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

58 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
and in terms of the radial coordinate defined by the resulting
static metric takes the simple explicit form
This solution contains the arbitrary real parameter for the
space–time becomes flat. At the axis the metric is regular; the
first derivatives of all metric functions vanish at the axis and the square
of the Killing vector
is zero at This behaviour is important for the interpretation of the
two–component null dust solution (18) as an axisymmetric gravitational
field. Notice that constants of integration have been chosen such that
the regularity condition at the axis is satisfied.
The null vectors and given by (13) together with
( – – Einstein’s gravitational constant) are geodesic, shearfree, non–
expanding, but twisting. They have zero radial and azimuthal contravariant
components.
An observer at rest in the metric (18), i.e. an observer with the 4–
velocity
measures the energy density
which is positive everywhere, has its maximal value at the axis
decreases monotonically with increasing distance from the symmetry
axis and goes to zero at which corresponds to an infinite physical
distance from the axis. The energy density cannot be
prescribed, it results from our ansatz K = U which makes the field
equations tractable. It is much more difficult to start the integration
procedure with a given
After a determined linear transformation (with constant coefficients)
of the ignorable coordinates our interior solution (18) can be smoothly
matched to the exterior Levi–Civita metric