Solution in the Matter Region

In the matter region Equation (4) for $\lambda(r)$ can be written as

  $$\left[r\,{\rm e}^{-2\lambda(r)}\right]' = 1-\kappa\rho_{0} r^{2},$$ (21)

which can be immediately integrated to

  $$\,{\rm e}^{-2\lambda(r)} = 1 + \frac{B}{r} - \frac{\kappa\rho_{0}}{3}\,r^{2},$$ (22)

where $B$ is an integration constant with dimensions of length, thus leading to the general solution for $\lambda(r)$ in the matter region,

  $$\lambda_{m}(r) = -\, \frac{1}{2}\, \ln\!\left(1+\frac{B}{r}-\frac{\kappa\rho_{0}}{3}\,r^{2}\right),$$ (23)

where the subscript $m$ denotes the matter region. This solution contains one integration constant, the constant $B$, and one parameter characterizing the system, namely $\rho_{0}$, which is not, however, a free input parameter of the problem, since it will depend on $M$ and thus on $r_{M}$.

In order to deal with $\nu(r)$ in the matter region, we consider the consistency condition given in Equation (7), which can be written in this region as

  $$\nu'(r) = -\, \frac{P'(r)}{\rho_{0}+P(r)},$$ (24)

thus allowing us to separate variables and hence to write $\nu(r)$ in terms of $P(r)$,

$$d\nu = -\, \frac{dP}{\rho_{0}+P}$$

$$= -d\ln\!\left(\rho_{0}+P\right).$$ (25)

If we integrate from the left end $r_{1}$ of the matter interval to a generic point $r$ within that interval, we get

  $$\nu(r)-\nu(r_{1}) = - \ln\!\left[\frac{\rho_{0}+P(r)}{\rho_{0}+P(r_{1})}\right].$$ (26)

However, the boundary conditions for $P(r)$ at the interfaces tell us that we must have $P(r_{1})=0$, and hence we get the general solution for $\nu(r)$ within the matter region, written in terms of $P(r)$,

  $$\nu_{m}(r) = \nu_{1} - \ln\!\left[\frac{\rho_{0}+P(r)}{\rho_{0}}\right],$$ (27)

where $\nu_{1}=\nu(r_{1})$. The solutions for $\lambda(r)$ and $\nu(r)$ within the matter region involve therefore two integration constants, $B$ and $\nu_{1}$. The solution for $\nu(r)$ is not yet completely determined, since it is given in terms of $P(r)$, which is also as yet undetermined. However, the information obtained so far already allows us to impose the boundary conditions at the interfaces, in order to determine the integration constants, which is what we turn to now.