Note on an elementary particle model with Bertotti-Robinson core

TL;DR

This paper examines a classical model of an elementary particle or black hole with a Bertotti-Robinson core, demonstrating its stability against linear radial perturbations, which supports its viability as a non-singular solution.

Contribution

It analyzes the stability of a spherically symmetric particle/black hole model with a Bertotti-Robinson core under linear radial perturbations, confirming its stability.

Findings

Model is stable against linear radial perturbations.

The stability holds with a linear equation of state.

Supports the model as a non-singular particle/black hole solution.

Abstract

Spherically symmetric classical model of an elementary particle or a black hole spacetime without central singularity had been constructed by O. B. Zaslavskii in PRD 70(2004)104017. In this model an extremal Reissner-Nordstr% \"{o}m (RN) black hole and a Bertotti-Robinson (BR) spacetime are glued at the horizon such that the inner/core spacetime is the regular BR while outside is the extremal RN. In this note we investigate the stability of such a particle / regular black hole against linear radial perturbations. The model turns out to be stable against such perturbations with a linear equation of state after the perturbation.

Equations

$$d{s}_{core}^2=-e^{2r/r_0}d{t}^2+d{r}^2+r_0^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right)$$

$$\begin{gathered} d{s}_{outer}^2=-{\left(1-\frac{r_h}{r}\right)}^{2}d{t}^2+ \\ \frac{d{r}^2}{{\left(1-\frac{r_h}{r}\right)}^2}+r^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right) \end{gathered}$$

$$F:=r-r_0\left(\tau \right)=0$$

$$d{s}_F^2=-e^{2}d{t}^2+r_0^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right)$$

$$d{s}_F^2=-{\left(1-\frac{r_h}{r_0}\right)}^{2}d{t}^2+r_0^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right)$$

$$\begin{gathered} -e^{2}d{t}_{core}^2+d{r}^2= \\ -{\left(1-\frac{r_h}{r_0}\right)}^{2}d{t}_{outer}^2+d{r}^2=-d{\tau }^2 \end{gathered}$$

$$K_{ij}=-n_{\mu }\left(\frac{{\partial }^2{x}^{\mu }}{\partial {\xi }^i\partial {\xi }^j}+{\Gamma }_{\alpha \beta }^{\mu }\frac{\partial x^{\alpha }}{\partial {\xi }^i}\frac{\partial x^{\beta }}{\partial {\xi }^j}\right)$$

$$n_{\mu }=\frac{1}{\sqrt{g^{\alpha \beta }\frac{\partial F}{\partial x^{\alpha }}\frac{\partial F}{\partial x^{\beta }}}}\frac{\partial F}{\partial x^{\mu }}$$

$$K_{icore}^j=diag\left(\frac{1+{\dot{r}}_0^2+r_0{\ddot{r}}_0}{r_0\sqrt{1+{\dot{r}}_0^2}},0,0\right)$$

$$\begin{gathered} K_{iouter}^j= \\ diag\left(\frac{\left(1-\frac{r_h}{r_0}\right)\frac{r_h}{r_0^2}+{\ddot{r}}_0}{\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}},\frac{\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}}{r_0},0\right). \end{gathered}$$

$$-8\pi G{S}_i^j=\left[K_i^j\right]-\left[K\right]{\delta }_i^j$$

$$\sigma =-\frac{2\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}}{r_0}$$

$$\begin{gathered} p=\frac{\left(1-\frac{r_h}{r_0}\right)\frac{r_h}{r_0^2}+{\ddot{r}}_0}{\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}}+ \\ \frac{\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}}{r_0}-\frac{1+{\dot{r}}_0^2+r_0{\ddot{r}}_0}{r_0\sqrt{1+{\dot{r}}_0^2}}. \end{gathered}$$

$$p_0=0$$

$${\sigma }_0=-\frac{2\left(1-\frac{r_h}{r_0}\right)}{r_0}$$

$$\begin{gathered} \frac{\left(1-\frac{r_h}{r_0}\right)\frac{r_h}{r_0^2}+{\ddot{r}}_0}{\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}}+ \\ \frac{\left(1+\omega \right)\sqrt{{\left(1-\frac{r_h}{r_0}\right)}^2+{\dot{r}}_0^2}}{r_0}-\frac{1+{\dot{r}}_0^2+r_0{\ddot{r}}_0}{r_0\sqrt{1+{\dot{r}}_0^2}}=0 \end{gathered}$$

