Using the above Lorentz transformation matrix, we may obtain the following transformation equations for quaternionic four-velocity
(W) of dyonic cold plasma which are an analogous to quaternionic potentials of dyons; i.e.,
the nonrelativistic four-velocity
(1, v), which is the time derivative of the four vector (t, r(t)), is an objective, observer independent quantity contrary to its spacelike projection, the relative velocity v .
The first one uses timelike observers with four-velocity [u.sup.a] and equates two quantities where this observer measures in the geometric as well as in the matter sector, leading to a scalar equation
Here [kappa] stands for the gravitational constant in four dimensions and enforcing this equation for all observers with four-velocity leading to Einstein's equations [G.sub.ab] = [kappa][T.sub.ab].
As a first step one must realize that the right hand side of Poisson's equation is intrinsically observer dependent, as it is the energy density of some matter field measured by an observer with some four-velocity [u.sub.i].
To derive the left hand side, that is, analogue of [[nabla].sup.2][phi], we note that this requires deriving a scalar object which involves two spatial derivatives of the metric (since this is what is responsible for the [[nabla].sup.2][phi] term in nonrelativistic limit) and necessarily depends on the four-velocity [u.sup.i].
where [U.sup.[mu]] is the unit timelike four-velocity vector and [[zeta].sub.[mu]] is the unit spacelike vector along the radial direction r.
For metric (1), the timelike unit four-velocity vector [U.sup.[mu]] is defined by
The kinematic parameters, the expansion [THETA], the acceleration vector [[??].sup.[mu]], the shear tensor [[sigma].sub.[mu]v], and the vorticity tensor [[omega].sub.[mu]v] associated with the fluid four-velocity vector are defined by