Existence and Stability of Static Spherical Fluid Shells in a Schwarzschild-Rindler-anti-de Sitter Metric

Boundary layers (shells of matter sources with zero thickness) play an important role in both electromagnetism and general relativity. They provide a useful laboratory for the exploration of new phenomena while at the same time they approximate smooth solutions such as domain walls (Vilenkin, 1981; Kraus, 1999; Ipser and Sikivie, 1984; Perivolaropoulos, 2018; Alestas and Perivolaropoulos, 2019) or braneworlds (Randall and Sundrum, 1999; Arkani-Hamed et al., 1998; Antoniadis et al., 1998; Arkani-Hamed et al., 1999; Langlois et al., 2002). Thin shells are also useful in describing gravitational collapse (Israel, 1967; Adler et al., 2005; Dokuchaev and Chernov, 2010) or in constructing spherically symmetric vacuum solutions that avoid the presence of singularities (e.g. gravastars (Mazur and Mottola, 2004; Visser and Wiltshire, 2004; Lobo, 2006; DeBenedictis et al., 2006; Ansoldi, 2008; Ray et al., 2020)).

Despite of the divergence of the stress-energy tensor on the thin shell, the corresponding singularities of the Einstein equations are mild and in fact they are easily integrable. Thus they lead to a simplification of the dynamical gravitational equations by converting the corresponding differential equations to finite difference equations known as ’junction conditions’ (Israel, 1966; Blau et al., 1987; Sen, 1924). These conditions lead to a determination of the discontinuities of various fields as the shell is crossed.

Thin spherical shells in General Relativity may be defined as 2+1 boundary hypersurfaces with energy momentum tensor $S^{i}_{j}\equiv\int^{R^{+}}_{R^{-}}T^{i}_{j}\;dr={\rm diag}(-\sigma,p,p)$, where $R$ is the shell radius, $r$ is the radial coordinate of the 3+1 dimensional spacetime, $\sigma$ is the surface energy density and $p$ is the surface pressure on the shell hypersurface with equation of state $p=p(\sigma)$. The thin shell interpolates between an interior and an exterior spherically symmetric metric. The exterior metric is related to the interior metric in the context of the Israel junction conditions Israel (1966); Blau et al. (1987); Sen (1924).

A well known spherical static stable thin shell configuration corresponds to the gravastar that interopolates between an interior de Sitter metric and an exterior Schwarzschild metric and constitutes an extension of the Schwarzschild metric with eliminated singularity (Mazur and Mottola, 2004; Visser and Wiltshire, 2004; Lobo, 2006; DeBenedictis et al., 2006; Ansoldi, 2008).

An alternative thin shell solution obtained using spherically symmetric scalar field dynamical equations in a non-trivial background geometry has been obtained in Ref. Alestas and Perivolaropoulos (2019). It was demonstrated that static metastable solutions can exist in the presence of a Schwarzschild-anti-deSitter curved spacetime Perivolaropoulos (2018); Alestas and Perivolaropoulos (2019) supplemented with the Rindler acceleration term. Thus the total metric is a Schwarzschild-Rindler-anti-deSitter (SRAdS) metric Grumiller (2010),

where $b$ is the Rindler acceleration parameter and $\Lambda$ is the cosmological constant.

The metric (1) has been constrained by solar system observations, indicating that $|b|<3nm/sec^{2}$ Carloni et al. (2011); Iorio (2011) and it has been shown that it can lead to the production of flat rotation curves as well as contribute to the explanation Grumiller and Preis (2011); Iorio (2012) of the Pioneer anomaly Anderson et al. (1998); Lammerzahl et al. (2008) for $b>0$.

Such a metric including a linear term in $r$ is non-standard but it has been widely considered in the literature previously and is physically motivated by at least three factors:

• •

  In the context of GR a linear term in r it emerges naturally in a spherically symmetric metric, in the context of a perfect fluid with density $\rho$ and pressure components $\rho=-p_{r}=-2p_{\theta}=-2p_{\phi}\sim 1/r$ (Alestas and Perivolaropoulos, 2019). In this sense it may be viewed as a generalization of the cosmological constant which gives a quadratic term in the metric and emerges in a spherically symmetric metric in the context of a homogeneous perfect fluid with $\rho=-p_{r}=-p_{\theta}=-p_{\phi}=constant$. In fact any spherically symmetric metric given as a power series

  $$f(r)=1-\sum^{N}_{n=-N}a_{n}r^{n}$$

  (2)

  is supported by an energy - momentum tensor of the form

  	$\displaystyle T^{0}_{0}$	$\displaystyle=\frac{1}{\kappa}\sum^{N}_{n=-N}a_{n}(1+n)r^{n-2}=\rho$		(3)
  	$\displaystyle T^{r}_{r}$	$\displaystyle=T^{0}_{0}=-p_{r}$		(4)
  	$\displaystyle T^{\theta}_{\theta}$	$\displaystyle=\frac{1}{2\kappa}\sum^{N}_{n=-N}a_{n}n(1+n)r^{n-2}=-p_{\theta}$		(5)
  	$\displaystyle T^{\phi}_{\phi}$	$\displaystyle=T^{\theta}_{\theta}=-p_{\phi}$		(6)

