On Franklin’s relativistic rotational transformation and its modification

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.

Fermi coordinates and modified Franklin transformation: a comparative study on rotational phenomena

“…The common practice in treating rotational phenomena is the employment of the so-called Galilean rotational transformation (GRT) between the rotating and non-rotating frames. Noting that in a rotating frame there are both inertial (centric) and non-inertial (eccentric) observers, one should be cautious with the restrictions in applying GRT which is only applicable to the former [3]. By the same token, its application to eccentric observers is also questionable on the grounds that for relativistic rotational velocities one requires a relativistic rotational transformation (RRT), very much in the same way as Lorentz transformation replacing the Galilean transformation among inertial frames at relativistic velocities.…”

The Unruh effect for eccentric uniformly rotating observers

“…But as we have discussed [19], FTs have all kinematical problems of GRT and can not be applied for relating eccentric rotating detector to centric laboratory observer. In addition, in [15] the Klein-Gordon's solution that has given in rotating frame is coordinate transformed solution of the inertial one.…”

“…In [14,15]using Trocheries-Takeno transformations, which we called Franklin transformations (FT) [19], it is shown that the rotating observer defines a vacuum state which is different from the Minkowski one. But as we have discussed [19], FTs have all kinematical problems of GRT and can not be applied for relating eccentric rotating detector to centric laboratory observer.…”

“…In [19], looking for a consistent relativistic rotational transformation between an inertial observer and an observer at a non-zero radius (eccentric observer) on a uniformly rotating disk, the following transformations (in cylindrical coordinates) were introduced…”

“…In addition to the usual ambiguities related to the rotating observers [19], the particle detection due to acceleration of rotational motion is also controversial and the agreement between canonical approach and detector approach seems not to occur for this observer. The problem of finding true rotational transformation between the rotating and non-rotating frames is one important aspect of conflict.…”