Property (3).

The third important property of the solutions is that, if there is a radial position $\xi=\xi_{b}$ where $\pi(\xi)=0$, then Equation (37) implies that $\pi'(\xi)=0$ at that radial position. Since the equation for $\gamma (\xi )$ is a second-order one, and in this situation we have that both $\pi'(\xi)=0$ and $\gamma'(\xi)=0$ when we approach that radial position integrating from one side, then on the other side of that radial position we will have that both $\pi (\xi )$ and $\gamma (\xi )$ are constant, and in particular that $\pi(\xi)=0$ is the null constant. Since we have that the energy density $\rho (\xi )$ is proportional to $\pi (\xi )$, this implies that on that other side we have a vacuum, so that the radial position $\xi_{b}$ represents the boundary of the matter. Note that this also implies that, if there is such a radial position $\xi_{b}$, then we must conclude that the dynamics of the system leads to the formation of a sharp boundary for the matter, even in the case of a gas, so long as it satisfies the polytropic equation of state exactly.

Let us now consider the consequences of Equation (37) regarding the manner in which such a radial position $\xi_{b}$ is approached by $\pi (\xi )$. We must consider separately the cases in which the radial position is approached from the left and from the right. We start by the case in which the radial position $\xi_{b}$ is approached by the integration process from the right, from values $\xi>\xi_{b}$. We will assume that the function $\pi (\xi )$ goes to zero at $\xi_{b}$ as a power $m\geq 1$, and verify whether we can find a solution of Equation (37) in a right-neighborhood of $\xi_{b}$. Since we know that $\pi (\xi )$ must be real and positive, while $m$ is not necessarily an integer, this means that we should assume that in a right-neighborhood of $\xi_{b}$ we have

  $$\pi(\xi) = B_{\oplus}^{m}(\xi-\xi_{b})^{m},$$ (54)

where we must have that $B_{\oplus}>0$ and that $m\geq 1$, given that $\pi (\xi )$ is real and non-negative and that $\pi '(\xi )$ must exist. The question is then whether or not we can find values of $m$ and $B_{\oplus}$ such that the equation is satisfied in that neighborhood. If we denote the difference shown by $\delta\xi=\xi-\xi_{b}$, we may then write for all the quantities involved in Equation (37), in a right-neighborhood of $\xi_{b}$,

$$\gamma(\xi) = \gamma(\xi_{b}) + \frac{B_{\oplus}^{m}}{m+1}\,(\delta\xi)^{m+1},$$ (55)

$$\pi(\xi) = B_{\oplus}^{m}(\delta\xi)^{m},$$ (56)

$$\pi'(\xi) = mB_{\oplus}^{m}(\delta\xi)^{m-1},$$ (57)

$$F(\xi,\pi) = D_{\oplus}\xi^{-2/n}(\delta\xi)^{m/n},$$ (58)

where $D_{\oplus}=CB_{\oplus}^{m/n}$. Substituting all the quantities in Equation (37), recalling that we are interested only in the $\delta\xi\to 0$ limit, assuming that $\gamma(\xi_{b})\neq 0$, which allows us to use for $\gamma (\xi )$ just its dominant term $\gamma_{b}=\gamma(\xi_{b})$, and dropping negligible terms where possible, we get

  $$m = -\, \frac{n}{n+1}\, \frac {\xi_{b}^{2/n-1}\gamma_{b}} {2D_{\oplus}(\xi_{b}-\gamma_{b})}\, (\delta\xi)^{1-m/n}.$$ (59)

It then follows from the $\delta\xi\to 0$ limit that the only possible solution with $m\neq 0$ is $m=n$, thus confirming that $m\geq 1$. Recalling the value of $D_{\oplus}$, we then have for $B_{\oplus}$

  $$B_{\oplus} = \frac{\xi_{b}^{2/n-1}}{2C(\xi_{b}-\gamma_{b})}\, \frac{(-\gamma_{b})}{n+1}.$$ (60)

