The condition of the continuity of at the interface
implies that we must have that
,
which from Equations (19) and (23) gives us the
following relation between the parameters
In addition to this, the condition of the continuity of at
the interface
implies that we must have
, which from
Equations (17) and (23) gives us the following relation
between the parameters
This last condition already determines the integration constant in
terms of the parameters of the problem,
and the difference of the two conditions just obtained determines the
integration parameter in terms of the parameters of the problem,
We have therefore the solution for in the matter region, in
terms of the parameters of the problem,
Let us point out that there is a consistency condition to be applied to
this result, since we must have that the cubic polynomial appearing in the
argument of the logarithm be strictly positive for all values of
within the matter region, that is
for all
. Note that the term with the cubes is
necessarily non-negative, but that the other term may be negative, if
is not smaller than
. Therefore, so long as
,
this strict positivity condition is automatically satisfied. If, however,
we have that
, then the condition must be actively
verified for all
. If it fails, then there is no
solution for that particular set of input parameters.
Since we have written in terms of
, and since we know
the interface boundary conditions for
in limits from within the
matter region, we are in a position to impose the boundary conditions on
across the interfaces, even without having available the complete
solution for
. To this end, let us note that from
Equation (27) we have that
. At the interface
the
condition of the continuity of
implies that we must have
, which from Equations (20)
and (27) gives us the following relation between the parameters,
In addition to this, the condition of the continuity of at the
interface
implies that we must have
, which from Equations (18)
and (27) gives us the following relation between the parameters,
This last condition gives us the integration constant in terms
of the parameters of the problem, and its difference with the previous one
determines the integration constant
,
Note that we have that for any positive values of
and
. This completes the determination of the solution for both
and
in the inner vacuum region, for which we now
have
with given by Equation (31). We also have the following
form for the solution for
within the matter region, still in
terms of
,
At this point the situation is as follows, in regard to the complete
solution of the problem. Given values of ,
and
,
which completely characterize the geometrical and physical nature of the
object under study, we have the complete solution for both
and
in the outer vacuum region. We also have the complete
solution for both
and
in the inner vacuum region,
except for the determination of the parameter
. We have as well
the complete solution for
in the matter region, again up to
the determination of the parameter
. The one element of the
solution still missing is the complete solution for
in the matter
region. However, since we have
determined in terms of
in
this region, this can also be accomplished by the complete determination
of
in this region, which is the task we tackle next. Let us
emphasize that the parameter
is not a free input parameter of
the problem, since it must be chosen so that the given value of
results, that is, the local value of the energy density must be chosen so
that the given value of the asymptotic gravitational mass
results at
radial infinity.