Main Properties of the Solutions

Let us start by recalling that, if it turns out that there are indeed inner and outer vacuum regions, then we already know the form of the solutions in those regions. From the derivations in [#!LiquidShells!#], we have that in the inner vacuum region, where $\xi<\xi_{1}$, the solution is given, in terms of our current set of dimensionless variables, by

$$\lambda_{i}(\xi) = -\, \frac{1}{2}\, \ln\!\left(\frac{\xi+\xi_{\mu}}{\xi}\right),$$ (39)

$$\nu_{i}(\xi) = A + \frac{1}{2}\, \ln\!\left(\frac{\xi+\xi_{\mu}}{\xi}\right),$$ (40)

where $\xi_{\mu}=r_{\mu}/r_{0}$, and where $r_{\mu}$ and $A$ are integration constants, and that in the outer vacuum region, where $\xi>\xi_{2}$, the solution is the exterior Schwarzschild solution, which can be written, in terms of our set of dimensionless variables, as

$$\lambda_{s}(\xi) = -\, \frac{1}{2}\, \ln\!\left(\frac{\xi-\xi_{M}}{\xi}\right),$$ (41)

$$\nu_{s}(\xi) = \frac{1}{2}\, \ln\!\left(\frac{\xi-\xi_{M}}{\xi}\right),$$ (42)

where $\xi_{M}=r_{M}/r_{0}$ and $r_{M}$ is the Schwarzschild radius. The constant $A$ can be determined with the use of the consistency condition and of the interface boundary conditions for $P(\xi)$, which as we saw before imply that we always have $\nu(\xi_{1})=\nu(\xi_{2})$ within the matter region. This fact allows us to determine the value of $A$, using also the fact that $\nu (\xi )$ is a continuous function, thus leading to $\nu(\xi_{1})=\nu_{i}(\xi_{1})$ and $\nu(\xi_{2})=\nu_{s}(\xi_{2})$, which imply that we have

$$\nu_{i}(\xi_{1}) = \nu_{s}(\xi_{2}) \;\;\;\Rightarrow$$ (43)

$$A + \frac{1}{2}\, \ln\!\left(\frac{\xi_{1}+\xi_{\mu}}{\xi_{1}}\right) = \frac{1}{2}\, \ln\!\left(\frac{\xi_{2}-\xi_{M}}{\xi_{2}}\right).$$ (44)

This gives us the solution for $A$ in terms of the other quantities,

  $$A = \frac{1}{2}\, \ln\! \left( \frac{\xi_{1}}{\xi_{2}}\, \frac{\xi_{2}-\xi_{M}}{\xi_{1}+\xi_{\mu}} \right).$$ (45)

We can therefore write out the complete solution in both vacuum regions,

$$\lambda_{i}(\xi) = -\, \frac{1}{2}\, \ln\!\left(\frac{\xi+\xi_{\mu}}{\xi}\right),$$ (46)

$$\nu_{i}(\xi) = \frac{1}{2}\, \ln\! \left( \frac{\xi_{1}}{\xi_{2}}\, \frac{\xi_{2... ...\xi_{\mu}} \right) + \frac{1}{2}\, \ln\!\left(\frac{\xi+\xi_{\mu}}{\xi}\right),$$ (47)

$$\lambda_{s}(\xi) = -\, \frac{1}{2}\, \ln\!\left(\frac{\xi-\xi_{M}}{\xi}\right),$$ (48)

$$\nu_{s}(\xi) = \frac{1}{2}\, \ln\!\left(\frac{\xi-\xi_{M}}{\xi}\right).$$ (49)

The quantities $\xi_{1}$ and $\xi_{2}$ are obtained during the resolution of the differential equation, as are the other two quantities, which once $\gamma (\xi )$ is determined are given by

$$\gamma(\xi_{1}) = -\xi_{\mu},$$ (50)

$$\gamma(\xi_{2}) = \xi_{M},$$ (51)

as consequences of the interface boundary conditions related to the continuity of $\lambda (\xi )$. We may now derive some crucially important properties of the solutions of the field equations analytically, by obtaining these properties directly from the equations.