Central to Irvine’s application of the OT method was the concept of the observer’s reference frame which, like the Lorentz frame of Special Relativity, has spatial axes that are mutually orthogonal and orthogonal to the tangent of the world line of the observer with a diagonal metric as in Equation (

6). An orthonormal set of basis vectors are thus defined. Clearly the metric of the rotating coordinate system given in Equation (

23) is not diagonal. Schiff’s analysis therefore derives a form of the electromagnetic fields in the

rotating coordinate system, but this is not the frame of a physical observer and therefore the fields cannot therefore be considered to be observable. As with Special Relativity the problem of the selection of the covariant or contravariant form of the Maxwell tensor does not arise in the OT method as there is no real distinction between these in a frame described by Orthogonal Tetrads. A system of reference is selected in an inertial frame and scale factors defined by a matrix

${\underline{\lambda}}_{\left(a\right)}^{i}$ are chosen that can be used to convert local tensor quantities into their tetrad scalar components. The index in brackets refers to the component in the OT frame, known as the Lorentz index, whereas the other is the tensor index that has been used elsewhere in this paper. For example, if we work in a specific spherical polar coordinate system in an inertial frame, we would transform the Maxwell tensor given in Equation (

A8) using the following matrix,

We can then determine the form of the Maxwell tensor in the OT inertial frame as

or, writing the components as a matrix, we find,

which recalls Equation (

5). To define a reference frame of an observer moving with respect to this inertial system we need to associate with that observer an orthonormal set of spacetime coordinate axes. This is done by solving a set of differential equations, (the Frenet-Serret equations), that match the time axis of the OT frame with the 4-velocity of the observer. This process involves finding the derivatives of the Christoffel symbols in the coordinate system used. This results in a transformation that is essentially an instantaneous Lorentz transformation. We can transform a covector, for example, in the

${\underline{\lambda}}_{\left(a\right)}^{i}$ OT system using the matrix

Notice the similarity of this to Equation (

15) but with

$\gamma $ and

$\beta $ now defined in Equations (

24) and (

25). We can find the OT form of the Maxwell tensor in the reference frame that accompanies the observer as

and this is analogous to the doubly covariant tensor transformation leading to Equation (

16) and it is not, therefore, surprising that the result is analogous to Equation (

17). We can then find the following relationships between the electric field in the inertial frame and the rotating reference frame of an observer,

Comparison with Equation (

28) shows that the

$\widehat{r}$ and

$\widehat{\theta}$ components of the electric fields predicted by the OT method differ from those predicted by Schiff’s method only by a factor of

$\gamma $. The explicit expressions for the electric fields due to the rotating shells can therefore be easily found from Equations (

9)–(

14). In contrast to Schiff’s method, the OT method predicts that the magnetic field no longer remains unchanged in the rotating system. We can write,

The relationships given in Equations (

42) and (

43), are formally identical with the standard results of Special Relativity as derived in (

17) but the components of the electromagnetic fields are now given in terms of the spherical polar rather than cartesian coordinates and the parameter

$\gamma $ is now in terms of the rotational speed rather than a linear one. The spherical polar coordinates are defined in the reference system in the inertial frame. To derive the form of Maxwell’s equations given in (

2) we need to express the fields and sources in terms of a coordinate system that co-rotates with the non-inertial observer (local coordinates). This is accomplished by first transforming

${F}_{\mu \nu}$ in Equation (

5) into the rotating coordinate system to find Equation (

26), in the way employed by Schiff. Now we use another transformation, which is again based on the Frenet-Serret equations, to obtain the OT components of the fields. This gives the following transformations,

and

where we denote the fields in the observer’s frame measured with local coordinates with the subscript

o and have defined them in terms of the fields defined in Equations (

42) and (

43) which are the field measured with the inertial coordinates. We see that to first order in

$\upsilon /c$ the fields in each of these frames are equal. We refer the reader to the original papers [

2,

17] for more details of the OT method and we note also that Corum used the OT method to derive equations for the fields due to a charged shell observed by a co-rotating observer using the coordinates defined in the inertial reference frame [

17].