Linear Stability of Mandal-Sengupta-Wadia Black Holes

TL;DR

This paper investigates the linear stability of non-extremal Mandal-Sengupta-Wadia black holes in (2+1)-dimensional gravity, demonstrating their stability against small, circularly symmetric perturbations through a Schrödinger-like analysis.

Contribution

It provides the first stability analysis of non-extremal MSW black holes using linear perturbation theory and Schrödinger-like equations.

Findings

MSW black holes are stable against small perturbations

Effective potential analysis confirms stability

Results apply to non-extremal configurations only

Abstract

In this letter, the linear stability of static Mandal-Sengupta-Wadia (MSW) black holes in $(2+1)$-dimensional gravity against circularly symmetric perturbations is studied. Our analysis only applies to non-extremal configurations, thus it leaves out the case of the extremal (2+1) MSW solution. The associated fields are assumed to have small perturbations in these static backgrounds. We then consider the dilaton equation and specific components of the linearized Einstein equations. The resulting effective Klein-Gordon equation is reduced to the Schr\"{o}dinger like wave equation with the associated effective potential. Finally, it is shown that MSW black holes are stable against to the small time-dependent perturbation

Equations

$$S_{EMD}=\int d^{3}x\sqrt{-g}\left[R-\frac{B}{2}{\left(\nabla \varphi \right)}^2-\exp \left(-4a\varphi \right)F_{\mu \nu }{F}^{\mu \nu }+2\exp \left(b\varphi \right)\Lambda \right]$$

$$R_{\mu \nu }=\frac{B}{2}{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi +\exp \left(-4a\varphi \right)\left(-g_{\mu \nu }{F}^2+2F_{\mu }^{\alpha }{F}_{\nu \alpha }\right)-2g_{\mu \nu }\exp \left(b\varphi \right)\Lambda ,$$

$${\nabla }^{\mu }\left(\exp \left(-4a\varphi \right)F_{\mu \nu }\right)=0,$$

$$\frac{B}{2}\left({\nabla }^{\mu }{\nabla }_{\mu }\varphi \right)+2a\exp \left(-4a\varphi \right)F^2+b\exp \left(b\varphi \right)\Lambda =0.$$

$$d{s}^2=-fd{t}^2+\frac{d{r}^2}{f}+\beta rd{\theta }^2,$$

$$f=8\Lambda \beta \left(\sqrt{r}-r_{+}\right)\left(\sqrt{r}-r_{-}\right),$$

$$r_{\pm }=\frac{m\pm \sqrt{m^2-64\Lambda \beta Q^2}}{8\Lambda \beta }.$$

$$T_H={\frac{1}{4\pi }\frac{d}{dr}\left(-g_{tt}\right)\sqrt{-g^{tt}{g}^{rr}}|}_{\sqrt{r}=r_{+},}$$

$$T_H=\frac{8\Lambda \beta r_{+}-m}{4\pi r_{+}}=\frac{\sqrt{m^2-64\Lambda \beta Q^2}}{4\pi r_{+}}.$$

$$d{s}^2=-\exp (2\gamma )d{t}^2+\exp (2\alpha )d{r}^2+\exp (2\eta )d{\theta }^2,$$

$$F_{tr}=q\exp \left(\alpha +\gamma -\eta +4\varphi \right),$$

$$R_{tr}=R_{rt}=4\stackrel{.}{\varphi }{\varphi }^{{}^{\prime }},$$

$$R_{\theta }^{\theta }=-2\left[\Lambda -q^2\exp (-2\eta )\right]\exp (4\varphi ),$$

$$\gamma \equiv \gamma \left(r,t\right)={\gamma }_0\left(r\right)+\in {\gamma }_1\left(r,t\right),$$

$$\alpha \equiv \alpha \left(r,t\right)={\alpha }_0\left(r\right)+\in {\alpha }_1\left(r,t\right),$$

$$\eta \equiv \eta \left(r,t\right)={\eta }_0\left(r\right)+\in {\eta }_1\left(r,t\right),$$

$$\varphi =\varphi \left(r,t\right)={\varphi }_0\left(r\right)+\in {\varphi }_1\left(r,t\right).$$

$$R_{tr}=\stackrel{.}{\alpha }{\eta }^{{}^{\prime }},$$

$$R_{\theta }^{\theta }=\left[{\eta }^{{}^{\prime }}({\alpha }^{{}^{\prime }}-{\gamma }^{{}^{\prime }})-({\eta }^{{}^{\prime }}{)}^2-{\eta }^{{}^{\prime \prime }}\right]\exp \left(-2\alpha \right),$$