  For a linear term in the metric ($n=1$), the corresponding energy-momentum term varies as $1/r$ which leads to asymptotic flatness since the energy-momentum tensor vanishes at infinity. For $n=2$ we have constant energy density-pressure via the cosmological constant term and for the case of $n=-1$ we have the vacuum solution which corresponds to zero energy momentum tensor.

• •

  It is the spherically symmetric vacuum solution in various modified gravity theories including Weyl (Mannheim and Kazanas, 1989) and in 2 dimensional scalar-tensor theories (Grumiller and Preis, 2011; Perivolaropoulos and Skara, 2019). In these theories, the proper sign of the linear term can lead to additional attractive gravity that can play the role of dark matter without actual existence of any form of energy-momentum Carloni et al. (2011); Iorio (2011); Mannheim and Kazanas (1989).

• •

  In view of the generic and natural existence of the terms proportional to $1/r$ (GR vacuum) and $r^{2}$ (cosmological constant) the presence of a linear term $\sim r$ emerges as a natural generalization with potentially interesting physical effects. One of these effects is the existence of scalar hair (stable spherical scalar domain wall (demonstrated in Ref. Perivolaropoulos (2018)).

The metric’s property of supporting metastable spherical domain walls motivates the search of additional stable shell solutions described as general fluid thin shells as opposed to scalar field vacuum energy shells (domain walls). Such an analysis would be based on the Israel junction conditions formalism as opposed to the solution of dynamical scalar field equations. The following questions therefore emerge:

• •

  Are there static, stable fluid shell solutions in a SRAdS background geometry?

• •

  If yes, what are the conditions for their stability given the equation of state of the fluid shell?

• •

  What is the metric parameter range for shell stability and how does the stability radius change as a function of these parameters?

These questions will be addressed in the present analysis. We implement the Israel junction conditions in the context of a fixed equation of state of the fluid shell and a SRAdS background metric with a discontinuous value of $m$ across the shell and fixed values of $b$ and $\Lambda$ with no discontinuity as the shell is crossed. We thus derive the stability conditions and identify the range of metric parameters $b,\Lambda$, that satisfy these conditions for given values of the shell coordinate radius $R$, shell surface density $\sigma$ and mass parameters inside and outside the shell ($m_{-}$, $m_{+}$). The conditions that need to be satisfied for stability by the shell density and shell radius are also determined.

The structure of this paper is the following: In the next section we develop the general formalism for the derivation of stability conditions by implementing the Israel junction conditions on the SRAdS metric for a shell with a general fluid equation of state. In section III we consider three specific applications of the method for corresponding shell fluid equations of state: vacuum shell, stiff matter shell and matter shell and find the particular stability conditions and parameter regions in each case. Finally in section IV we conclude summarize and discuss possible extensions of this analysis.

In what follows we set $G=c=1$. In most cases we will also set the interior mass parameter $m_{-}=1$. Thus in this context, a dimensionless form of $\Lambda$ corresponds to the dimensionless combination $m_{-}^{2}\Lambda$. Furthermore, whenever Greek letters are used as indices they correspond to spacetime ones, while Roman (Latin) indices range over the coordinates of the 2+1-surface of the shell. Also, the radius of the shell is always considered in the region outside the event horizon of the black hole. Notice that for the parameter values considered (AdS) there is no cosmological horizon but only an event horizon.

Consider a thin spherical shell with coordinate radius R interpolating between an interior ($g_{\mu\nu}^{-}$) and an exterior metric ($g_{\mu\nu}^{+}$). Let the interior and exterior metrics be of the form Visser and Wiltshire (2004); Mazur and Mottola (2004); Frauendiener et al. (1990),

where,

and

We now impose the following conditions:

1. 1.

   Continuity of the metric on the shell $(r_{-}=r_{+}=R)$. This implies

   $\displaystyle f_{+}(r_{+})dt_{+}^{2}-\frac{dr_{+}^{2}}{f_{+}(r_{+})}=f_{-}(r_{-})dt_{-}^{2}-\frac{dr_{-}^{2}}{f_{-}(r_{-})}.$

   (10)

   which leads to

   	$\displaystyle t_{-}$	$\displaystyle=\frac{f_{+}(R)}{f_{-}(R)}t_{+}$		(11)
   	$\displaystyle\frac{dr_{-}}{dr_{+}}$	$\displaystyle=\frac{f_{-}(R)}{f_{+}(R)}.$		(12)

2. 2.