Note that, since $B_{\oplus}$ must be strictly positive, it follows from this that we must have $\gamma_{b}<0$, thus confirming a posteriori that $\gamma_{b}\neq 0$, so long as we are not dealing with just the identically null solution for $\pi (\xi )$. Since in the resulting internal vacuum we have that $\gamma_{b}=-\xi_{\mu}=-r_{\mu}/r_{0}$, we once more conclude that we must have $r_{\mu}>0$, just as we did in [#!LiquidShells!#], but in a very different way. We may also identify that in this case we have $\xi_{b}=\xi_{1}$, the inner boundary of the matter, and hence we may write that

  $$B_{\oplus} = \frac{\xi_{1}^{(2-n)/n}}{2C(n+1)}\, \frac{\xi_{\mu}}{\xi_{1}+\xi_{\mu}}.$$ (61)

We now consider the case in which the radial position $\xi_{b}$ is approached by the integration process from the left, from values $\xi<\xi_{b}$. Again we will assume that the function $\pi (\xi )$ goes to zero at $\xi_{b}$ as a power $m\geq 1$, and verify whether we can find a solution of Equation (37) in a left-neighborhood of $\xi_{b}$. Since we know that $\pi (\xi )$ must be real and positive, while $m$ is not necessarily an integer, this means that this time we should assume that in a left-neighborhood of $\xi_{b}$ we have

  $$\pi(\xi) = B_{\ominus}^{m}(\xi_{b}-\xi)^{m},$$ (62)

where once again we must have that $B_{\ominus}>0$ and that $m\geq 1$. If we now denote the difference shown by $\delta\xi=\xi_{b}-\xi$, we may then write for all the quantities involved in Equation (37), in a left-neighborhood of $\xi_{b}$,

$$\gamma(\xi) = \gamma(\xi_{b}) - \frac{B_{\ominus}^{m}}{m+1}\,(\delta\xi)^{m+1},$$ (63)

$$\pi(\xi) = B_{\ominus}^{m}(\delta\xi)^{m},$$ (64)

$$\pi'(\xi) = -mB_{\ominus}^{m}(\delta\xi)^{m-1},$$ (65)

$$F(\xi,\pi) = D_{\ominus}\xi^{-2/n}(\delta\xi)^{m/n},$$ (66)

where $D_{\ominus}=CB_{\ominus}^{m/n}$. As before, substituting all the quantities in Equation (37), recalling once again that we are interested only in the $\delta\xi\to 0$ limit, assuming that $\gamma(\xi_{b})\neq 0$, which allows us to use for $\gamma (\xi )$ just its dominant term $\gamma_{b}=\gamma(\xi_{b})$, and dropping negligible terms where possible, we get

  $$m = \frac{n}{n+1}\, \frac {\xi_{b}^{2/n-1}\gamma_{b}} {2D_{\ominus}(\xi_{b}-\gamma_{b})}\, (\delta\xi)^{1-m/n}.$$ (67)

It follows once more from the $\delta\xi\to 0$ limit that the only possible solution with $m\neq 0$ is $m=n$, once again confirming that $m\geq 1$. Recalling the value of $D_{\ominus}$, we then have for $B_{\ominus}$

  $$B_{\ominus} = \frac{\xi_{b}^{2/n-1}}{2C(\xi_{b}-\gamma_{b})}\, \frac{\gamma_{b}}{n+1}.$$ (68)

Note that, since $B_{\ominus}$ must be strictly positive, in this case it follows that we have $\gamma_{b}>0$, once again confirming a posteriori that $\gamma_{b}\neq 0$. Besides, the quantity $\gamma_{b}$ can be written in terms of the parameters of the resulting external vacuum as $\gamma_{b}=\xi_{M}=r_{M}/r_{0}$, which also implies that it cannot be zero. We may also identify that in this case we have $\xi_{b}=\xi_{2}$, the outer boundary of the matter, and hence we may write that

  $$B_{\ominus} = \frac{\xi_{2}^{(2-n)/n}}{2C(n+1)}\, \frac{\xi_{M}}{\xi_{2}-\xi_{M}},$$ (69)

where we necessarily have that $\xi_{2}>\xi_{M}$ since the integration cannot produce the singularity of a horizon. This gives us the exact asymptotic behavior of $\pi (\xi )$, and hence also of $\gamma (\xi )$, as we approach the radial positions $\xi_{1}$ and $\xi_{2}$ that delimit the region containing the matter, from within that region.

