On Fast Linear Gravitational Dragging

A new formula is given for the fast linear gravitational dragging of the inertial frame within a rapidly accelerated spherical shell of deep potential. The shell is charged and is electrically accelerated by an electric field whose sources are included in the solution.

💡 Research Summary

The paper “On Fast Linear Gravitational Dragging” presents a novel theoretical treatment of frame‑dragging that arises not from rotation, as in the classic Lense‑Thirring effect, but from rapid linear acceleration of a massive, charged spherical shell. The authors consider a shell of radius R, mass M, and total charge Q that is accelerated by an external electric field E₀. The field’s sources are explicitly included in the solution, modeled as a distant charged plane that produces a uniform field across the shell. The central novelty lies in treating the shell’s gravitational potential as “deep” (φ = GM/(Rc²) ≈ O(1)), thereby abandoning the usual weak‑field approximation, and allowing the acceleration to be “fast” in the sense that the product a·Δt can exceed the light‑crossing time of the shell (R/c).

To obtain the solution the authors solve the coupled Einstein‑Maxwell equations under spherical symmetry but with a time‑dependent radial acceleration a(t). The stress‑energy tensor includes both the material contribution of the shell (T_ab) and the electromagnetic contribution (T_em). Unlike standard linearised gravity, they retain second‑order terms in φ because the deep potential makes these terms comparable to the first‑order acceleration terms. The resulting metric contains a non‑zero g₀r component, which directly encodes linear frame‑dragging.

From this metric they derive an explicit expression for the inertial velocity change experienced by a test particle inside the shell: \