   The Israel junction conditions Israel (1966) expressed through a discontinuity of the extrinsic curvature on the shell hypersurface $\Sigma$. The extrinsic curvature (second fundamental form) at either side of the three-dimensional (2+1) hypersurface $\Sigma$ swept by a spherically symmetric shell, embedded in the four-dimensional spacetime are:

   $\displaystyle K_{ij}^{\pm}=\left(h_{\mu}^{\lambda}n_{\nu;\lambda}\frac{dx^{\mu}}{dx^{i}}\frac{dx^{\nu}}{dx^{j}}\right)^{\pm}_{\Sigma},$

   (13)

   where $x^{i}$ are coordinates on $\Sigma$, $h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$, and $(;)$ denote covariant derivative with respect to $g_{\mu\nu}^{\pm}$. Let

   $\displaystyle g(x^{\alpha}(x^{i}))=0,$

   (14)

   denote the parametric equation for $\Sigma$ as embedded in the 4-dimensional spacetime. The unit 4-normals to $\Sigma$ in the four-dimensional spacetime are given by Sen (1924),

   $\displaystyle n_{\alpha}=\pm\left(\left|g^{\beta\gamma}\frac{\partial g}{\partial x^{\beta}}\frac{\partial g}{\partial x^{\gamma}}\right|\right)^{-1/2}\frac{\partial g}{\partial x^{\alpha}},$

   (15)

   We assume $n_{\alpha}\not=0$ and label $\Sigma$ as timelike for $n_{\alpha}n^{\alpha}=1$ (a spacelike normal). The Israel junction conditions are expressed as discontinuities of the extrinsic curvature of the shell

   $$\left[\left[K_{ij}\right]\right]=-8\pi\left[S_{ij}-\frac{1}{2}Sh_{ij}\right]$$

   (16)

   where $\left[\left[X\right]\right]\equiv\left[X_{+}\right]-\left[X_{-}\right]$ denotes the discontinuity of the quantity $X$ as the shell is crossed.

   In the case of a static shell in the SRAdS metric (1) the extrinsic curvature tensor takes the form

   $\displaystyle K_{ij}=\sqrt{f_{\pm}(r_{\pm})}{\rm diag}\left(\frac{\frac{1}{2}f_{\pm}^{\prime}(r_{\pm})}{f_{\pm}(r_{\pm})},\frac{1}{r_{\pm}},\frac{1}{r_{\pm}}\right).$

   (17)

   and the Israel junction conditions for a dynamic shell are of the form Visser and Wiltshire (2004)

   	$\displaystyle\sigma$	$\displaystyle=-\frac{1}{4\pi R}\left[\left[\sqrt{1-2m_{\pm}(R)/R+\dot{R}^{2}}\right]\right]$		(18)
   	$\displaystyle p$	$\displaystyle=\frac{1}{8\pi R}\left[\left[\frac{1-m_{\pm}(R)/R-m_{\pm}(R)^{\prime}+\dot{R}^{2}+R\ddot{R}}{\sqrt{1-2m_{\pm}(R)/R+\dot{R}^{2}}}\right]\right]$		(19)

   where (^′) denotes derivative of $m_{\pm}(r)$ with respect to $r$ at $r=R$ and dot denotes derivative with respect to the proper time of the shell defined as

   $\displaystyle d\tau^{2}$

   $\displaystyle=$

   $\displaystyle\;\left[1-{2m_{\pm}(R)\over R}\right]\;{dt}^{2}-\;{1\over 1-2m_{\pm}(R)/R}\left[{dR\over dt}\right]^{2}dt^{2}$

   (20)

These equations lead also to the energy conservation equation on the shell,

The eq. (18) may also be expressed as,

where,

and $E=0$.

Clearly, eq. (22) is identical with the energy conservation equation of a particle moving in one dimension with coordinate $R(\tau)$ and zero energy. Thus, the conditions for the existence of a static, stable shell may be written as,

These conditions, along with the equation of state $p(\sigma)$ and the energy conservation eq. (21) may be used to identify constraints on the metric parameters appearing in the expressions of $m_{-}(R)$ and $m_{+}(R)$ required for the existence of a stable spherical shell with given radius R. In the present analysis we consider the particular forms of $m_{\pm}(r)$ given by eq. (9) corresponding to the SRAdS metric. In this case the potential of eq. (23) takes the form,

In the context of a constant shell fluid equation of state we have $p=w\sigma$ and it is easy to show that energy conservation (21) leads to

where $\sigma_{0}^{\prime}$ is the surface density of a shell of radius $R_{0}$. In what follows we define

For example in the special case of a pressureless matter shell ($w=0$) we obtain the expected result $\sigma(R)\sim R^{-2}$ while for a vacuum shell we have $\sigma(R)=\sigma_{0}=const$. The dimensionality of $\sigma_{0}$ is therefore depended on the equation of state parameter $w$, resulting in different dimensions for $\sigma_{0}$ regarding each case of $w$ discussed in the next section.

