The problem of ultracompact rotating gravastars

Gravastars, proposed by Mazur and Mottola (Mazur and Mottola, 2004) as an alternative to black holes, have been studied extensively in the recent years ((Cattoen et al., 2005; Carter, 2005; Chirenti and Rezzolla, 2007; Cardoso et al., 2008; Chirenti and Rezzolla, 2008; Pani et al., 2009, 2010; Chirenti and Rezzolla, 2016)). One of the issues concerning gravastars is to find a rotating gravastar solution. So far only perturbative versions of such a solution exist ((Uchikata and Yoshida, 2015; Uchikata et al., 2016; Posada, 2017)). These studies indicate that in the ultracompact limit ((Mazur and Mottola, 2015)) the rotating gravastar can be a source of the Kerr metric (i.e. I, Love, Q numbers tend to those of Kerr in this limit). Similar perturbation-type sources (thin shells) of the Kerr metric were studied earlier by e.g, (Cohen, 1967; De La Cruz and Israel, 1968; Pfister and Braun, 1986). On the other hand, constructing perturbation sources of the Kerr metric have been criticised by Krasiński (1978).

In this work, we take perturbation approach to check if the matching of the gravastar with the Kerr spacetime survives at higher orders. It means that we want to construct a rotating analogue of (Mazur and Mottola, 2015) with the Kerr spacetime outside. We use slightly different framework to ((Uchikata and Yoshida, 2015; Uchikata et al., 2016; Posada, 2017)) and instead of solving Einstein equations both for interior and exterior, we a priori assume that an exterior solution is the Kerr metric. Then we seek for an interior solution and try to match it with the Kerr metric.

Most of the work on rotating gravastars was based on Hartle’s structure equations Hartle (1967) (see also (Reina and Vera, 2015; Mars et al., 2020a, b)). Hartle’s framework allows to study slowly rotating perfect fluid objects up to the second order in angular momentum. To go beyond the second order, we find it easier to follow Rostworowski (2017), who provided a nonlinear extension of Regge-Wheeler and Zerilli formalisms. Formalism given by Rostworowski (2017) is dedicated to ($\Lambda$-) vacuum spacetimes and can be easily adapted to our needs. The difference between Hartle’s framework and our approach is only on the level of ansatz on metric perturbation form and they are physically equivalent within the range of applicability of Hartle’s framework. We find that (Rostworowski, 2017) provides a very powerful tool for dealing with nonlinear perturbations. Although in the present article we describe perturbation analysis only up to the third order, we solved Einstein equations up to the sixth order to calculate Kretschmann scalar and we think it’s possible to go further if needed.

The paper is organised as follows: in Sections II, III and IV we provide preliminaries, in Section V we discuss the matching, in Section VI we expand the Kerr metric, in sections VII and VIII we solve interior Einstein equations and try to match interior and exterior metrics and in Section IX we summarise and discuss our calculations.

As a background, we take the ultracompact gravastar model (Mazur and Mottola, 2015). In static coordinates ($t$,$r$,$u$,$\varphi$), where $u=\cos\theta$, it’s metric is given by:

$\displaystyle\bar{g}=f(r)dt^{2}+\frac{1}{h(r)}dr^{2}+r^{2}\left(\frac{du^{2}}{1-u^{2}}+(1-u^{2})d\varphi^{2}\right)\,,$

(1)

where

	$\displaystyle f(r)=$	$\displaystyle\left\{\begin{array}[]{ll}\frac{1}{4}\left(1-\frac{r^{2}}{4M^{2}}\right)&\quad r\leq R\,,\\ 1-\frac{2M}{r}&\quad r>R\,,\end{array}\right.$		(4)
	$\displaystyle h(r)=$	$\displaystyle\left\{\begin{array}[]{ll}1-\frac{r^{2}}{4M^{2}}&\quad r\leq R\,,\\ 1-\frac{2M}{r}&\quad r>R\,.\end{array}\right.$		(7)

