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

60 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
not vanish in the stationary solution (27). In any case rotational effects
are responsible for balancing the gravitational attraction such that
time–independent configurations can exist at all and mutual focusing
does not occur.
The derivation of the exact null fluid solutions (18) and (27) was
first given in our paper [7]. For a discussion and illustration of the
trajectories (screwed motion), see [9]. Therein Fermi coordinates along
the world line of a fixed point on the cylinder mantle are introduced to
confirm the interpretation of the photon trajectories.
3.4. OTHER SOLUTIONS
We have mentioned that the quantity S defined in (14) factorizes for
K = U. This holds also for the special choice K = –2U. In both cases
one can put either of the two factors in S equal to zero. In this way
one obtains four different static solutions which are listed in [8]. The
resulting mass density profiles plotted in [8] are well-behaved only in the
solution treated here.
The attempt to start with a metric admitting an Abelian group
where none of the Killing vectors is hypersurface–orthogonal did not
lead to more general solutions. The existence of three twisting Killing
vectors is not compatible with the structure (13) of the propagation
vectors. Hence there are only two possibilities: the non–twisting Killing
vector is either timelike or spacelike, i.e. the gravitational field is either
static or stationary and the corresponding metrics are (12) and (27),
respectively.
One can interchange the coordinates and in the static solution
(18) to obtain a metric for counter–rotating null dust components, but
then the axis is not regular.
4. DISCUSSION
Under certain conditions the superposition of two null dust components
propagating in opposite spatial directions gives rise to static (or
stationary) gravitational fields in the interaction region. Examples considered
in this article are the static spherically symmetric solution (10)
derived in [6], the static cylindrically symmetric solution (18) derived in
[7], and its stationary counterpart (27). These three metrics are given
in closed form by introducing suitable coordinates.
The superposition of counter–moving pencils of light implies a rotational
effect which compensates for the gravitational attraction to prevent
mutual focusing of the rays. An important point is the fact that
the propagation null vectors of the two null dust components are both