  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.
