Effect of tidal curvature on dynamics of accelerated probes

We obtain a remarkable semianalytic expression concerning the role of purely tidal curvature on accelerated probes, revealing some novel insights into the role of absolute vs tidal acceleration in the response of such probes. The key quantity we evaluate is the relation between geodesic (${\tau }_{\mathrm{geod}}$) and proper time (${\tau }_{\mathrm{acc}}$) intervals between events on the probe trajectory. This is obtained as a covariant power series in curvature using a combination of analytical and numerical tools. A serendipitous observation then reveals that one can exactly sum all terms involving the purely tidal component ${ℰ}_n=R_{abcd}{ϵ}^{ab}{ϵ}^{cd}$ of curvature, with ${ϵ}^{ab}$ the binormal to the plane of motion, ${\tau }_{\mathrm{geod}}=\frac{2}{\sqrt{{-ℰ}_n}}{\sinh }^{-1}[\sqrt{\frac{-{ℰ}_n}{a^2-{ℰ}_n}}\sinh (\frac{1}{2}\sqrt{a^2-{ℰ}_n}{\tau }_{\mathrm{acc}})]$. For classical clocks, the above result represents an interesting closed form contribution of tidal curvature to the differential ageing of twins in the classic twin paradox. For quantum probes, it gives a thermal contribution to the detector response with a modified Unruh temperature, $[k_{\mathrm{B}}T{]}_{{ℰ}_n}=\frac{\hslash \sqrt{a^2-{ℰ}_n}}{2\pi }$. As an operational tool, the computational framework we present and the corresponding results should find applications to a wide range of physical problems that involve measurements and observations by use of accelerated probes in curved spacetimes.

Figure 1

The geometric setup for the problem. Two events on the hyperbolic trajectory $\mathcal{C}(\tau )\equiv z^i(\tau )$ are connected by a geodesic. The relation between the corresponding proper times holds the key information relevant for classical and quantum probes.

Figure 2

Comparing the relation between the ${\tau }_{\mathrm{acc}}$ and ${\tau }_{\mathrm{geod}}$ for static observer and radial geodesic observers in Schwarzschild ($r_s=2GM/c^2$). The parameters involved are the radius of the static observer, $r_0$, and the radius of the turning point for the geodesic observer, $r_{\mathrm{tp}}$. Left plot is for $r_0=1.2r_s$, $r_{\mathrm{tp}}=0.5r_0$, and the right one for $r_0=5r_s$, $r_{\mathrm{tp}}=2r_0$.

Figure 3

Comparing the relation between the ${\tau }_{\mathrm{acc}}$ and ${\tau }_{\mathrm{geod}}$ for different spacetimes. The values chosen are, $r_0=2r_s$, $r_{\mathrm{tp}}=2r_0$ for Schwarzschild and $|\mathrm{\Lambda }|=a^2/2$ for de Sitter and anti–de Sitter where $a$ is the magnitude of acceleration for Schwarzschild observer.