whence a component of the energy-momentum tensor, say the energydensity, gives 10 ergs/ ( µm ) , an unphysical value. Similar conclusions can bedrawn for the axially-symmetric trap and for the asymmetric trap.To summarize, although the paper is interesting and innovative, we havebrought evidences that the analogue gravity applied to a Bose-Einstein conden-sate in a harmonic-oscillator trap would not work and the system may not beregarded as an accelerated observer. References [1] C. J.Pethick and H. Smith,