An induced metric is continuous across the (null) matching surface $r=2M$. There is a nonzero stress-energy tensor induced on this shell, see Mazur and Mottola (2015) for the details. The exterior metric is a solution to vacuum Einstein equations and the interior metric is a solution to Einstein equations with a cosmological constant $\Lambda=\frac{3}{4M^{2}}$. Both interior and exterior metrics are singular at $r=2M$. To keep them regular, also in higher perturbation orders, we use Eddington–Finkelstein (EF) coordinates ($v$,$r$,$u$,$\varphi$). Interior metric in EF coordinates reads:

$\displaystyle\bar{g}=\frac{1}{4}\left(1-\frac{r^{2}}{4M^{2}}\right)dv^{2}+drdv+r^{2}\left(\frac{du^{2}}{1-u^{2}}+(1-u^{2})d\varphi^{2}\right)\,.$

(8)

and exterior metric in EF coordinates reads:

$\displaystyle\bar{g}=\left(1-\frac{2M}{r}\right)dv^{2}+2drdv+r^{2}\left(\frac{du^{2}}{1-u^{2}}+(1-u^{2})d\varphi^{2}\right)\,.$

(9)

In a spherically symmetric background, in 3+1 dimensions, vector and tensor components split into two sectors: polar and axial (for the details see e.g. (Regge and Wheeler, 1957; Zerilli, 1970a, b; Nollert, 1999; Mukohyama, 2000)). Symmetric tensors have 7 polar and 3 axial components. Below we list the expansion of the components of symmetric tensors in axial symmetry ($P_{\ell}$ denotes the $\ell$-th Legendre polynomial).

The symmetric tensor, polar sector:

	$\displaystyle S_{ab}(r,u)=\sum\limits_{0\leq\ell}{S_{\ell}}_{ab}(r)P_{\ell}(u)\,,\quad a,b=v,r\,,$		(10)
	$\displaystyle S_{au}(r,u)=-\sum\limits_{1\leq\ell}{S_{\ell}}_{au}(r)\partial_{u}P_{\ell}(u)\,,\quad a=v,r\,,$		(11)
	$\displaystyle\frac{1}{2}\left((1-u^{2})S_{uu}(r,u)+\frac{S_{\varphi\varphi}(r,u)}{(1-u^{2})}\right)=\sum\limits_{0\leq\ell}{S_{\ell}}_{+}(r)P_{\ell}(u)\,,$		(12)
	$\displaystyle\frac{1}{2}\left((1-u^{2})S_{uu}(r,u)-\frac{S_{\varphi\varphi}(r,u)}{(1-u^{2})}\right)=$
	$\displaystyle=\sum\limits_{2\leq\ell}{S_{\ell}}_{-}(r)(-\ell(\ell+1)P_{\ell}(u)+2u\partial_{u}P_{\ell}(u))\,.$		(13)

The symmetric tensor, axial sector:

	$\displaystyle S_{a\varphi}(r,u)=\sum\limits_{1\leq\ell}{S_{\ell}}_{a\varphi}(r)(-1+u^{2})\partial_{u}P_{\ell}(u)\,,\quad a=v,r\,,$		(14)
	$\displaystyle S_{u\varphi}(r,u)=\sum\limits_{2\leq\ell}{S_{\ell}}_{u\varphi}(r)\left(\ell(\ell+1)P_{\ell}(u)-2u\partial_{u}P_{\ell}(u)\right)\,.$		(15)

We assume that there exists an exact, stationary and axially symmetric solution to Einstein equations, which we expand into series in a parameter $a$ (which will be an angular momentum per unit mass of a an exterior metric) around the static metric (4):

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+\sum\limits_{i=1}^{\infty}\frac{a^{i}}{i!}{}^{(i)}h_{\mu\nu}$

(16)

After perturbation expansion we polar–expand metric perturbations according to (10) - (15). Thus, apart from the perturbation index $i$, all perturbations gain an index $\ell$ corresponding to the $\ell$-th Legendre polynomial.