$$a=\left(1-\frac{\omega \left|v\right|}{\sqrt{1+v^2}-\left|v\right|}\right)\left|v\right|\sqrt{1+v^2}$$

$$d{s}^2=-f_i\left(r_i\right)d{t}_i^2+\frac{d{r}_i^2}{f_i\left(r_i\right)}+r_i^2\left(d{\theta }_i^2+{\sin }^2{\theta }_{i}d{\varphi }_i^2\right)$$

$$\sigma =-\frac{1}{4\pi G}\left(\frac{\sqrt{f_2\left(R\right)+{\dot{R}}^2}-\sqrt{f_1\left(R\right)+{\dot{R}}^2}}{R\left(\tau \right)}\right)$$

$$\begin{gathered} p=\frac{1}{8\pi G}(\frac{2\ddot{R}\left(\tau \right)+f_2^{\prime }\left(R\right)}{2\sqrt{f_2\left(R\right)+{\dot{R}}^2}}-\frac{2\ddot{R}\left(\tau \right)+f_1^{\prime }\left(R\right)}{2\sqrt{f_1\left(R\right)+{\dot{R}}^2}}+ \\ \frac{\sqrt{f_2\left(R\right)+{\dot{R}}^2}-\sqrt{f_1\left(R\right)+{\dot{R}}^2}}{R\left(\tau \right)}) \end{gathered}$$

$${\sigma }_0=-\frac{1}{4\pi G}\left(\frac{\sqrt{f_2\left(R_0\right)}-\sqrt{f_1\left(R_0\right)}}{R_0}\right),$$

$$\begin{gathered} p_0=\frac{1}{8\pi G}(\frac{f_2^{\prime }\left(R_0\right)}{2\sqrt{f_2\left(R_0\right)}}-\frac{f_1^{\prime }\left(R_0\right)}{2\sqrt{f_1\left(R_0\right)}}+ \\ \frac{\sqrt{f_2\left(R_0\right)}-\sqrt{f_1\left(R_0\right)}}{R_0}). \end{gathered}$$

$$f_2\left(R_0\right)=f_1\left(R_0\right)$$

$$f_2^{\prime }\left(R_0\right)=f_1^{\prime }\left(R_0\right).$$

$$1-\frac{R_0^2}{{\ell }^2}={\left(1-\frac{e}{R_0}\right)}^2$$

$$-\frac{R_0}{{\ell }^2}=\frac{e}{R_0^2}\left(1-\frac{e}{R_0}\right)$$

$$R_0=\frac{2}{3}e$$

$$\frac{R_0^2}{{\ell }^2}=\frac{3}{4}.$$

$$f_2\left(r\right)\simeq f_2\left(R_0\right)+f^{\prime }\left(R_0\right)\left(r-R_0\right).$$

$$f_1\left(r\right)=1+\frac{r^2}{{\ell }^2}$$

Full text

I Introduction

With the advent of general relativity, the geometric theory of gravitation due to Einstein, attempts to establish geometric model of an elementary particle took start. Upon discovering a model spacetime in which the radius of spacetime had a minimum encouraged Einstein and Rosen to propose such a model of elementary particle ER . Later on the model turned out to be interpreted as an Einstein-Rosen bridge, and in present day’s terminology as a wormhole connecting two asymptotically flat spacetimes. Initially what Einstein and Rosen called the radius of the elementary particle coincides with the throat of the wormhole MT ; MV . The two spacetimes that are connected at the throat may host black holes which are thought recently to be entangled from the view of quantum interpretation Maldacena . Existence of singularities at the center of black holes has been a serious obstacle in the construction of a self-consistent geometric model of a particle. From physical grounds we observe / detect no singularity at the location of a particle such as an electron. This automatically eliminates, the models of black holes as viable particle models. Next, we must have a smooth matching of the inner and outer geometries at the surface of the so called particle model with well-defined energy scale that doesn’t contradict experiments. To overcome these requirements certain boundary conditions are imposed which came to be known as the Israel’s junction conditions Israel . Such a construction was considered by Vilenkin and Fomin in Vilenkin1 ; Vilenkin2 and recently by Zaslavskii in Zaslavskii where he attempted to match different spacetimes to fit the physical requirements. Naturally the choice of the regular Minkowski space as the inner core region to external spacetimes such as Schwarzschild or Reissner-Nordström (RN) deserved a special attention. Such a choice, however, failed to satisfy the matching conditions. The next attempt by Zaslavskii was to test another regular candidate for the inner region for an elementary particle. The choice has been the regular Bertotti-Robinson (BR) BR spacetime that represents a uniform electromagnetic field which is manifestly regular, to be matched with the external RN geometry. The matching works consistently provided the surface of the particle coincides with the event horizon of the extremal RN geometry. As shown in Zaslavskii the proper mass of the particle turns out to be negative which vanishes identically with the vanishing radius of the particle.

