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.