Final Component Equation

Up to this point in our development all the quantities related to our new solution have been left written in terms of the single function $\beta (r)$. The only equation still to be satisfied is the $\theta$ component of the field equation, shown in Equation (58), the one that, therefore, will become the equation that determines $\beta (r)$. The rather long detailed calculations needed in order to write that equation explicitly in terms of $\beta (r)$, as well as its subsequent simplification, can be found in Subsection A.2.2 of Appendix A. The result is a rather complicated non-linear differential equation that, together with the limiting conditions discussed before, determines the function $\beta (r)$, for all $r$, and that for convenience we will write with an extra overall factor of $r_{M}$, as

$$2\omega r_{M} \left[ r - r_{M}\beta(r) \right] \left\{r\left[r\beta'(r)\right]'\right\} + \omega(1+\omega) r_{M}^{2} \left[r\beta'(r)\right]^{2} +$$

$$- 6\omega r r_{M} \left[r\beta'(r)\right] + (1+7\omega) r_{M}^{2} \beta(r) \left[r\beta'(r)\right] = 0,$$ (74)

with the boundary conditions at $r=0$ and the asymptotic conditions for $r\to\infty$ given by

$$\beta'(0) = 0,$$

$$\beta''(0) = 0,$$

$$\lim_{r\to\infty}\beta(r) = 1,$$

$$\lim_{r\to\infty}\left[r\beta'(r)\right] = 0,$$

$$\lim_{r\to\infty}\left[r^{2}\beta''(r)\right] = 0.$$ (75)

It is an interesting fact that this same equation for $\beta (r)$, given in Equation (74), can also be derived from the consistency condition shown in Equation (27). This serves as an independent verification of the correctness of the derivation mentioned above and detailed in Appendix A. The detailed calculation for this second derivation can be found in Subsection A.2.3 of Appendix A. If we use in the consistency condition the results obtained before for $\nu'(r)$ and $\mathfrak{T}(r)$, we find that it becomes exactly the same expression shown in Equation (74). Therefore, once we have determined $\beta (r)$ we will have satisfied all the relevant equations, including all the component equations and the consistency condition as well.

We can simplify Equation (74) somewhat by means of a simple change of variables. For this purpose we define a new dimensionless radial variable by $\xi=r/r_{0}$ where $r_{0}$ is simply an arbitrary reference point for measuring the radial positions. We then have the particular value $\xi_{M}=r_{M}/r_{0}$, which corresponds therefore to the total mass $M$. It would also be possible to define another particular value, given by $\xi_{S}=r_{S}/r_{0}$, but this value will not appear in the equations, due to the difficulty in defining in general, and with precision, the position $r_{S}$ of the sphere that contains essentially all the fluid matter. It is not difficult to see that the definition of $\xi$ implies for the corresponding derivatives that we have

$$r\partial_{r} = \xi\partial_{\xi}.$$ (76)

With this, and using, from now on, the primes to denote derivatives with respect to $\xi$, we can write our final equation in the form

$$2\omega \xi_{M} \left[ \xi - \xi_{M}\beta(\xi) \right] \left\{\xi... ...ight]'\right\} + \omega(1+\omega) \xi_{M}^{2} \left[\xi\beta'(\xi)\right]^{2} +$$

$$- 6\omega \xi \xi_{M} \left[\xi\beta'(\xi)\right] + (1+7\omega) \xi_{M}^{2} \beta(\xi) \left[\xi\beta'(\xi)\right] = 0.$$ (77)

We may further simplify this equation by defining a new dimensionless function $\gamma (\xi )$, in terms of the equally dimensionless function $\beta(\xi)$, by

$$\gamma(\xi) = \frac{r_{M}}{r_{0}}\,\beta(\xi)$$

$$= \xi_{M}\beta(\xi).$$ (78)