Our aim in this study is to investigate the stability of such a geometrical model of an elementary particle. We perturb the junction shell which sits at the horizon of the extremal RN black hole with BR spacetime as its inner partner. Following the small radial perturbation a highly non-linear equation is obtained which is plotted with chosen initial conditions. The plots indicate that for tuned parameters the particle exposes stable behaviours whereas its energy density remains negative. To develop analogues models we investigate also the model with inner metrics other than BR. For instance the choice of de Sitter serves our purpose well. The outer metric is RN, which is more general than the extremal RN, and matches well with the inner de Sitter spacetime. We have proposed this model recently as a method to cut / eliminate the time like singularity of the RN geometry MH . It remains to be seen whether such an approach of cut and paste of spacetimes on a thin-shell removes the singularities of other black holes. It should be added also that in such a mechanism the physical parameters of the external RN geometry are derived from geometrical properties, which is reminiscent of the Wheeler’s geometrodynamics Wheeler .

II Stability of the Zaslavskii’s particle model

The geometrical particle model introduced by Zaslavskii in Zaslavskii consists of the core and outer spacetimes given by

[TABLE]

and

[TABLE]

respectively. These two regions are glued at $r=r_{0}$ on a timelike hypersurface defined by

[TABLE]

where $\tau$ stands for the proper time, as described below. Straightforward calculation gives the induced metric on the shell due to $ds_{core}^{2}$ as

[TABLE]

and $ds_{outer}^{2}$

[TABLE]

respectively. These clearly match provided the angular coordinates of both spacetimes are identified on the shell and the coordinate time satisfies the relations

[TABLE]

with proper time $\tau$ on the shell. Having (6), the first fundamental form of the shell is continuous across the shell. To see the situation of the second fundamental form we start from the definition of the extrinsic curvature

[TABLE]

in which

[TABLE]

is the normal $4-$vector which points from the shell outward for the outer-side of the shell and inward for the inner-side of the shell. Explicit calculation yields

[TABLE]

while

[TABLE]

Applying the Israel junction condition Israel

[TABLE]

in which the bracket implies $\left[X\right]=X_{outer}-X_{core},$ $K=trK_{i}^{j}$ and $S_{i}^{j}=diag\left(-\sigma,p,p\right)$ we obtain ($8\pi G=1$)

[TABLE]

and

[TABLE]

Let’s note that at the static equilibrium condition where $r_{0}=const.\geq r_{h}$ we get

[TABLE]

and

[TABLE]

which vanishes at $r_{0}=r_{h}$ Zaslavskii .

Hence we assume that the system is at equilibrium at $r_{0}=r_{h}.$ Any radial perturbation causes $r_{0}\neq r_{h}$ and therefore $p$ and $\sigma$ are obtained by (12) and (13) which means that on the perturbed shell there would be a perfect fluid presented. Considering a linear equation of state of the form $p=\omega\sigma$ ($\omega=const.$) one finds an equation of motion given by

[TABLE]

which is a differential equation giving the behaviour of $r_{0}$ after the perturbation. This equation is not solvable analytically but one may purse a numerical solution provided the initial conditions are known. As we consider a radial perturbation about $r_{0}=r_{h}$ we set the following to be the initial conditions: at $\tau=0,$ $r_{0}=r_{h}=1$ and $\dot{r}_{0}\neq 0.$ In Figs. 1 and 2 we plot the solution of Eq. (16) for $\dot{r}_{0}=0.1$ and $-0.2,$ respectively, and $\omega=0.1,0.2,0.4,0.6,0.8,1.0$ and $2.0.$ As we observe here; for a positive $\omega$ and a small perturbation the particle remains stable. In both cases, the shell spends more time with $r_{0}>r_{h}$ although in Fig. 2 initially the shell is perturbed toward the center. In such a particle model, however, both the energy density $\sigma$ and angular pressure $p$ are negative. They both vanish at the equilibrium radius while perturbation drives the system to attain $\sigma<0$ and $p<0.$ Assuming that the parameter $\omega<0,$ yields a model with $\sigma<0$ and $p>0$ which results unfortunately in an unstable particle model. At any cost Eq. (12) implies that the model will always have $\sigma<0.$

From (16) at $r_{0}=r_{h}=1$ one finds the acceleration

[TABLE]

in which $v=$ $\dot{r}_{0}$ and $a=\ddot{r}_{0}$ both at $r_{0}=r_{h}=1.$ The direction of acceleration $a$ depends not only on the magnitude of $v$ but also on the value of $\omega.$ Hence for the case that directions of $v$ and $a$ are both negative the shell is more likely to become unstable. For the negative $\omega$, with positive initial velocity the shell is also unstable and that is why we chose $\omega>0$ in Figs. 1 and 2.

