An Analysis of a Regular Black Hole Interior Model

Abstract

We analyze the thermodynamical properties of the regular static and spherically symmetric black hole interior model presented by Mbonye and Kazanas. Equations for the thermodynamical quantities valid for an arbitrary density profile are deduced, and from them we show that the model is thermodynamically unstable. Evidence is also presented pointing to its dynamical instability. The gravitational entropy of this solution based on the Weyl curvature conjecture is calculated, following the recipe given by Rudjord, Grøn and Sigbjørn, and it is shown to have the expected behavior.

Appendix: Dynamical Stability of the Interior Region

In order to study the stability of a system with tangential pressures against radial perturbations, both Einstein field equations and the equation of state must be perturbed following the standard procedure (see for instance [34]). The goal is to obtain a differential equation for the radial dependence of the quantity ξ(r,t), which represents a small radial displacement of the fluid respect to its position of equilibrium at time t, that is, r(r,t)=r ₀+ξ(r,t).

In what follows, the unperturbed metric coefficients A(r), B(r), and the unpertubed thermodynamics variables will be denoted by a zero subindex. The corresponding perturbations are denoted by δA(r,t), δB(r,t), δp _r (r,t), δp _⊥(r,t) and δρ(r,t). A long but straightforward calculation (which parallels that presented in [34]) shows that the equations for the perturbations are given by:

$$\begin{aligned} 8\pi r^{2}\delta\rho = & \frac{\partial}{\partial r} \biggl( \frac {r\delta A}{A_{0}^{2}} \biggr), \end{aligned}$$

(44)

$$\begin{aligned} 8\pi\dot{\xi}\bigl(\rho_{0}+p_{r}^{0} \bigr) = & -\frac{1}{r}\frac {\partial\delta A}{\partial t}\frac{1}{A_{0}^{2}}, \end{aligned}$$

(45)

$$\begin{aligned} 8\pi r^{2}\delta p_{r} = & \frac{r}{A_{0}} \biggl[\frac{\partial }{\partial r} \biggl(\frac{\delta B}{B_{0}} \biggr)-\frac{dB_{0}}{dr} \frac {\delta A}{A_{0}} \biggr]-\frac{\delta A}{A_{0}^{2}}, \end{aligned}$$

(46)

$$\begin{aligned} \delta p_{\perp} = & \frac{r}{2}\frac{\partial\delta p_{r}}{\partial r}+ \delta p_{r}+\frac{r}{4B_{0}}\frac {dB_{0}}{dr}(\delta p_{r}+\delta\rho) \\ &{} + \biggl[\frac{r}{4}\frac{\partial}{\partial r} \biggl(\frac{\delta B}{B_{0}} \biggr)+\frac{r}{2}\frac{A_{0}}{B_{0}}\frac{\partial^{2}\xi }{\partial t^{2}} \biggr] \bigl(p_{r}^{0}+\rho_{0}\bigr). \end{aligned}$$

(47)

Notice that there are six unknowns in this system, a consequence of the choices made by MK (one EOS and the explicit form of ρ(r) to determine the system).

Equation (45) can be integrated respect to time,⁹ yielding:

$$ \frac{\delta A}{A_{0}}=-8\pi r\bigl(\rho_{0}+p_{r}^{0} \bigr)A_{0}\xi. $$

(48)

By isolating δB from (46) we get:

$$ \frac{\partial}{\partial r} \biggl(\frac{\delta B}{B_{0}} \biggr)=8\pi rA_{0} \delta p_{r}+\frac{\delta A}{A_{0}} \biggl(\frac {1}{B_{0}} \frac{dB_{0}}{dr}+\frac{1}{r} \biggr). $$

(49)

The final set of equations takes the form:

$$\begin{aligned} 8\pi r^{2}\delta\rho = & \frac{\partial}{\partial r} \biggl( \frac {r\delta A}{A_{0}^{2}} \biggr), \end{aligned}$$

(50)

$$\begin{aligned} \delta p_{r}+\bigl(p_{r}^{0} \bigr)'\xi = & \biggl(\frac{dp_{r}}{d\rho} \biggr)_{0} \bigl(\delta\rho+\rho_{0}'\xi\bigr), \end{aligned}$$

