Effect of tidal curvature on dynamics of accelerated probes

Abstract

We obtain a remarkable semi-analytic 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 ($\tau_{\rm geod}$) and proper time ($\tau_{\rm acc}$) 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 ${\mathscr E}_n= R_{abcd} \varepsilon^{ab} \varepsilon^{cd}$ of curvature, with $\varepsilon^{ab}$ the bi-normal to the plane of motion: $$ \tau_{\rm geod} = \frac{2}{\sqrt{{-\mathscr E}_n}} \sinh ^{-1}\Biggl[\sqrt{\frac{-{\mathscr E}_n}{a^2-{\mathscr E}_n}} \sinh \left(\frac{1}{2} \sqrt{a^2-{\mathscr E}_n} \; \tau_{\rm acc} \right) \Biggl] $$ 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 $Unruh\;temperature$ $$ [k_{\rm B} T]_{{\mathscr E}_n} = \frac{\hbar \sqrt{a^2- {\mathscr E}_n }}{2 \pi} $$ 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.