Quantum Energy Inequalities with a twist: rotating frames along stationary worldlines

Abstract

The energy densities of the quantised scalar and electromagnetic fields need not be positive, but their averages along timelike worldlines are constrained by Quantum Energy Inequalities (QEIs) that hold in flat and curved spacetimes for arbitrary Hadamard quantum field states. However the general QEI bound is not easy to compute in closed form. Here, we compute closed form expressions for the massless scalar and electromagnetic fields along stationary worldlines in Minkowski spacetime. These worldlines are solutions to the four-dimensional Frenet-Serret equations with constant curvature invariants and include inertial, uniformly linearly accelerated, and uniformly rotating worldlines as special cases. The QEI bound depends on a choice of tetrad along the worldline, and we compute it for any tetrad with constant angular velocity relative to the Frenet-Serret tetrad. Optimising over the angular velocity, we show that the tightest electromagnetic QEI bound occurs when the tetrad is invariant under Fermi-Walker tetrads. Surprisingly, the Fermi-Walker tetrad does not produce the tightest QEI bound for massless scalar fields, and we investigate the optimal bound both analytically and numerically. Our results and methodology significantly improve on those in [CJ Fewster & J Thompson, Class. Quantum Grav. 40 (2023) 175008], in which only Frenet-Serret tetrads were considered for massless scalar fields.