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Lorentz force free charged fluids in general relativity 317
their kinematic counterparts; moreover, when an electric field is present
in a rotating frame (this is not the case for the comoving frame of fluid
in our solutions), there appears a kinematic magnetic charge distribution
whose dynamic counterpart in classical theory does not exist (the
magnetic monopole). More on this subject see in [19]. For earlier studies
(in an approximated approach) of the kinematic charges see [7, 11].
3. THE ANSATZ OF ZERO LORENTZ
FORCE IN SPECIAL RELATIVISTIC
ELECTRODYNAMICS
Since a charged particle in absence of Lorentz force in the Minkowski
spacetime should have a straight (geodesic) worldline, this situation is
most naturally described in Cartesian coordinates. We denote them by
and (the same sub/superscripts, when necessary, will be used
as tensor indices). Let all functions depend on only, with a prime
for the corresponding ordinary differentiation. We consider the charged
fluid moving with a constant three–velocity along axis, so that the
four–velocity is
Choosing the electromagnetic field tensor components as
where E and B are magnitudes of electric and magnetic field vectors E
and B in the overall reference frame pointing in the
positive directions of axes and respectively, we may reformulate the
ansatz of zero Lorentz force as
(this is obviously equivalent to existence of only magnetic field in the
local comoving reference frame of the fluid). Since the field tensor (17)
represents a simple bivector, the second invariant of the field vanishes
identically, while the first invariant is, as usual,
It is obvious that the homogeneous system of Maxwell’s equations
(structure equations) is satisfied automatically, while the
inhomogeneous (dynamical) equations read
Here is proper charge density of the fluid. From (18) it
follows that and the natural condition
1 leads to This means that the electromagnetic