Low-energy Lorentz violation from high-energy modified dispersion in inertial and circular motion

We consider an Unruh-DeWitt detector in inertial and circular motion in Minkowski spacetime of arbitrary dimension, coupled to a quantized scalar field with the Lorentz-violating dispersion relation $\omega =|\mathbf{k}|f(|\mathbf{k}|/M_{\star })$, where $M_{\star }$ is the Lorentz-breaking scale. Assuming that $f$ dips below unity somewhere, we show that an inertial detector experiences large low-energy Lorentz violations in all spacetime dimensions greater than two, generalizing previous results in four dimensions. For a detector in circular motion, we show that a similar low-energy Lorentz violation occurs in three spacetime dimensions, and we lay the analytic groundwork for examining circular motion in all dimensions greater than three, generalizing previous work by Stargen, Kajuri and Sriramkumar in four dimensions. The circular motion results may be relevant for the prospects of observing the circular motion Unruh effect in analogue laboratory systems.

Figure 1

A perspective plot of the $n=2$ transition rate $\mathcal{F}(E)$ [Eq. (3.5)], as a function of $\beta$ and $h=E/M_{\star }$, for the dispersion relation and density of states (3.6). In the low-energy regime, $|E|/M_{\star }\ll 1$, large deviations from the Lorentz-invariant transition rate (3.4) are apparent when $\beta$ increases above the critical value ${\beta }_c\approx 1.3675$.

Figure 2

As in Fig. 1 but in a top-down color (grayscale) plot.

Figure 3

Plot of the function $gf(g)\cosh \beta -g\sinh \beta$, with $f$ given in Eq. (3.6). The middle curve is for $\beta ={\beta }_c\approx 1.3675$, the top curve is for $\beta =1.363<{\beta }_c$, and the bottom curve is for $\beta =1.372>{\beta }_c$.

Figure 4

Plots of $\mathcal{F}(E)$ [Eq. (4.5)] as a function of $\tilde{E}=RE$ and $v$, for $f$ and $d$ given by Eq. (3.6), with $\tilde{M}$ increasing from 1 to 250 as shown.

Figure 5

The first two plots continue from Fig. 4, with $\tilde{M}$ increasing to 1000. The third plot shows the $\tilde{M}\to \infty$ limit, given by Theorem 4.1. The last plot shows the transition rate for the ordinary massless scalar field, given by ${\mathcal{F}}_0^{circ}(E)$ [Eq. (4.7)]. For large $\tilde{M}$, a large low-energy deviation from the ordinary massless scalar field transition rate is clear from the plots when $v$ increases above the critical value $f_c=0.8781$.