On Franklin's relativistic rotational transformation and its modification

TL;DR

This paper introduces a modified version of Franklin's relativistic rotational transformation, resolving previous drawbacks and ensuring consistency with Galilean rotational transformation near the rotation axis, with implications for non-inertial observers.

Contribution

The paper proposes a modified Franklin transformation that improves the spacetime metric description for rotating frames, addressing limitations of the original transformation.

Findings

The modified transformation resolves issues with the original Franklin transformation.

The new metric aligns with Galilean rotational transformation near the axis.

Spatial and temporal measurements are consistent with the modified transformation.

Abstract

Unlike the Lorentz transformation which replaces the Galilean transformation among inertial frames at high relative velocities, there seems to be no such a consensus in the case of coordinate transformation between inertial frames and uniformly rotating ones. There have been some attempts to generalize the Galilean rotational transformation to high rotational velocities. Here we introduce a modified version of one of these transformations proposed by Philip Franklin in 1922. The modified version is shown to resolve some of the drawbacks of the Franklin transformation, specially with respect to the corresponding spacetime metric in the rotating frame. This new transformation introduces non-inertial eccentric observers on a uniformly rotating disk and the corresponding metric in the rotating frame is shown to be consistent with the one obtained through Galilean rotational transformation for points close to the rotation axis. Employing the threading formulation of spacetime decomposition, spatial distances and time intervals in the spacetime metric of a rotating observer's frame are also discussed.