Therefore, we get for our final equation, now for the function $\gamma (\xi )$,

$$2\omega \left[ \xi - \gamma(\xi) \right] \left\{\xi\left[\xi\gamma'(\xi)\right]'\right\} + \omega(1+\omega) \left[\xi\gamma'(\xi)\right]^{2} +$$

$$- 6\omega \xi \left[\xi\gamma'(\xi)\right] + (1+7\omega) \gamma(\xi) \left[\xi\gamma'(\xi)\right] = 0,$$ (79)

where we see that the parameter $\xi _{M}$ no longer appears in the equation, which now depends only on the parameter $\omega$. On the other hand, $\xi _{M}$ now appears in one of the relevant boundary and asymptotic conditions on $\gamma (\xi )$. The boundary conditions at $\xi=0$ and the $\xi\to\infty$ asymptotic conditions for $\gamma (\xi )$ are given by

$$\gamma'(0) = 0,$$

$$\gamma''(0) = 0,$$

$$\lim_{\xi\to\infty} \gamma(\xi) = \xi_{M},$$

$$\lim_{\xi\to\infty} \left[\xi\gamma'(\xi)\right] = 0,$$

$$\lim_{\xi\to\infty} \left[\xi^{2}\gamma''(\xi)\right] = 0.$$ (80)

Note that with these changes of variable we have clearly separated the roles of the parameter $\xi _{M}$, which represents here the total mass $M$, and of the parameter $\omega$, which represents here the state of the fluid matter. While the parameter $\omega$ appears in the differential equation and therefore modulates the propagation of the solution along $\xi$, the parameter $\xi _{M}$ appears only in the asymptotic condition involving $\gamma (\xi )$. We may now write all our previous results in terms of $\gamma (\xi )$, rather that $\beta (r)$,

$$\mathfrak{T}(\xi) = \gamma'(\xi),$$ (81)

$$\,{\rm e}^{2\lambda(\xi)} = \rule{0em}{5ex} \frac{\xi}{\xi-\gamma(\xi)},$$ (82)

$$\left[\xi\nu'(\xi)\right] = \frac{1}{2}\, \frac {\gamma(\xi)+\omega\left[\xi\gamma'(\xi)\right]} {\xi-\gamma(\xi)},$$ (83)

$$\left[\xi\lambda'(\xi)\right] = -\, \frac{1}{2}\, \frac {\gamma(\xi)-\left[\xi\gamma'(\xi)\right]} {\xi-\gamma(\xi)},$$ (84)

$$\xi\left[\xi\nu'(\xi)\right]'(\xi) = \frac{1}{2}\, \frac... ... \gamma(\xi) \end{array} \right) } {[\xi-\gamma(\xi)]^{2}}.$$ (85)

For use in the arguments that follow, it is important to record here that Equation (79) can also be written in the form

$$2\omega \xi^{2} \left[ \xi - \gamma(\xi) \right] \gamma''(\xi) + \omega(1+\omega) \xi^{2} \left[\gamma'(\xi)\right]^{2} +$$

$$- 4\omega \xi^{2} \gamma'(\xi) + (1+5\omega) \xi \gamma(\xi) \gamma'(\xi) = 0,$$ (86)

where we have used the fact that

$$\xi \left[ \xi\gamma'(\xi) \right]' = \xi^{2}\gamma''(\xi) + \xi\gamma'(\xi).$$ (87)

We must now consider how to solve the equation for the function $\beta (r)$ that appears in the invariant interval, or equivalently the equation for $\gamma (\xi )$ given in Equations (79) or (86). For the time being the complete determination of $\gamma (\xi )$ has to be made numerically, but enough can be established about the properties of the solutions to give us a fairly complete picture of the physics involved. Given the general qualitative behavior of the function $\beta (r)$, which we already know, and given the simple relation between $\beta (r)$ and $\gamma (\xi )$, we can state at once that the function $\gamma (\xi )$ must have the general qualitative behavior shown in Figures 2 and 3. Just like $\beta (r)$, the function $\gamma (\xi )$ must have non-negative derivative everywhere, must have a finite negative value at $\xi=0$ and a finite positive limit when $\xi\to\infty$.

Figure 2: The qualitative behavior of the dimensionless function $\gamma (\xi )$, showing the main features: the inflection point $\xi _{i}$, the root $\xi _{r}$, the critical point $\xi _{c}$ and the asymptotic limit $\xi _{M}$; the identity function $\xi$ is also shown; the configuration show is that of the middle-density regime.

Certain special values of $\xi$ play an important role in this analysis. As shown in the figures, $\xi _{i}$ is the point of inflection of the function $\gamma (\xi )$, where we have that $\gamma''(\xi_{i})=0$, and which will play an important role in the description and classification of the solutions. The point $\xi _{r}$ is the root of the function $\gamma (\xi )$, the single point where we have that $\gamma(\xi_{r})=0$. The critical point $\xi _{c}$ is the point where $\exp[2\lambda(\xi)]$ may develop a singularity, and it is the point which may become an event horizon in a certain limit. It is defined as the point where $\gamma'(\xi_{c})=1$, and it is also the point where the difference $[\xi-\gamma(\xi)]$ has its minimum value. This is a consequence of the fact that the singularity in the metric factor $\exp[2\lambda(\xi)]$ comes about when the graph of $\gamma (\xi )$ just touches the graph of the identity function, so that $[\xi_{c}-\gamma(\xi_{c})]$ becomes zero.

