Dawood Kothawala

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 (τgeod) and proper time (τacc) 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 n=Rabcd ab cd of curvature, with ab the binormal to the plane of motion, τgeod=2-nsinh-1[-na2-nsinh(12a2-nτacc)]. 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, [kBT]n=a2-n2π. 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.

Riemann normal coordinates (RNC) at a regular event p0 of a spacetime manifold M are constructed by imposing (i) gab|p0=ηab, and (ii) Γabc|p0=0. There is, however, a third, independent, assumption in the definition of RNC which essentially fixes the density of geodesics emanating from p0 to its value in flat spacetime, viz.: (iii) the tangent space Tp0(M) is flat. We relax (iii) and obtain the normal coordinates, along with the metric gab, when Tp0(M) is a maximally symmetric manifold MΛ with curvature length |Λ|-1/2. In general, the "rest"frame defined by these coordinates is noninertial with an additional acceleration a=-(Λ/3)x depending on the curvature of tangent space. Our geometric setup provides a convenient probe of local physics in a universe with a cosmological constant Λ, now embedded into the local structure of spacetime as a fundamental constant associated with a curved tangent space. We discuss classical and quantum implications of the same.

We analyze several aspects of detectors with uniform acceleration a and uniform rotation ω in de Sitter (Λ>0) and anti-de Sitter (Λ<0) spacetimes, focusing particularly on the periodicity, in (Euclidean) proper time τtraj, of geodesic interval τgeod between two events on the trajectory. For Λ<0, τgeod is periodic in iτtraj for specific values of a and ω. These results are used to obtain numerical plots for the response rate Ḟ of Unruh-DeWitt detectors, which display nontrivial combined effects of rotation and curvature through the dimensionless parameter Λc2/ω2. In particular, periodicity does not imply thermality due to additional poles in the Wightman function away from the imaginary axis. We then present some results for stationary rotational motion in arbitrary curved spacetime, as a perturbative expansion in curvature.