$$\exp (-2\alpha )\left[{\varphi }^{{}^{\prime \prime }}-{\varphi }^{{}^{\prime }}({\alpha }^{{}^{\prime }}-{\gamma }^{{}^{\prime }}-{\eta }^{{}^{\prime }})\right]-\exp (-2\gamma )\left[\stackrel{..}{\varphi }+\stackrel{.}{\varphi }(\stackrel{.}{\alpha }-\stackrel{.}{\gamma })\right]+$$

$$\exp (4\varphi )\left[-\Lambda -q^2\exp (-2\eta )\right]=0.$$

$$\stackrel{.}{{\alpha }_1}+2\stackrel{.}{{\varphi }_1}=0,$$

$$4\left(\Lambda \beta r-Q^2\right)\left({\alpha }_1+2{\varphi }_1\right)+r\left(4\Lambda \beta r-m\sqrt{r}+4Q^2\right)({\alpha }_1^{{}^{\prime }}-{\gamma }_1^{{}^{\prime }})=0,$$

$$({\alpha }_1^{{}^{\prime }}-{\gamma }_1^{{}^{\prime }}+4r{\varphi }_1^{{}^{\prime \prime }})(4\Lambda \beta r-m\sqrt{r}+4Q^2)+4({\alpha }_1+2{\varphi }_1)\left(\Lambda \beta -\frac{Q^2}{r}\right)-$$

$$\frac{r}{(4\Lambda \beta r-m\sqrt{r}+4Q^2)}\stackrel{..}{{\varphi }_1}+4(6\Lambda \beta r-m\sqrt{r}+2Q^2){\varphi }_1^{{}^{\prime }}=0.$$

$$4\left(\Lambda \beta r-Q^2\right)\left({\alpha }_1+2{\varphi }_1\right)+r\left(4\Lambda \beta r-m\sqrt{r}+4Q^2\right)({\alpha }_1^{{}^{\prime }}-{\gamma }_1^{{}^{\prime }})+4r^2{\varphi }_1^{{}^{\prime }}\left(4\Lambda \beta r-m\sqrt{r}+4Q^2\right)$$

$$-\frac{r^2}{(4\Lambda \beta r-m\sqrt{r}+4Q^2)}\stackrel{..}{{\varphi }_1}+4r(6\Lambda \beta r-m\sqrt{r}+2Q^2){\varphi }_1^{{}^{\prime }}=0,$$

$${\varphi }_1^{{}^{\prime \prime }}+\left[\frac{6\Lambda \beta r-m\sqrt{r}+2Q^2}{r(4\Lambda \beta r-m\sqrt{r}+4Q^2)}\right]{\varphi }_1^{{}^{\prime }}-\frac{\stackrel{..}{{\varphi }_1}}{(8\Lambda \beta r-2m\sqrt{r}+8Q^2{)}^2}=0.$$

$${\varphi }_1\left(r,t\right)={\varphi }_1\left(r\right)\exp \left(-ikt\right),$$

$${\varphi }_1^{{}^{\prime \prime }}\left(r\right)+h{\varphi }_1^{{}^{\prime }}\left(r\right)-j{\varphi }_1\left(r\right)=0,$$

Full text

Linear Stability, Dilaton, MSW Black Hole, Perturbation, Klein-Gordon equation

I Introduction

One of the compelling problems in black hole physics is stability (for a review see Dafermos and references therein). The stability of a black hole presents an ideal theoretical test-bed for any gravity theory Berti . In the pioneering works on black hole stability STABL1 ; STABL2 ; STABL3 ; STABL4 , the linear stability of gravitational perturbations of the Schwarzschild black hole was comprehensively studied. The mode-stability of the Kerr black hole was first proved in KERRstb , which employed the Teukolsky equation Chandram . Recent developments have shown that there is a conserved and positive definite energy in the axially symmetric linear gravitational perturbations of the extreme Kerr black hole extreme . This work stipulates the framework of linear stability in axial symmetry. Although there exist worthwhile studies in literature regarding the non-linear stability of black holes Hintz1 ; Hintz2 , further progress in the concerned topic is required. In fact, linear stability studies are expected to clarify the non-linear stability problem. However, this is conditional on techniques that can be suitably extended to the non-linear regime instability . In general, no-hair theorems (NHTs) NHT are about the existence of black hole solutions and they do not consider the stability of black holes. However, we know from the literature that some black hole solutions have been rejected due to their instability under perturbation Kanti . As a matter of fact, there is a sort of consensus among physicists that a new NHT is more likely to be accepted when the stability of a black hole solution is assured. On the other hand, many black holes that are well known in the literature and proved by NHTs have not been tested for stability. For example, from the literature, the linear stability of the famous three dimensional MSW black hole solution MSW , which is a solution in three dimensions from the classical dilaton system of Chan and Mann ChanMann , has not been studied before. Like the Ba$\widetilde{n}$ados-Teitelboim-Zanelli (BTZ) black hole model BTZbh , which is a well-known toy model of three-dimensional black holes using general relativity theory, MSW black holes have attracted much attention. For instance, the problems of spectroscopy, thermodynamics, Hawking radiation, quasinormal modes, and scalar perturbations for this black hole have been extensively studied in msw1 ; msw2 ; msw3 ; msw4 ; msw5 . It is worth noting that the studies on the $(2+1)$-dimensional black holes have the potential to generate valuable insight into many conceptual issues that arise in $(3+1)$-dimensional black holes. Among those studies, Cardoso and Lemos CLBTZ showed that quasinormal modes of the BTZ black hole for which the frequencies are no longer pure real which means that the system is losing energy. Therefore, the QNMs dominate the signal during the intermediate stages of the perturbation, being therefore extremely important. In line with the study Clem , the main aim of the present letter is to address the stability problem of MSW black holes, as hinted above. To this end, MSW dilaton black holes are examined to see if they are stable against small spacetime-dependent perturbations. The letter is laid out as follows. In Sect. II, we review MSW black holes and their characteristic features. The stability of the charged MSW static solutions against small perturbations is studied in Sect. III. Finally, in Sect. IV we draw conclusions. (Throughout the letter, natural units with $c=G=k_{B}=\hbar=1$ are used).