For axial perturbations we take:

$\displaystyle{}^{(i)}h_{\ell}=\begin{pmatrix}0&0&0&{}^{(i)}h_{\ell\,v\varphi}(r)(-1+u^{2})\partial_{u}P_{\ell}(u)\\ 0&0&0&{}^{(i)}h_{\ell\,r\varphi}(r)(-1+u^{2})\partial_{u}P_{\ell}(u)\\ 0&0&0&0\\ {}^{(i)}h_{\ell\,v\varphi}(r)(-1+u^{2})\partial_{u}P_{\ell}(u)&{}^{(i)}h_{\ell\,r\varphi}(r)(-1+u^{2})\partial_{u}P_{\ell}(u)&0&0\end{pmatrix}\,.$

(17)

Using the gauge freedom, we set ${}^{(i)}h_{\ell\,u\varphi}(r)=0$, what corresponds to the Regge-Wheeler (RW) gauge.

For the polar perturbations we take:

$\displaystyle{}^{(i)}h_{\ell}=\begin{pmatrix}{}^{(i)}h_{\ell\,vv}(r)P_{\ell}(u)&{}^{(i)}h_{\ell\,vr}(r)P_{\ell}(u)&0&0\\ {}^{(i)}h_{\ell\,vr}(r)P_{\ell}(u)&{}^{(i)}h_{\ell\,rr}(r)P_{\ell}(u)&0&0\\ 0&0&{}^{(i)}h_{\ell\,+}(r)\frac{P_{\ell}(u)}{1-u^{2}}&0\\ 0&0&0&{}^{(i)}h_{\ell\,+}(r)\left(1-u^{2}\right)P_{\ell}(u)\end{pmatrix}\,.$

(18)

Using the gauge freedom, we set ${}^{(i)}h_{\ell\,ru}={}^{(i)}h_{\ell\,vu}={}^{(i)}h_{\ell\,-}=0$, what also corresponds to the RW gauge. Note that in Hartle (1967) there are no ${}^{(i)}h_{\ell\,vr}$ and ${}^{(i)}h_{\ell\,r\varphi}$ coefficients in the metric ansatz. This fact arises from the fact that Hartle uses static coordinates. For EF coordinates in the background both ${}^{(i)}h_{\ell\,vr}$ and ${}^{(i)}h_{\ell\,r\varphi}$ turn out to be nonzero in most cases.

In the interior, we solve perturbation Einstein equations with a cosmological constant $\Lambda=\frac{3}{4M^{2}}$. For a given order $i$ and a given multipole $\ell$, they have the following form:

$\displaystyle\delta G_{\ell\,\mu\nu}={}^{(i)}S_{\ell\,\mu\nu}\,,$

(19)

where $\delta G_{\ell\,\mu\nu}$ denotes the components of the Einstein tensor expansion. ${}^{(i)}S_{\mu\nu}$ denotes a source for the $i$-th order Einstein equations consisting of metric perturbations of orders lower that $i$. We provide an explicit form of equations (19) in the Appendix A.

We match the exterior metric with the interior metric on a three-dimensional hypersurface located at $r^{\pm}=r_{b}^{\pm}$, where “+” and “-” stand for exterior and interior, respectively. From the first Israel junction condition ((Israel, 1966; Barrabès and Israel, 1991)) we demand continuity of the induced metric at the matching hypersurface:

$\displaystyle[[\mathfrak{g}^{\pm}_{ab}]]=0\,,$

(20)

where $[[E]]=E^{+}(r_{b}^{+})-E^{-}(r_{b}^{-})$. Following (Uchikata et al., 2016), we introduce intrinsic coordinates on the three-dimensional hypersurface: $y^{a}=(V,U,{\Phi})$. Then we express interior and exterior coordinates ${x}^{\pm\,\mu}$ on a hypersurface in terms of $y^{a}$:

	$\displaystyle x^{-\mu}\big{|}_{r_{b}^{-}}$	$\displaystyle=\left(A^{-}\,V,r_{b}^{-}(U),F^{-}(U),\Phi\right)\,,$		(21)
	$\displaystyle x^{+\mu}\big{|}_{r_{b}^{+}}$	$\displaystyle=\left(A^{+}\,V,r_{b}^{+}(U),F^{+}(U),\Phi\right)\,,$		(22)

where $r_{b}^{\pm}(U)=2M+\frac{a^{2}}{M^{2}}\eta^{\pm}(U)+\mathcal{O}(a^{4})$, $F^{\pm}(U)=U+\frac{a^{2}}{M^{2}}\lambda^{\pm}(U)+\mathcal{O}(a^{4})$. We expand $\eta^{\pm}$ into $\eta^{\pm}(U)=\eta_{0}^{\pm}+\eta_{2}^{\pm}P_{2}(U)$.