Figure 3: The qualitative behavior of the dimensionless function $\gamma (\xi )$, showing the main features: the root $\xi _{r}$, the inflection point $\xi _{i}$, the critical point $\xi _{c}$ and the asymptotic limit $\xi _{M}$; the identity function $\xi$ is also shown; the configuration show is that of the high-density regime.

We know that the function $\gamma (\xi )$ has non-negative derivative at all points, and that it goes from some negative finite value at $\xi=0$ to some positive finite value for $\xi\to\infty$. It follows that the function cannot have any local maxima or minima and that it must have at least one inflection point. In fact, as we will see in what follows, it has exactly one inflection point. At this inflection point we have that $\gamma''(\xi_{i})=0$, and that the functions $\gamma(\xi_{i})$ and $\gamma'(\xi_{i})$ have some definite values. In principle, since the differential equation determining $\gamma (\xi )$ is of the second order, given the values of these two functions at this point, a solution for $\gamma (\xi )$ is completely determined. Interestingly, we can easily obtain from the equation itself a definite relation between these two values, leaving us with only one arbitrary parameter for the determination of the solution. We can write an equation for the inflection point $\xi _{i}$ of the function $\gamma (\xi )$, by simply putting $\gamma''(\xi_{i})=0$ in Equation (86), which then results in

$$\left[ \rule{0em}{2.5ex} - 4\omega \xi_{i}^{2} + (1+5... ...xi_{i}^{2} \gamma'(\xi_{i}) \right] \gamma'(\xi_{i}) = 0.$$ (88)

Since at the inflection point $\xi_{i}\neq 0$, this implies that either we have that $\gamma'(\xi_{i})=0$, which in fact only happens at $\xi=0$ and for $\xi\to\infty$, or we must have that

$$\omega(1+\omega) \xi_{i} \gamma'(\xi_{i}) = 4\omega \xi_{i} - (1+5\omega) \gamma(\xi_{i}).$$ (89)

This equation involves $\omega$, $\xi _{i}$, $\gamma(\xi_{i})$ and $\gamma'(\xi_{i})$. Given values of $\omega$ and $\xi _{i}$, it relates the values of $\gamma(\xi_{i})$ and $\gamma'(\xi_{i})$ for a solution which has its inflection point at $\xi _{i}$. We may therefore describe and classify all the possible solution of this equation by the use of two parameters, one being the parameter $\omega$ that describes the state of the matter. We will choose the other to be the positive parameter given by $\pi(\xi_{i})=\gamma'(\xi_{i})$, and then the value $\gamma(\xi_{i})$ is given by

$$\gamma(\xi_{i}) = \omega\xi_{i}\, \frac{4-(1+\omega)\pi(\xi_{i})}{1+5\omega}.$$ (90)

Note that there is a single solution for this quantity, and therefore a single inflection point. Note also that the actual value of $\xi _{i}$ can be chosen arbitrarily because, since we have that $\xi=r/r_{0}$, choosing this value just corresponds to choosing a position for the arbitrary reference point $r_{0}$. We now draw attention to the fact that the linear function

$$\gamma_{0}(\xi) = \frac{4\omega}{1+6\omega+\omega^{2}}\, \xi$$ (91)

is a particular solution of the equation that determines $\gamma (\xi )$, show in Equation (86). However, this solution clearly does not satisfy the correct asymptotic conditions. Is we consider the inflection point of $\gamma (\xi )$, which corresponds to the maximum value of the derivative $\gamma'(\xi)$, and where $\gamma''(\xi)$ is zero, then we must also impose that the second derivative of $\gamma'(\xi)$ be negative there, thus characterizing a local maximum of the derivative $\gamma'(\xi)$. This is therefore a condition on the third derivative of $\gamma (\xi )$, and we may calculate it explicitly using the form of that equation shown in Equation (86), which can be written as

$$2\omega \left[ \xi^{2} - \xi\gamma(\xi) \right] \gamma''(\xi) + \omega(1+\omega) \xi \left[\gamma'(\xi)\right]^{2} +$$

$$- 4\omega \xi \gamma'(\xi) + (1+5\omega) \gamma(\xi) \gamma'(\xi) = 0.$$ (92)