Whether or not these points exist in each particular case, for various values of the parameters that define the physical system, has to be determined numerically. If they do, then there is more that can be established about $\gamma (\xi )$. In this case, not only is this a monotonically increasing function that is limited from above, but we now know that it is also limited from below, since it is a constant within the inner vacuum region. In fact, it must go from a constant negative value within the inner vacuum region, to a constant positive value in the outer vacuum region. In addition to this, so long as $n>1$ it is a continuous and differentiable function on the whole positive real semi-axis. In particular, it follows from this that $\gamma (\xi )$ must have a single zero in the open interval $(\xi_{1},\xi_{2})$, which therefore is always within the matter region. This completes the discussion of the third property.

It is worth the trouble exploring now some of the consequences of the facts just established about $\gamma (\xi )$ and $\pi (\xi )$. Note that these two boundary points are soft singular points of both functions $\gamma (\xi )$ and $\pi (\xi )$, since these functions cannot be analytic at these points. This follows from the fact that they are constant on one side of the points and behave as a strictly positive power on the other side. Due to this there is no power series centered at $\xi_{b}$ that can converge to these functions on a neighborhood of these points, and therefore they are not analytic. However, both functions are still well-defined at the boundary points, thus characterizing the singularities as soft ones.

Added to the fact that the position of the boundary points is not known beforehand, this makes it impossible to integrate the equation numerically starting at these points. Since the equation for $\gamma (\xi )$ is a second-order one, and since in this situation we have that both $\pi'(\xi)=0$ and $\gamma'(\xi)=0$ hold at the positions of the boundaries, if we try to integrate from these points into the matter region we will simply get a constant for $\gamma (\xi )$ and the constant value $\pi(\xi)=0$, which corresponds to a vacuum rather than to the matter that is there. Therefore, the solution within the matter region is not determined by its values at the boundaries, and hence this problem cannot be characterized as a typical boundary value problem.

Since $n>1$, the two functions $\pi (\xi )$ and $\gamma (\xi )$ and the second derivative $\pi'(\xi)=\gamma''(\xi)$ will all be continuous at these boundary points, but there will be some higher-order derivative that does not exist there. For example, if $n=3/2$, which is typical of the convective layer of a star, then $\pi '(\xi )$ will not be differentiable at these points, although it is still continuous there, so that the third derivative $\pi''(\xi)$ of $\gamma (\xi )$ will not exist, since it diverges at these points. As another relevant example, if $n=3$, which is typical of the radiative layer of a star, then the fourth derivative $\pi^{'''}(\xi)$ of $\gamma (\xi )$ will be discontinuous at these points, so that its fifth derivative $\pi^{''''}(\xi)$ will not exist there.

This same singular character of the interface boundary points has the global consequence that, given definite boundary conditions at radial infinity, the complete solution of the problem is not determined, and is therefore not unique. On the one hand, if one integrates inwards from a point $\xi>\xi_{2}$, then one produces just a continuation of the exterior Schwarzschild solution of the outer vacuum region, until one reaches its horizon, and never any solution associated to the matter. On the other hand, if one integrates outwards from a point $\xi<\xi_{2}$, then there will be many sets of values of the parameters that describe the state and character of the matter which are such that the solution arrives at $\xi_{2}$ with $\pi(\xi_{2})=0$ and $\gamma(\xi_{2})=\xi_{M}$. There are therefore many interior solutions that correspond to the same exterior solution.

The numerical analysis indicates that the boundary points $\xi_{1}$ and $\xi_{2}$ exist for all values of the parameters of the system, within a wide range of variation of these parameters, within physically reasonable bounds. It seems to indicate that, if a static solution exists at all, then it has the property that these points are present. Next we will describe in detail the numerical approach that leads to this conclusion.