(51)

$$\begin{aligned} \delta p_{\perp} = & \frac{r}{2}\frac{\partial\delta p_{r}}{\partial r}+ \delta p_{r}+\frac{r}{4B_{0}}\frac {dB_{0}}{dr}(\delta p_{r}+\delta\rho) \end{aligned}$$

(52)

$$\begin{aligned} &{} + \biggl(\frac{r}{4}\frac{\partial}{\partial r} \biggl( \frac{\delta B}{B_{0}} \biggr)+\frac{r}{2}\frac{A_{0}}{B_{0}}\frac{\partial^{2}\xi }{\partial t^{2}} \biggr) \bigl(p_{r}^{0}+\rho_{0}\bigr), \\ \frac{\delta A}{A_{0}} = & -8\pi r(\rho_{0}+p_{0})A_{0} \xi, \end{aligned}$$

(53)

$$\begin{aligned} \frac{\partial}{\partial r} \biggl(\frac{\delta B}{B_{0}} \biggr) = & 8\pi rA_{0}\delta p_{r}+\frac{\delta A}{A_{0}} \biggl( \frac {1}{B_{0}}\frac{dB_{0}}{dr}+\frac{1}{r} \biggr). \end{aligned}$$

(54)

Standard manipulations using the expression ξ(r,t)=σ(r)expiωt in these equations lead to the following second order ordinary inhomogeneous differential equation for the function σ(r):

$$ A_{\star}(r)\sigma''(r)+C_{\star}(r) \sigma'(r)+ \bigl[B_{\star }(r)+\omega D_{\star}(r) \bigr]\sigma(r)= \varPi(r), $$

(55)

where we have also used δp _⊥(r,t)=Π(r)expiωt.

This equation defines an inhomogeneous Sturm-Liouville (SL) problem, differently from the common case which yields a homogeneous one, the inhomogeneity being a direct consequence of the way the MK model solution was obtained, as mentioned before.

Equation (55) is to be solve for r∈[0,1], using the boundary conditions σ(r)=0, σ(1)=0, and $[\delta\rho(1,t)+\rho '_{0}(1,t) \xi(1,t) ] = 0$ [34]. The coefficients A _⋆(r), B _⋆(r), C _⋆(r), and D _⋆(r) are given by:

$$\begin{aligned} A_{\star}(r) =& \frac{r}{2} \bigl(p_{r}^{0}+ \rho_{0} \bigr) \biggl(\frac{dp}{d\rho} \biggr)_{0} , \end{aligned}$$

(56)

$$\begin{aligned} B_{\star}(r) =& B_{1}(r)+B_{2}(r)+B_{3}(r)+B_{4}(r), \end{aligned}$$

(57)

where:

$$\begin{aligned} B_{1}(r) = & \frac{1}{8\pi r^2}q'(r)\alpha, \end{aligned}$$

(58)

$$\begin{aligned} q'(r) = & \frac{\partial}{\partial r} \bigl[-8\pi r^{2} \bigl( \rho _{0}+p_{r}^{0} \bigr) \bigr], \\ \alpha = & \frac{r}{2} \frac{\partial}{\partial r} \biggl(\frac {dp}{d\rho} \biggr)_{0}+ \biggl(\frac{dp}{d\rho} \biggr)_{0} \varSigma(r) +\frac{r}{4B_{0}} \frac{dB_{0}}{dr}, \\ \varSigma(r) = & 1+ \frac{r}{4B_{0}} \frac{dB_{0}}{dr}+4 \pi r^2 A_{0} \bigl(p_{r}^{0}+ \rho_{0} \bigr), \\ B_{2}(r) = & \frac{r}{2} \tau(r), \end{aligned}$$

(59)

$$\begin{aligned} \tau(r) = & \frac{\partial}{\partial r} \biggl(\frac{dp}{d\rho} \biggr)_{0} \rho'_{0}-\frac{\partial}{\partial r} \bigl(p_{r}^{0} \bigr)'+ \biggl(\frac{dp}{d\rho} \biggr)_{0} \frac{\partial\rho '_{0}}{\partial r}, \\ B_{3}(r) =& \frac{r}{2} \frac{\partial}{\partial r} \biggl( \frac{1}{8 \pi r^2} q'(r) \biggr) \biggl(\frac{dp}{d\rho} \biggr)_{0}, \end{aligned}$$

