The Equation for the Pressure

The equation determining the pressure $P(r)$ in the matter region can be obtained by eliminating $\nu'(r)$ from Equations (5) and (7), which gives us

  $$\rho_{0}+P(r)-2\left[rP'(r)\right] = \,{\rm e}^{2\lambda_{m}(r)} \left[1+\kappa r^{2}P(r)\right]\left[\rho_{0}+P(r)\right].$$ (40)

In this equation the quantity $\exp[2\lambda_{m}(r)]$ is a known function, since we have already determined $\lambda(r)$ in the matter region. This is a first-order non-linear differential equation determining $P(r)$, with the boundary conditions $P(r_{1})=0$ and $P(r_{2})=0$. Since the equation is first-order and there are two boundary conditions to be satisfied, it is clear that the parameter $\rho_{0}$ will have to be adjusted so that the second condition can be satisfied. This will therefore determine the parameter $\rho_{0}$ in terms of the other parameters of the problem. This equation can be simplified by a series of transformations on the variables and parameters. First we define the parameter $\Upsilon_{0}$, which has dimensions of inverse length and is such that

  $$\Upsilon_{0}^{2} = \kappa\rho_{0},$$ (41)

and the dimensionless pressure

  $$p(r) = \frac{P(r)}{\rho_{0}},$$ (42)

in terms of which Equation (40) becomes

  $$\left[rp'(r)\right] = \frac{1}{2}\left[1+p(r)\right] \left\{ 1-\,{\rm e}^{2\lambda_{m}(r)}\left[1+\Upsilon_{0}^{2}r^{2}p(r)\right] \right\}.$$ (43)

Substituting the known value of $\lambda_{m}(r)$ from Equation (32) we get

  $$p'(r) = \frac{1}{2r}\left[1+p(r)\right] \frac { \Upsilon_{0... ...} { \Upsilon_{0}^{2} \left(r_{2}^{3}-r^{3}\right)+3\left(r-r_{M}\right) }.$$ (44)

This has the form of a Riccati equation, and can be linearized by the transformation of variables

  $$p(r) = \frac{1}{z(r)}-1,$$ (45)

thus resulting in the equation for $z(r)$,

  $$z'(r) + \frac {\Upsilon_{0}^{2}\left(r_{2}^{3}+2r^{3}\right)-... ...lon_{0}^{2}\left(r_{2}^{3}-r^{3}\right) + 3\left(r-r_{M}\right) \right] }.$$ (46)

This equation has an integrating factor given by $\exp[F(r)]$, where $F(r)$ is defined as an integral of the coefficient of the second term from $r_{2}$ to some arbitrary $r$ within $[r_{1},r_{2}]$,

$$F(r) = \int_{r_{2}}^{r}ds\, \frac {\Upsilon_{0}^{2}\left(r_{2}^{3}+2s^{3... ... \Upsilon_{0}^{2}\left(r_{2}^{3}-s^{3}\right) + 3\left(s-r_{M}\right) \right] }$$

$$= \frac{1}{2} \int_{r_{2}}^{r}ds\, \frac{1}{s} - \frac{1}{2} \int_{... ...{2}+3} { \Upsilon_{0}^{2} \left(r_{2}^{3}-s^{3}\right)+3\left(s-r_{M}\right) }.$$ (47)

One can see now that both integrals can be done, and thus we obtain

  $$e^{F(r)} = \sqrt{\frac{r}{r_{2}}} \; \sqrt { \frac {3\lef... ...{ \Upsilon_{0}^{2} \left(r_{2}^{3}-r^{3}\right)+3\left(r-r_{M}\right) } },$$ (48)

in terms of which Equation (46) for $z(r)$ can be written as

  $$\left[\,{\rm e}^{F(r)}z(r)\right]' = \frac{3}{2}\, \frac {\U... ...{F(r)}} {\Upsilon_{0}^{2}\left(r_{2}^{3}-r^{3}\right)+3\left(r-r_{M}\right)},$$ (49)

which can then be integrated over the interval $[r,r_{2}]$ giving

  $$z(r) = e^{-F(r)}+\frac{3}{2}\,e^{-F(r)}\! \int_{r_{2}}^{r}ds\... ...{F(s)}} {\Upsilon_{0}^{2}\left(r_{2}^{3}-s^{3}\right)+3\left(s-r_{M}\right)},$$ (50)

where we used the fact that by definition $F(r_{2})=0$, and the fact that $P(r_{2})=0$ implies $z(r_{2})=1$.

