Trans-Planckian physics and signature change events in Bose gas hydrodynamics

We present an example of emergent spacetime as the hydrodynamic limit of a more fundamental microscopic theory. The low-energy, long-wavelength limit in our model is dominated by collective variables that generate an effective Lorentzian metric. This system naturally exhibits a microscopic mechanism allowing us to perform controlled signature change between Lorentzian and Riemannian geometries. We calculate the number of quasiparticles produced from a finite-duration Euclidean-signature event, where we take the position that to a good approximation the dynamics is dominated by the evolution of the linearized perturbations, as suggested by Calzetta and Hu [Phys. Rev. A 68, 043625 (2003)]. We adapt the ideas presented by Dray et al. [Gen. Relativ. Gravit. 23, 967 (1991)], such that the field and its canonical momentum are continuous at the signature-change event. We investigate the interplay between the underlying microscopic structure and the emergent gravitational field, focussing on its impact on quasiparticle production in the ultraviolet regime. In general, this can be thought of as the combination of trans-Planckian physics and signature-change physics. Further we investigate the possibility of using the proposed signature-change event as an amplifier for analogue “cosmological particle production” in condensed matter experiments.

Figure 1

Two sudden variations in the strength of the atomic interactions is analogous to two spacelike hypersurfaces connecting three different Minkowski spacetimes emerging from a Bose gas.

Figure 2

Two sudden variations in the strength of the atomic interactions are analogues to two spacelike hypersurfaces connecting three different spacetimes emerging from a Bose gas. While the initial and final geometries are Lorentzian, we allow the atomic interactions to be attractive in the intermediate region, corresponding to a Euclidean geometry.

Figure 3

The figure shows a quasiparticle production process starting from the vacuum in region $\underline{1}$. The bold line corresponds to quasiparticles produced in a single Lorentzian step from region $\underline{1}\to \underline{2}$, while the dotted line shows quasiparticles produced during region $\underline{2}$. Region $\underline{3}$ depicts a subsequent amplification process of a finite time Euclidean interval. Altogether the OUT signal in region $\underline{4}$ can be viewed as an amplified IN signal.

Figure 4

The figure qualitatively displays the mean number of quasiparticles $N_k$ produced from two sudden variations in the strength of the atomic interactions. The different curves correspond to the time period $T$ for which the system is kept in the intermediate regime. The left column focusses on purely collective effects, while in the right column microscopic corrections are taken into account. The top two diagrams show variations between spacetimes emerging from a Bose gas with repulsive interactions ($a>0$). In the bottom row of the diagrams the intermediate regime exhibits a Euclidean geometry, where the underlying Bose gas shows attractive atomic interactions ($a<0$).