The metric induced on this hypersurface is given by:

$\displaystyle\mathfrak{g}^{\pm}_{ab}=\left(\begin{array}[]{ccc}\left(A^{\pm}\right)^{2}g^{\pm}_{vv}&A^{\pm}g^{\pm}_{vr}{r^{\pm}_{b}}^{\prime}(U)+A^{\pm}g^{\pm}_{vu}{F^{\pm}}^{\prime}(U)&A^{\pm}g^{\pm}_{v\varphi}\\ A^{\pm}g^{\pm}_{vr}{r^{\pm}_{b}}^{\prime}(U)+A^{\pm}g^{\pm}_{vu}{F^{\pm}}^{\prime}(U)&\left({F^{\pm}}^{\prime}(U)\right)^{2}g^{\pm}_{uu}+\left({r^{\pm}_{b}}^{\prime}(U)\right)^{2}g^{\pm}_{rr}+2{F^{\pm}}^{\prime}(U){r^{\pm}_{b}}^{\prime}(U)g^{\pm}_{ru}&{F^{\pm}}^{\prime}(U)g^{\pm}_{u\varphi}+{r^{\pm}_{b}}^{\prime}(U)g^{\pm}_{r\varphi}\\ A^{\pm}g^{\pm}_{v\varphi}&{F^{\pm}}^{\prime}(U)g^{\pm}_{u\varphi}+{r^{\pm}_{b}}^{\prime}(U)g^{\pm}_{r\varphi}&g^{\pm}_{\varphi\varphi}\\ \end{array}\right)\,.$

(26)

Using the freedom in a choice of coordinates $V,U,\Phi$, we set $F^{+}(U)=U$ and $A^{+}=1$ (see e.g. Uchikata and Yoshida (2015)). For simplicity, we denote $A^{-}=A$.

The location of the matching hypersurface is not known a priori and $\eta^{\pm}(U)$ and $\lambda^{-}(U)$ are unknown functions that need to be found. Our procedure of matching interior and exterior metrics for a given perturbation order is the following:

1. 1.

   We solve perturbation Einstein equations for the interior. These solutions contain two constans per $\ell$ in every perturbation order, but most of these constants need to be set to zero to keep Kretschmann scalar regular at $r=0$ and $r=2M$. However, this is not straightforward to apply, because in our case singularities of the Kretschmann scalar occur in higher perturbation orders than the singularities of the metric itself (in the opposition to the exterior case, e.g. Raposo et al. (2019)). Therefore, to settle constants in the third order, we solved Einstein equations up to the sixth perturbation order to study behaviour of the Kretschmann scalar. Since these expressions are too long to be listed in this paper, we make them available in the Mathematica notebook (Kre, ).

2. 2.

   We act with the general gauge transformation on the interior metric, and then we solve matching conditions (20) for constants arising from Einstein equations, for $\eta^{\pm}(U),\,\lambda(U),$ and for gauge components. Finding a proper gauge is a part of the matching problem and using the result of Bruni et al. (1997), we are able to control the impact of the gauge from the lower perturbation order on the metric functions in the higher perturbation order.

3. 3.

   If the matching is successful, we go to the higher perturbation order.

The second junction condition tells about the energy content of the matching hypersurface - already in the background solution there is a thin shell located at $r=2M$ (since this is a null hypersurface, second junction condition needs to be modified, see (Barrabès and Israel, 1991), (Mazur and Mottola, 2015) for the details). However, in the next sections we show that even the first junction condition is not possible to fulfil, therefore we don’t find it necessary to discuss second junction condition at all.