II MSW Black Holes

MSW black holes redound to investigations in which the concepts of general relativity and string theory can be linked. Examining the actions of Einstein-Maxwell-dilaton and string theories Horow , and using conformal field transformations to relate them to each other, the unification of these two theories can be achieved ChanMann . Throughout this section, we focus on MSW black hole solutions from the general relativity perspective only.

$(2+1)$-dimensional Einstein-Maxwell-dilaton action is given by ChanMann

[TABLE]

in which $\phi$ and $F_{\mu\nu}$ represent the fields of concern (dilaton and Maxwell fields, respectively), $a$, $b,$ and $B$ are arbitrary constants, and $\Lambda$ stands for the cosmological constant. If we perform variations in the metric, gauge, and dilaton field, we obtain the following equations of motion:

[TABLE]

[TABLE]

[TABLE]

The above field equations with $b=4a=\frac{B}{2}=4$, and $\phi=-\frac{1}{4}\ln\left(\frac{r}{\beta}\right)$ {$\beta:$constant} lead to the charged MSW black hole solutions, which can be stated as ChanMann

[TABLE]

where $f=8\Lambda\beta r-2m\sqrt{r}+8Q^{2}$, which can be rewritten as

[TABLE]

in which $r_{+}$ and $r_{-}$ correspond to the event (outer) and inner horizons of the charged MSW black holes:

[TABLE]

For the MSW black hole ChanMann , the dilaton field is given $\phi=-\frac{1}{4}ln(\frac{r}{\beta})$ and the non-zero Maxwell tensor components are $F_{tr}=-F_{rt}=e^{4\phi}\frac{Q}{\sqrt{\beta{r}}}=\frac{Q\sqrt{\beta}}{r^{\frac{3}{2}}}$ in which $Q$ denotes the electric charge belonging to the electromagnetic vector potential $A_{\mu}=2Q\sqrt{\frac{\beta}{r}}\delta{{}_{\mu}^{t}}$. To have a black hole solution, ${m}\geqslant 8Q\sqrt{\Lambda\beta}$. For the uncharged MSW black holes (at the limit of $Q\rightarrow 0$), the horizons reduce to $r_{+}=\frac{m}{4\Lambda\beta}$ and $r_{-}=0$.

Using Eq. (7), one can obtain the Hawking temperature of the MSW black holes via msw5

[TABLE]

which results in

[TABLE]

At the $Q=0$ limit, the Hawking temperature of an uncharged MSW black hole becomes $\left.T_{H}\right|_{Q=0}=\frac{\Lambda\beta}{\pi}$, which means that having an ordinary black hole (with positive-valued temperature) is conditional on $\Lambda\beta\geq 0$. Moreover, it is obvious from Eq. (8) that for a real physical temperature of the charged MSW black hole, the condition of ${m}\geqslant 8Q\sqrt{\Lambda\beta}$ should also be stipulated. On the other hand, it is worth noting that a negative temperature is a well-known issue in spin systems with some upper energy level limits Kittel , and corresponds to a non-blackbody spectrum for exotic black holes Park .

III Stability of MSW Black Holes

Our purpose is to explore the linear stability of the charged MSW black holes. To this end, we borrowed some ideas from Clem by considering the breakneck perturbation: the $s$-mode.