(60)

$$\begin{aligned} B_{4}(r) =& - 8 \pi r \bigl(\rho_{0}+p_{r}^{0} \bigr) A_{0} \biggl[r \frac{d}{dr} (\ln B_{0} ) + 1 \biggr], \end{aligned}$$

(61)

$$\begin{aligned} C_{\star}(r) =& C_{1}(r)+C_{2}(r)+C_{3}(r), \end{aligned}$$

(62)

$$\begin{aligned} C_{1}(r) = &\frac{1}{8 \pi r^{2}}q \alpha, \end{aligned}$$

(63)

$$\begin{aligned} q = & -8\pi r^{2} \bigl(\rho_{0}+p_{r}^{0} \bigr), \\ C_{2}(r) =& \frac{r}{2} \biggl[ \biggl(\frac{dp}{d \rho} \biggr)_{0} \rho '_{0}-p'_{0} \biggr], \end{aligned}$$

(64)

$$\begin{aligned} C_{3}(r) =& \frac{1}{8 \pi r} \biggl(-\frac{1}{r}q+ q' \biggr), \end{aligned}$$

(65)

$$\begin{aligned} D_{\star}(r) =& - \frac{r}{2} \frac{A_{0}}{B_{0}} \bigl(p_{r}^{0}+\rho_{0} \bigr). \end{aligned}$$

(66)

For a given Π(r), (55) might be solved using the Green’s function method (see for instance [35]). However, as we shall show next, the SL problem defined by (55) has a singular point. Consider the equation [36]:

$$ - \bigl(py' \bigr)'+qy= \lambda wy, $$

(67)

which corresponds to the homogeneus SL problem defined on J=(a,b), with −∞≤a≤b≤∞. The eigenvalue λ is such that $\lambda\in\mathbb{C}$, and p,q,w and y are functions of x. It is also assumed that:

$$ \frac{1}{p}, q, w \in L_{\mathrm{loc}} (J,\mathbb{C} ), $$

(68)

where $L_{\mathrm{loc}} (J,\mathbb{C} )$ denotes the linear manifold of functions y satisfying $y \in L ( [\alpha,\beta ],\mathbb{C} )$ for all compact intervals [α,β]⊆J. $L (J,\mathbb{C} )$ denotes the linear manifold of complex valued Lebesgue measurable functions y defined on J for which: $\int_{a}^{b} \vert y(t)\vert dt \equiv\int_{J} \vert y(t)\vert dt \equiv\int_{J} \vert y(t)\vert < \infty$.

Here J is any interval of the real line, open, closed, half open, bounded or unbounded. Following [36], we have the next definitions:

• The (finite or infinite) endpoint a is regular if

  $$ \frac{1}{p}, q, w \in L\bigl((a,d),\mathbb{C}\bigr), $$

  (69)

  holds for some (hence any) d∈J.

• An endpoint is called singular if it is not regular,

with similar definitions at r=b. To analyse whether a=0 is a regular or singular endpoint for our problem, defined by (55), we show in Fig. 17 the plot of the coefficient A _⋆ as a function of radial coordinate. We can see that A _⋆(0)=A _⋆(0.4)=0, A _⋆(r)<0 for r∈(0,0.4), and A _⋆(r)>0 for r∈(0.4,1). Hence, $\frac {1}{A}_{\star} \notin L((0,d),\mathbb{C})$ for any d∈(0,1). Therefore, the endpoint a=0 is singular. If an endpoint is singular it could be a limit point or a limit circle. According to [36], “there is hardly any literature on the LP/LC (limit point-limit circle) dichotomy when all three coefficients are present in the SL equation”. In particular, “there seems to be no literature on LP/LC criteria when p changes sign”. Due to these complications, we shall attack this problem in a future publication.

Fig. 17

Coefficient A _⋆ as a function of the radial coordinate