Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

318 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
field has here to be of purely magnetic type, having the meaning of the
(constant) proper magnetic field intensity. Then it is natural to assume
so that and
There are only two natural approaches to solve the problem under
consideration: either (i) one may choose the behaviour of and find
via straightforward integration of (19), or (ii) postulate the velocity
distribution and find merely by a differentiation. Thus our task
proves to be a trivial one. The general properties of the fluid now are
formulated as follows: Its particles are moving inertially (the velocity is
constant in the direction of motion though it depends on the concrete
layer, thus being a function of only); if the velocity vanishes on two
opposite boundaries, the charge density nearby has to take opposite
signs; finally, the constant proper magnetic field is directed along axis.
As illustrations, we just mention four special solutions:
1.
2. when e.g.,
Then,
3. now,
4. The next special case meets the following alternative condition:
This solution can be easily joined with a constant sourceless magnetic
field (the exterior solution) at one or two (not necessarily consecutive)
nodes.
4. CONCLUSIONS
We have shown in this communication that LFFS exist not only in
general relativity, but also in flat spacetimes and even in the case when
charged fluids are nonrelativistic. These latter cases are of importance
since then a superposition of electromagnetic fields is applicable (the
Maxwell theory is completely linear, and there is no self–interaction of
electromagnetic fields due to their contribution to curvature characteristic
for general relativity). In these flat–spacetime solutions the Lorentz
force is compensated by influence of a source–free (usually, purely magnetic)
field. Our interpretation of the general relativistic LFFS is that
they should contain these source–free electromagnetic part which cannot
be separated from that created by charged fluid proper in the nonlinear
realm of general relativity. Even an overcompensation of the Lorentz