The electrically charged circularly symmetric static solution of the Einstein-Maxwell-dilaton field equations can be described by

[TABLE]

with the Maxwell tensor component

[TABLE]

where $\gamma$, $\alpha$, and $\eta$ are ($r$,$t$) dependent metric functions and $q=Q$. The field equation (2) admits the following non-zero $R_{tr}$ and $R_{\theta}^{\theta}$ components:

[TABLE]

[TABLE]

where prime and dot denote derivatives with respect to $r$ and $t$, respectively. One can assume that the metric functions and the dilaton field have the following small perturbations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Setting $\eta_{1}\left(r,t\right)=0$ gauge and comparing the metrics (5) and (10), one obtains the following identity: $\exp(2\eta)=\beta{r}$. Thus, after a straightforward calculation, $R_{tr}$ and $R_{\theta}^{\theta}$ components of the Ricci tensor and the Klein-Gordon equation (4) become

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Matching Eqs. (12) and (13) with Eqs. (18) and (19) and performing the perturbations considering Eqs. (14-17) with the gauge of $\eta_{1}\left(r,t\right)=0$, the linearized forms of the Einstein equations for the $R_{tr}$ and $R_{\theta}^{\theta}$ components and the Klein-Gordon equation (20) yield the following expressions:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Multiplying Eq. (23) by $r$ leads to

[TABLE]

[TABLE]

and the first two terms vanish due to Eq. (22). This implies that the linearized Klein-Gordon equation is independent of the solution ($\alpha_{1}=-2\phi_{1}+f(r)$) of Eq. (21), since Eq. (24) comprises Eq. (22). Ultimately, one can get

[TABLE]

Using the Fourier transformation with respect to the time coordinate, one can introduce

[TABLE]

where $k$ (real) represents the frequency. Therefore, Eq. (25) reduces to the following effective Klein-Gordon equation:

[TABLE]

where h\and $j$ are the functions of $r$, which are provided as

[TABLE]

[TABLE]

One can introduce the tortoise coordinate as follows.

[TABLE]

The range $r_{+}<\sqrt{r}<\infty$ corresponds to $−\infty<u<\infty$ (since $u\rightarrow-\infty$ as $\sqrt{r}\rightarrow r_{+}$). As one approaches to the MSW black hole, the radial coordinate changes more and more slowly with $u$, since $\frac{d{r}}{du}\rightarrow 0$. Hence, we only look at the spacetime outside the horizon. The tortoise coordinate (30) indicates that once the MSW black hole is extremal ($r_{+}=r_{-}$), the analysis performed cannot be straightforwardly applied; as $u$ becomes undefined. After some manipulations, one can show that Eq. (27) recasts in

[TABLE]

where

[TABLE]

Setting

[TABLE]

and substituting Eq. (33) into Eq. (31), after a straightforward calculation, one can obtain a Schrödinger like one-dimensional wave equation (see for example Chandrasekhar’s exhaustive book Chandram ):

[TABLE]

where $V_{eff}(r)$ is the effective potential, given by

[TABLE]

As discussed in the study of Dennhardt and Lechtenfeld Dennhardt , in order for proving the stability of the black hole spacetime, it is required to show the nonexistence of the $k^{2}<0$ solution, i.e. non-existence of an exponentially growing solution Kimura . In the black hole case (${m}\geqslant 8Q\sqrt{\Lambda\beta}$), one can see that $V_{eff}\left(r\right)$ is positive definite for all $\sqrt{r}>r_{+}$. This is because when $\sqrt{r}>r_{+},$ $f>0$, one obtains

[TABLE]

Non-negative $V_{eff}\left(r\right)$ implies $k^{2}\geqslant 0$ so that there are no bounded solutions Dennhardt . Namely, the positiveness of the effective potential governing the perturbations forbids any unstable modes. Thus, the MSW black holes are linearly stable in the $s$-sector.

IV Conclusion

We applied linear stability analysis to MSW black holes. For this purpose, we followed a semi-analytical method used in Kanti ; Clem which is mainly based on the Fubini-Sturm theorem Fubini . Then, we considered small spacetime-dependent perturbations in both $R_{tr}$ and $R_{\theta}^{\theta}$ components of the Einstein equations and in the Klein Gordon equation (4). The key feature of the applied method was reduction of the linearized equations to a single one-dimensional Schrödinger type differential equation. As a result, the effective potential of the MSW black holes was derived. In the black hole case (${m}\geqslant 8Q\sqrt{\Lambda\beta}$), the effective potential (35) was found to be positive through $\sqrt{r}>r_{+}$. This means that there are no bounded solutions of Eq. (34) Dennhardt . Thus, we concluded that the MSW black hole solutions are linearly stable for the potentially most dangerous perturbation, which is the $s$-mode. Our findings give support to SMSW , in which it was shown that MSW black holes are stable against small time-dependent perturbation.

Our future plan is to extend our linear stability analysis to higher dimensional ($d\geq 3$) rotating black holes in Einstein-Maxwell-dilaton gravity Sheykhi . Through this, we aim to see the effects of rotation parameters and the number of dimensions on the stability of black holes.

Acknowledgements.

We are thankful to the Editor and anonymous Referees for their constructive suggestions and comments.