Comment on 'Accelerated observers emerging from a Bose-Einstein condensate through analogue gravity'

TL;DR

This paper critiques the applicability of the analogue gravity model to Bose-Einstein condensates in anisotropic traps, questioning its realism in simulating accelerated observers and highlighting issues with physical units.

Contribution

It provides critical analysis and evidence challenging the validity of the analogue gravity analogy in certain BEC configurations.

Findings

Analogue gravity does not work for anisotropic BEC traps

The analogy with accelerated observers is not realistic

Physical units introduce loopholes in the analogy

Abstract

Few comments upon Gonzalez-Fernandez and Camacho paper (arXiv: 1904.02299) are pointed out. We bring evidences that the analogue gravity recipe 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.

Equations

$$g_{tt}-\sum \frac{g_{ti}^2}{g_{ii}}=0,i=1,2,3$$

$$\kappa =\sqrt{a^b{a}_b}\sqrt{-g_{00}}{|}_{{\xi }^1=-1/g}=\frac{g}{1+g{\xi }^1}(1+g{\xi }^1)=g,$$

Full text

Most textbooks on Statistical Mechanics treat the phenomenon of Bose-Einstein condensate (BEC) in an uniform, non-interacting gas of bosons. In the semi-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 a study on moving fluids and an analog spacetime based on the acoustics of the fluid. The BEC is trapped in an anisotropic 3-dimensional harmonic-oscillator potential. In the acoustic representation, the authors’ effective metric $g_{\mu\nu}(t,\vec{r})$ (their Eq.4) contains the prefactor $n_{c}/mc_{s}$, where $c_{s}$ is the phonons speed in the medium, $m$ is the mass of the particles of the BEC and $n_{c}$ is the density distribution.

From authors’ Gross-Pitaevskii equation (1) (or from their Eq.2) we notice that the coupling constant $k(a)$, characterising the effective interaction between the particles, has units of energy $\times$ volume while $n_{c}$ is 1/(volume). Therefore, the prefactor $n_{c}/mc_{s}$ from their Eq.4 has the dimension $TM^{-1}L^{-4}$ ( T- time, M - mass, L - length) (see also Eq.13). But $g_{\mu\nu}$ - the metric tensor in General Relativity (GR) should be dimensionless or a length squared if we take the coordinates to be dimensionless. In either case, it turns out that the units are erronous, even though the authors use geometrical units ($C=1$, where $C$ is the velocity of light in vacuo). Having dimensions, we cannot absorb the prefactor in the velocities $c_{s}$ and $v$.111At p.6, the authors introduced a $g_{\mu\nu}$ with the right dimensions (Eq.16) by simply getting rid of a (non dimensionless) conformal factor. In addition, the spacetime (4) has no an event horizon because there is no any value of $v^{2}=v_{i}v^{i}$ to satisfy the equation [3, 4]

[TABLE]

which is the condition to have a horizon (the l.h.s. of (0.1) equals $-c_{s}^{2}$, that is nonzero).

Let us observe that $K$ from Eq.17 has the correct dimensions - a velocity squared. However, when $x<<b,y<<c$ and $z<<d$, the exponential factor tends to unity and $K=c_{s}^{2}$, using (10). Therefore, the metric is not Minkowskian since the constant $c_{s}$ is not Lorentz-invariant, like the speed of light ($ds^{2}$ will not be invariant under the Lorentz transformation). In other words, we may not define the fourth coordinate as $x^{0}=c_{s}t$. In our view, there is no way to get the Minkowski geometry from the metric (17) because $K$ cannot become the velocity 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 flat but covers only a part of Minkowski’s spacetime. As the authors use geometrical units - see below Eq.27 - their $g_{00}$ in (18) is, in fact, $g_{00}=-1-g\xi^{1}/C^{2}$. Same is valid for Eq. (20) and (21). $g$ is the constant acceleration of an observer located at the origin $\xi^{1}=0$ of the accelerated reference system or the surface gravity

[TABLE]

with $a^{b}=(0,g/(1+g\xi^{1}),0,0)$ the acceleration of a static observer and $\xi^{1}=-1/g$ is the location of the Rindler horizon.

As far as the generalized form (21) of the approximated line-element (20) is concerned, it is worth noting that the spatial components of the acceleration are not $g_{i}$, as the authors of [2] claim, but $g_{i}/(1+2g_{i}\xi^{i})$. Moreover, we stress that the 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 means of an approximation ($g\xi^{1}/C^{2}<<1)$, which is not reasonable from the physical point of view. We also notice that $t^{\prime}$ from $dt^{\prime}=\sqrt{K}dt$ is not a time, but a length ($\sqrt{K}$ is a velocity). The points inside the harmonic-oscillator trap obey, of course, $(x_{i}/b_{i})<1$ but from here we cannot conclude that $(x_{i}^{2}/b_{i}^{2})<<1$, as the authors have claimed in Eq.23 ( from, say, $x_{i}/b_{i}=0.9<1$ one may not deduce that $x_{i}^{2}/b_{i}^{2}=0.81<<1$).

As another evidence that the implication (23) is not always valid, one observes that the spatial accelerations of a static observer in the spacetime (21) are $g_{i}/(1+2g_{i}\xi^{i})$, which have a very different behavior compared to $-x_{i}/2b_{i}^{2}$. 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 geometrical units, the l.h.s. should be $g_{i}/C^{2}$. But there is no any $C^{2}$ in the metric coefficients of (24), since $dt^{\prime 2}=Kdt^{2}$. It turns out there is a contradiction here.

Our conclusion is therefore that the analogue-gravity system to that of a BEC in an anisotropic harmonic-oscillator trap cannot be considered as an accelerated observer.

Concerning the Sec.3, the stress tensor (32) for a spherically-symmetric trap does not seem to be realistic. For instance, taking $r<<b$ in (32) (a case of a more interest), we find that $T_{\mu\nu}^{s}=(C^{4}/8\pi G)G_{\mu\nu}^{s}$ is of the order of $C^{4}/8\pi Gb^{2}$, where the width of the wave function $b=\sqrt{\hbar/m\omega}$ (see [1], Eq.2.34), with $\omega$ - the trap frequency. If we consider $m=1amu$ and $\omega=10^{5}$ Hz, one obtains $b=1\mu m$, whence a component of the energy-momentum tensor, say the energy density, gives $10^{44}ergs/(\mu m)^{3}$, an unphysical value. Similar conclusions can be drawn for the axially-symmetric trap and for the asymmetric trap.

To summarize, although the paper is interesting and innovative, we have brought evidences that the analogue gravity applied to a Bose-Einstein condensate in a harmonic-oscillator trap would not work and the system may not be regarded as an accelerated observer.