Velocity-dependent deformations of the energy spectrum of a quantum cavity from Lorentz symmetry violations

Several approaches to quantum gravity suggest that Lorentz invariance will be broken at high energy. This can lead to modified dispersion relations for wave propagation, which can be concretely realized in effective field theories where the equation of motion involves higher order spatial derivatives in a preferred frame. We consider such a model in the presence of a finite cavity whose walls follow parallel inertial trajectories of speed $v$ with respect to the preferred frame. We find evidence that when the cavity wall speed exceeds the phase velocity, the system becomes classically unstable. For dispersion relations that do not lead to an instability, the energy levels of the cavity are non-trivial functions of $v$. In other words, an observer could in principle measure their velocity with respect to the preferred frame by studying the energy spectra of a quantum cavity, which is a stark violation of the principle of relativity. We also find that the energy levels of the cavity become infinitely large as its velocity approaches light speed.

Figure 1

Cavity geometry in various coordinate systems.

Figure 2

Plot demonstrating the convergence of the lowest ten eigenvalues of $\mathbf{D}$ as $N$ is increased. We have taken $\widehat{F}$ to be the cubic polynomial given in (109) and selected $L_0=20\pi L_{\star }$ with $v=0.9$.

Figure 3

Numerical approximations to the positive eigenvalues ${\mu }_n$ of ${\mathbf{D}}_{\infty }$ and the normal mode frequency ${\omega }_n$ as a function of cavity velocity $v$. We have take $\widehat{F}$ to the cubic polynomial given in (109).

Figure 4

Numerical approximations to the positive eigenvalues ${\mu }_n$ of ${\mathbf{D}}_{\infty }$ as a function of cavity rapidity $v$ (solid lines). Also plotted is the analytic approximations for the high velocity eigenvalue behavior derived in Sec. 6b (dashed lines). The squares on the solid lines where the characteristic wavelength of a given mode equals the exotic physics length scale. This plot assumes that $\widehat{F}$ is given by (109) with $L_0=100\pi L_{\star }$.

Figure 5

Numerical approximations to the positive eigenvalues ${\mu }_n$ of ${\mathbf{D}}_{\infty }$ in the Lorentz-invariant case $\widehat{F}={\partial }_X$ (points). Shown for comparison are the curves ${\mu }_n=n(1-v^2)$, which represent our expectations from Lorentz symmetry considerations (lines).

Figure 6

Numeric simulations of the solution of the wave equation (17) for the choice (123). The top panels show $F^2(k_n)/k_n^2-v^2$ versus $n$ for the first few Fourier modes of the system and three different sets of parameter values. The sign of this quantity determines whether or not a given mode is fast or slow. If there exist any slow modes, the classical Hamiltonian to be unbounded from below and we expect the system to be classically unstable. The lower panels show numeric simulations for the various parameter choices shown in the top row. We indeed see that when the system supports even one slow mode, the numerical solution the wave equation appears to exhibit a classical instability.

Figure 7

A rotating cavity experiment.