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

The superposition of null dustbeams in General Relativity 59
For matching it is convenient to use the physical distance L from the
symmetry axis as the radial coordinate in the interior and exterior metrics
instead of the coordinates and which are related to L by the
relations
The boundary conditions at the surface imply the relation
between the real parameters in the interior, in the exterior, and the
radius of the boundary surface.
3.3. THE STATIONARY SOLUTION
Now we will treat the stationary case when the Killing vectors
(timelike) and (spacelike) are not hypersurface–orthogonal. In this
case we can start with the non–diagonal line element
The derivation of the stationary solution is very similar to the procedure
applied in the static case. The resulting stationary metric
can be obtained from its static counterpart (18) with the aid of the
complex substitution
The interior solution (27) can be matched to an exterior vacuum solution
which is also related to (24) by a complex substitution.
The two solutions (18) and (27) describe physically different situations.
In both cases the photons which form the source of the gravitational
field move along screwed lines on the cylinder mantle (
and have no ) and surround the axis However,
in the static case the photons of the beam with propagation vector
and those with move on the same trajectories in opposite directions
whereas in the stationary case they move on different screwed curves
with the same sense of revolution. Thus the total rotation of the source
is zero because of compensation in the static solution (18) and it does