If we simply differentiate this equation, we get

$$2\omega \left[ \xi^{2} - \xi\gamma(\xi) \right] \gamma'''(\xi) + 2\omega \left[ 2\xi - \gamma(\xi) - \xi\gamma'(\xi) \right] \gamma''(\xi) +$$

$$+ \omega(1+\omega) \left[\gamma'(\xi)\right]^{2} + \omega(1+\omega) \xi\, 2\gamma'(\xi)\gamma''(\xi) +$$

$$- 4\omega \gamma'(\xi) - 4\omega \xi \gamma''(\xi) +$$

$$+ (1+5\omega) \left[\gamma'(\xi)\right]^{2} + (1+5\omega) \gamma(\xi) \gamma''(\xi) = 0.$$ (93)

Applying this at the inflection point, where we have that $\gamma''(\xi_{i})=0$, we are left with

$$2\omega \left[ \xi_{i}^{2} - \xi_{i}\gamma(\xi_{i}) \right] \gamma'''(\xi_{i}) + \omega(1+\omega) \left[\pi(\xi_{i})\right]^{2} +$$

$$- 4\omega \pi(\xi_{i}) + (1+5\omega) \left[\pi(\xi_{i})\right]^{2} = 0 \;\;\;\Rightarrow$$

$$2\omega \left[ \xi_{i}^{2} - \xi_{i}\gamma(\xi_{i}) \right] \gamma'''(\xi_{i}) - 4\omega \pi(\xi_{i}) +$$

$$+ \left(1+6\omega+\omega^{2}\right) \left[\pi(\xi_{i})\right]^{2} = 0.$$ (94)

Isolating the third derivative we get

$$\gamma'''(\xi_{i}) = \frac { 4\omega - \left(1+6\omeg... ...eft[ \xi_{i} - \gamma(\xi_{i}) \right] }\, \pi(\xi_{i}).$$ (95)

All the quantities appearing on this expression except the numerator are manifestly strictly positive at the inflection point, so that in order for the third derivative to be strictly negative we must impose that the numerator be strictly negative, thus leading to

$$\pi(\xi_{i}) > \frac{4\omega}{1+6\omega+\omega^{2}}.$$ (96)

The particular solution $\gamma_{0}(\xi)$ seems to represent a situation in which the gravitational field and the pressure are such as to cause the matter to escape the gravitational attraction well and thus spread out to infinity. It is not difficult to determine that for this particular solution we have that $\mathfrak{T}(\xi)$ is in fact a constant, corresponding to an energy density $T_{0}(r)$ that goes to zero at infinity slowly, as $1/r^{2}$, rather than exponentially fast. Therefore, in this case there is no sphere at some radial coordinate $r_{S}$ that contains essentially all the matter. In other words, the matter fails to be localized. There seems to be no independent solutions for $\gamma (\xi )$ if $\pi(\xi_{i})$ is smaller that this limiting value. This means, of course, that in this case there can be no static solution of the field equation. We have therefore the first and most important limit of our energy-density parameter $\pi(\xi_{i})$,

$$\pi(\xi_{i}) > \frac{4\omega}{1+6\omega+\omega^{2}}.$$ (97)

We now shift our attention to the relation in Equation (90), that determines the value of $\gamma (\xi )$ at the inflection point. We observe that the sign of $\gamma(\xi_{i})$ will be determined by the value of $\pi(\xi_{i})$. Since all the other factors in the right-hand side of that equation are positive, we conclude that we will have $\gamma(\xi_{i})\geq 0$ when

$$\pi(\xi_{i}) \leq \frac{4}{1+\omega}.$$ (98)

The value given by the equality corresponds, of course, to $\gamma(\xi_{i})=0$, which means that the inflection point $\xi _{i}$ coincides with the root $\xi _{r}$ of $\gamma (\xi )$. In a complementary way, we will have $\gamma(\xi_{i})\leq 0$ when

$$\pi(\xi_{i}) \geq \frac{4}{1+\omega}.$$ (99)