III A general overview of the particle model

To complete our study we consider the static spherically symmetric spacetimes in general form. The line element inside and outside the shell of the particle are given by

[TABLE]

with $i=1,2$ indicating the inside and outside, respectively. The energy momentum tensor components on the shell are found to be from $S_{\mu}^{\nu}=diag\left(-\sigma,p,p\right)$

[TABLE]

and

[TABLE]

in which $r=R\left(\tau\right)$ represents the surface of the particle. In static condition one finds

[TABLE]

and

[TABLE]

Considering $\sigma_{0}=0$ imposes

[TABLE]

while $p_{0}=0$ yields

[TABLE]

Having these conditions satisfied, the particle model becomes physical. In MH we have shown that the outer RN black hole can be glued to the inner de-Sitter spacetime consistently and due to that we proposed to remove the singularity of the RN black hole.

For instance, if we set $f_{1}\left(r\right)=1-\frac{r^{2}}{\ell^{2}}$ and $f_{2}\left(r\right)=\left(1-\frac{e}{r}\right)^{2}$ the latter equations become

[TABLE]

and

[TABLE]

respectively. Simultaneous solution of these equations reveals that

[TABLE]

which is smaller than the horizon radius i.e., $r_{h}=e$. In addition to that one obtains the geometrical condition

[TABLE]

One simple solution for these equations can be found by considering Taylor expansion of the outer metric about $R_{0}$ i.e.,

[TABLE]

In this case we set

[TABLE]

and due to (24) and (25) we find $f_{2}\left(R_{0}\right)=f_{1}\left(R_{0}\right)=1+\frac{R_{0}^{2}}{\ell^{2}}$ and $f_{2}^{\prime}\left(R_{0}\right)=f_{1}^{\prime}\left(R_{0}\right)=\frac{2R_{0}}{\ell^{2}}$ so that to the first order we find

[TABLE]

The spacetime expressed by the latter metric is singular GM at $r=0$. This singularity may be naked with $\ell^{2}\geq R_{0}^{2}$ or hidden behind a horizon for $\ell^{2}<R_{0}^{2}$ located at

[TABLE]

As one expects for the particle model $R_{0}\geq r_{h}$ and one finds

[TABLE]

which is trivially satisfied. Let’s add that the energy momentum tensor of the spacetime outside the shell is found to be of the perfect fluid form given by

[TABLE]

in which

[TABLE]

[TABLE]

and

[TABLE]

In Fig. 3 we plot $f\left(r\right)$ versus $r$ for $\ell^{2}=1$ and $R_{0}=0.5,1.0$ and $1.5$ from bottom to the top. The smooth matching is clearly seen from the plots.

IV Conclusion

We revisit the geometrical model of an elementary particle Zaslavskii with the purpose to investigate its stability and search for alternative models. The suggested model in Zaslavskii glued spacetimes BR (inside) with the extremal RN (outside). The radius of the elementary particle coincides with the horizon of the extremal RN. At the static equilibrium the shell has no surface energy-momentum, i.e., $\sigma_{0}=p_{0}=0.$ Upon radial perturbation the shell absorbs energy and behaves as a perfect fluid satisfying $p=\omega\sigma.$ When the shell is perturbed a master equation (i.e., Eq. (16) for the general spherical symmetry configuration, given in Sec. III), governing the perturbations is derived and plotted for different initial conditions. The oscillatory behaviours in Figs. 1 and 2 support the stability of the shell and therefore the particle is stable against such perturbations. An inward motion is observed to reverse immediately outward following a small penetration of the shell below the horizon. The energy density attained by the shell in this process is negative. The particle model constructed from BR and RN can be modified by invoking other regular metrics to replace the BR. One immediate choice is the de Sitter metric to be matched with the outer RN metric. In this venture naturally the mass and charge of the RN metric are defined entirely from the geometrical parameters.