In the special case when $\sigma$ is independent of R discussed in the next section (vacuum shell) it is straightforward to show that a minimum of the potential (25) exists for $b<0$ and $\Lambda<0$ due to the attractive nature of the linear potential term $2\;b\;R$ which dominates at large $R$ competing with the repulsive effects of the quadratic potential term $-\Lambda R^{2}/3$ which dominates at even larger $R$.

For more general metrics or fluid equations of state than the one considered here it is clearly possible to have several minima for the potential corresponding to configurations of more than one stable concentric shells.

In the next section we identify the metric parameter ranges of $b$ and $\Lambda$ that allow for stable shells in the spacial cases of three shell fluid equations of state.

We then proceed by finding the ranges of the parameters $b,\Lambda$ which allow for stable spherical shell solutions, via the implementation of these conditions for different cases of interior and exterior equations of state.

We have demonstrated the existence of static, stable spherically symmetric thin fluid shells in a Schwarzschild-Rindler-anti-de Sitter (SRAdS) metric. We have found analytically the conditions for stability and the corresponding range of values of metric parameters that admit stable fluid shells for different forms of fluid equation of state. These structures have similarities with the well known gravastar shell structures Mazur and Mottola (2004); Visser and Wiltshire (2004); Martin Moruno et al. (2012); Uchikata and Yoshida (2016); Broderick and Narayan (2007). In our shell structures the interior de Sitter term of the gravastars is replaced by a combination of Rindler-anti-de Sitter terms present in a continuous form (same values both in the interior and in the exterior of the shell) allowing for the existence of a minimum of the stability effective potential.

Interesting extensions of this analysis include the following:

• •

  The investigation of alternative forms of metrics that may admit stable shell solutions. For example an interesting alternative simple metric would be one with a Rindler term inside the shell and a Schwarzschild term outside. Such a metric would be free of singularities and would differ from a gravastar in the replacement of the de Sitter interior by a Rindler interior. Other types of metrics could accept multiple concentric shell structures if the corresponding stability potential has multiple minima at different radii R.

• •

  The investigation of observational effects of such shell structures. Since the radius of the shell is always considered in the region outside the event horizon of the black hole, lensing can be considered in a straightforward manner by studying lightlike geodesics in the SRAdS spacetime along the lines of Refs Lim and Wang (2017); Cutajar and Adami (2014) where the lensing of similar metrics is considered. For example signatures of such SRAdS shell structures in typical lensing patterns could be identified and compared to observed lensing patterns around black holes (Hannuksela et al., 2019; Bowman et al., 2004; Rahman and Sen, 2019; Ishak et al., 2008; Park, 2008; Sereno, 2008). Signatures of SRAdS shells in such optical images could be specified and compared with predicted signatures of other similar exotic objects like gravastars Sakai et al. (2014).

• •

  The investigation of non-spherical junctions and shells. An interesting problem would be the study of joining rotating spacetimes in the presence of the cosmological constant.

• •

  The consideration of more general fluid shell equations of state. In the case of phantom shells it may be possible to have stable shells in a pure Schwarzschild background due to the tendency of such shells to expand rather than contract (negative tension). This is easily shown using the energy conservation equation (21) with $w<-1$ which leads to a surface density $\sigma(R)=\sigma_{0}R^{-2(w+1)}$ which increases with $R$. The positive value of the exponent for $w<-1$ indicates that it is energetically favourable for such phantom shell to expand rather than contract leading to a negative tension (pressure) that would tend to stabilize the shell even in a pure Schwarzschild background.

• •

  The investigation of the dynamical evolution of the shell in the context of spherical symmetry and beyond. Non-spherical dynamical excitations of the shell could also lead to interesting gravitational wave signatures.

Numerical Analysis Files: The numerical files for the reproduction of the figures can be found in num .

The authors would like to extend a thank you to prof. Demetrios Papadopoulos for the stimulating discussions and ideas he provided towards the completion of this paper. This research is co-financed by Greece and the European Union (European Social Fund - ESF) through the Operational Programme ”Human Resources Development, Education and Lifelong Learning 2014-2020” in the context of the project ”Scalar fields in Curved Spacetimes: Soliton Solutions, Observational Results and Gravitational Waves” (MIS 5047648).