As an exterior metric, we take the Kerr solution. In the advanced EF coordinates it reads:

	$\displaystyle ds^{2}=$	$\displaystyle-\left(1-\frac{2Mr}{a^{2}u^{2}+r^{2}}\right)dv^{2}+2dvdr+\frac{a^{2}u^{2}+r^{2}}{1-u^{2}}du^{2}+\left(1-u^{2}\right)\left(\frac{2a^{2}Mr\left(1-u^{2}\right)}{a^{2}u^{2}+r^{2}}+a^{2}+r^{2}\right)d\varphi^{2}+$
		$\displaystyle+\frac{4aMr\left(1-u^{2}\right)}{a^{2}u^{2}+r^{2}}dvd\varphi+2a\left(1-u^{2}\right)drd\varphi\,.$		(27)

Since we solve the interior equations in RW gauge, we prefer to use the Kerr metric in RW gauge as well. To do this, we expand (27) into series in $a$ up to the 3rd order, and then act with the gauge transformations (115)-(117) to move to the RW gauge. Finally, we obtain:

	$\displaystyle ds^{2}=$	$\displaystyle-\left(\left(1-\frac{2M}{r}\right)-\frac{a^{2}M\left(u^{2}\left(6M^{2}-Mr-3r^{2}\right)-2M^{2}+Mr+r^{2}\right)}{r^{5}}\right)dv^{2}+$
		$\displaystyle+\left(\frac{2a^{2}M\left(1-3u^{2}\right)}{r^{3}}\right)dr^{2}+\left(\frac{r^{2}}{1-u^{2}}+\frac{a^{2}M\left(3u^{2}-1\right)(2M+r)}{r^{2}\left(u^{2}-1\right)}\right)du^{2}$
		$\displaystyle+\left(r^{2}\left(1-u^{2}\right)+\frac{a^{2}M\left(u^{2}-1\right)\left(3u^{2}-1\right)(2M+r)}{r^{2}}\right)d\varphi^{2}+$
		$\displaystyle+2\left(1+\frac{a^{2}M\left(3u^{2}-1\right)(M+r)}{r^{4}}\right)dvdr+2\left(\frac{a^{3}M\left(1-u^{2}\right)\left(5u^{2}-1\right)(9M+5r)}{5r^{4}}\right)drd\varphi+$
		$\displaystyle+2\left(\frac{2aM\left(1-u^{2}\right)}{r}-\frac{a^{3}M\left(u^{2}-1\right)\left(M^{2}\left(6u^{2}-2\right)+M\left(r-5ru^{2}\right)+r^{2}\left(1-5u^{2}\right)\right)}{r^{5}}\right)dvd\varphi\,+\mathcal{O}(a^{4}),$		(28)

For simplicity, we omit “+” and “-” coordinate superscripts and use them only when it’s necessary to differentiate the interior from the exterior. We expand (28) into series in $a$. Below we list nonzero components of this expansion after the polar decomposition.

$\displaystyle\begin{split}{}^{(1)}h^{+}_{1,\,v\varphi}&=-\frac{2M}{r}\,,\\ {}^{(2)}h^{+}_{0,\,vv}&=\frac{4M^{2}}{3r^{4}}\,,\\ {}^{(2)}h^{+}_{2,\,vv}&=\frac{4M\left(6M^{2}-Mr-3r^{2}\right)}{3r^{5}}\,,\\ {}^{(2)}h^{+}_{2,\,vr}&=\frac{4M(M+r)}{r^{4}}\,,\\ {}^{(2)}h^{+}_{2,\,rr}&=-\frac{8M}{r^{3}}\,,\end{split}\begin{split}{}^{(2)}h^{+}_{2,\,+}&=-\frac{4M(2M+r)}{r^{2}}\,,\\ {}^{(3)}h^{+}_{1,\,v\varphi}&=\frac{24M^{3}}{5r^{5}}\,,\\ {}^{(3)}h^{+}_{3,\,v\varphi}&=\frac{4M\left(-6M^{2}+5Mr+5r^{2}\right)}{5r^{5}}\,,\\ {}^{(3)}h^{+}_{3,\,r\varphi}&=-\frac{4M(9M+5r)}{5r^{4}}\,.\end{split}$

(29)