Note that once more the existence of the solutions for $F(r)$ and for $z(r)$ is conditioned by the strict positivity of the same cubic polynomial that we discussed before in Equation (33), which we can now write as

  $$\Upsilon_{0}^{2}\left(r_{2}^{3}-r^{3}\right) + 3\left(r-r_{M}\right) > 0,$$ (51)

for all $r\in[r_{1},r_{2}]$. Substituting the value of $\exp[F(r)]$ we have the solution for $z(r)$ written in terms of a real integral,

$$z(r) = \sqrt { \frac { \Upsilon_{0}^{2}\left(r_{2}^{3}-r^{3}\right) + 3\left(r-r_{M}\right) } {r} }$$

$$\times \left\{ \sqrt{\frac{r_{2}}{3\left(r_{2}-r_{M}\right)}} + \... ...}\left(r_{2}^{3}-s^{3}\right) + 3\left(s-r_{M}\right) \right]^{3/2} } \right\}.$$ (52)

In most cases this remaining integral is elliptic and therefore cannot be written in terms of elementary functions, so that in general this remaining last step of the resolution procedure has to be performed by numerical means. However, as we are going to show in Section 4, for some values of the parameters it is possible to express this integral in terms of elementary functions.

After determining $z(r)$ in the matter region, Equations (45) allows us to calculate the dimensionless pressure $p(r)$ which, according to Equation (42), is equal to the pressure divided by the energy density $\rho_{0}$,

$$p(r) = \frac{1}{z(r)}-1 \;\;\;\Longrightarrow$$ (53)

$$P(r) = \frac{\rho_{0}}{z(r)}-\rho_{0}.$$ (54)

Note that $z(r)$ also determines $\nu(r)$ in the matter region, since in Equation (39) we have $\nu_{m}(r)$ in terms of $P(r)$, and therefore we have for the exponential of $\nu_{m}(r)$,

  $$\,{\rm e}^{\nu_{m}(r)} = \sqrt{\frac{r_{2}-r_{M}}{r_{2}}}\, \frac{\rho_{0}}{\rho_{0}+P(r)},$$ (55)

which, using Equation (54), implies that

  $$\,{\rm e}^{\nu_{m}(r)} = \sqrt{\frac{r_{2}-r_{M}}{r_{2}}}\, z(r),$$ (56)

so that, up to a constant factor, $z(r)$ turns out to be the square root of the temporal coefficient of the metric. This completes the determination of the solution in all three regions, in terms of the parameters of the problem. Given certain values of $r_{1}$, $r_{2}$ and $r_{M}$, one must still find a value of the parameter $\rho_{0}$, and hence of $\Upsilon_{0}$, such that the boundary conditions for $P(r)$ at the two interfaces are satisfied. One can obtain an equation determining this value of $\Upsilon_{0}$ by considering the value of $z(r_{1})$. Since $P(r_{1})=0$, we have that $z(r_{1})=1$, so that from Equation (52) we get

$$\sqrt{\frac{r_{2}}{3\left(r_{2}-r_{M}\right)}} = \sqrt { \frac{r_{1}} { \Upsilon_{0}^{2}\left(r_{2}^{3}-r_{1}^{3}\right) + 3\left(r_{1}-r_{M}\right) } }$$

$$+ \frac{3}{2} \int_{r_{1}}^{r_{2}}dr\, \frac {\Upsilon_{0}^{2}r^{... ...on_{0}^{2}\left(r_{2}^{3}-r^{3}\right) + 3\left(r-r_{M}\right) \right]^{3/2} }.$$ (57)

The solution of this algebraic equation gives the value of $\Upsilon_{0}$, and hence the value of $\rho_{0}$, for which the two interface boundary conditions for $P(r)$ will be satisfied. The solution of this equation necessarily includes the consistency check of the solution obtained, since the calculation of the integral is dependent on the strict positivity of the polynomial in Equation (51), for all $r$ within $[r_{1},r_{2}]$. This is the same condition that guarantees the consistency of the results for $F(r)$ and $z(r)$, and hence the consistency of the results for $P(r)$ and $\nu(r)$ within the matter region.