There is therefore an interval of values of $\pi(\xi_{i})$ given by

$$\frac{4\omega}{1+6\omega+\omega^{2}} < \pi(\xi_{i}) \leq \frac{4}{1+\omega},$$ (100)

for which $\gamma(\xi_{i})\geq 0$. For the allowed values of $\omega$ the left limit is in the interval $(0,3/7]$, and the right limit is in the interval $[3,4)$, which are two intervals that do not intersect. In this case, since $\gamma(\xi_{i})\geq 0$, the root $\xi _{r}$ of $\gamma (\xi )$ is necessarily to the left of the inflection point $\xi _{i}$. This is the situation depicted in Figure 2. On the other hand, the critical point $\xi _{c}$ will only exist if $\pi(\xi_{i})\geq 1$, because otherwise, since $\pi(\xi_{i})$ is the largest value of the derivative, it will never be equal to one, so that the critical point, which is defined as the point $\xi _{c}$ where $\pi(\xi_{c})=1$, will not exist. The complementary interval of values of $\pi(\xi_{i})$, to the one given in the equation above, is that for which we have that $\gamma(\xi_{i})\leq 0$, and which is given by

$$\frac{4}{1+\omega} \leq \pi(\xi_{i}) < \infty.$$ (101)

In this case, since we always have $\pi(\xi_{i})\geq 1$, the critical point always exists, and the root $\xi _{r}$ of $\gamma (\xi )$ is necessarily to the right of the inflection point $\xi _{i}$. This is the situation depicted in Figure 3. We may therefore classify the possible values of $\pi(\xi_{i})$, and the corresponding possible solutions for $\gamma (\xi )$, in the following way.

Low Energy-Density Regime:

if we have that

$$\frac{4\omega}{1+6\omega+\omega^{2}} < \pi(\xi_{i}) < 1,$$ (102)

then there is no critical point $\xi _{c}$, and we have that $\xi_{r}<\xi_{i}$. We will call this the low energy-density regime.

Middle Energy-Density Regime:

if we have that

$$1 \leq \pi(\xi_{i}) \leq \frac{4}{1+\omega},$$ (103)

then there is a critical point $\xi _{c}$, and we have that $\xi_{r}\leq\xi_{i}<\xi_{c}$. We will call this the middle energy-density regime.

High Energy-Density Regime:

if we have that

$$\frac{4}{1+\omega} \leq \pi(\xi_{i}),$$ (104)

then there is a critical point $\xi _{c}$, and we have that $\xi_{i}<\xi_{r}<\xi_{c}$. We will call this the high energy-density regime.

Black-Hole Limit:

the limit in which we make $\pi(\xi_{i})\to\infty$ we will name the black hole limit, since it can be shown that in this limit, as seen from outside the horizon, the general character of the solutions does in fact approach that of a black hole.

Note that the case of the middle energy-density regime includes the special case

$$\pi(\xi_{i}) = \frac{4}{1+\omega},$$ (105)

in which case we have $\gamma(\xi_{i})=0$, so that the points $\xi _{r}$ and $\xi _{i}$ coincide. Note also that in the case of the high energy-density regime we can easily prove that $\gamma(0)$ must be negative. Since we have that at the inflection point $\gamma(\xi_{i})<0$, and since we also have that the derivative $\gamma'(\xi)$ is always positive, it follows that at every point to the left of $\xi _{i}$ the function $\gamma (\xi )$ must be smaller than $\gamma(\xi_{i})$, and hence negative, including at the point $\xi=0$.

Finally, note that the classification of the solutions in these various regimes does not mean that the solutions always exist, for every pair of values of the physical parameters $\omega$ and $\pi(\xi_{i})$ within each one of the regimes. They may fail to exist, mostly for the larger values of $\omega$, and in particular for the case $\omega=1/3$, that corresponds to pure radiation, since in this case we have a maximum intensity of the expanding tendency of the pressure, as compared to the compressing tendency of gravity, thus leading to the possibility of the fluid matter being non-localized. The way in which a solution fails to exist is that the asymptotic conditions fail to hold. In particular, the $\xi\to\infty$ limit of $\gamma (\xi )$ fails to be finite, and instead increases without bound, much like what happens in the particular solution $\gamma_{0}(\xi)$ shown in Equation (91). In other words, unlike what is the typical situation for the case of linear differential equations, in this case the available parameters of the model cannot always be used to adjust the boundary conditions. In fact, most often they cannot be used in this way. The situation is simply that there are pairs of values of the two physical parameters $\omega$ and $\pi(\xi_{i})$ for which a solution exists, and other for which a solution does not exist.