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Stability of the massive graviton around a BTZ black hole in three dimensions

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Abstract

We investigate the massive graviton stability of the BTZ black hole obtained from three dimensional massive gravities which are classified into the parity-even and parity-odd gravity theories. In the parity-even gravity theory, we perform the \(s\)-mode stability analysis by using the BTZ black string perturbations, which gives two Schrödinger equations with frequency-dependent potentials. The \(s\)-mode stability is consistent with the generalized Breitenlohner-Freedman bound for spin-2 field. It seems that for the parity-odd massive gravity theory, the BTZ black hole is stable when the imaginary part of quasinormal frequencies of massive graviton is negative. However, this condition is not consistent with the \(s\)-mode stability based on the second-order equation obtained after squaring the first-order equation. Finally, we explore the black hole stability connection between the parity-odd and parity-even massive gravity theories.

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Notes

  1. It is well-known that the Einstein gravity in three dimensions has no propagating degrees of freedom (DOF). This is clearly shown by counting a massless graviton \(h_{\mu \nu }\): \(D(D-3)/2\). One has zero DOF for \(D =3\). For a massive graviton, it is changed into \( (D-2)(D + 1)/2\) which gives 2 DOF for \(D = 3\). Thus, massive generalizations of the Einstein gravity [13, 18], allow propagating degrees of freedom. The three dimensional massive gravity is regarded as a toy model of perturbative quantum gravity, since we expect to have less severe short-distance behavior than four dimensional gravity with non-renormalizability. We note that if a black hole solution in three dimensional massive gravity is found, a first issue is to examine its classical stability properties as will be performed in the present paper.

  2. We note that the action \(S_\mathrm{FP}\) (2.6) has no diffeomorphism invariance, while the TT gauge condition is imposed only when considering diffeomorphism invariant actions of \(S_\mathrm{SDG},S_\mathrm{TMG}\), and \(S_\mathrm{GMG}. \)

  3. In order to eliminate scalar graviton, we require three conditions as [19]

    $$\begin{aligned} a_1=-3a_2/8,~~\alpha =\Lambda a_2/8-3\beta /8,~~-\!\sigma /2+3\Lambda ^2 a_2/4-\Lambda \beta /4\ne 0. \end{aligned}$$

  4. In the decoupling limit of \(m^2\rightarrow 0\), however, NMG action (2.10) reduces to massless NMG [20] where the fourth order equation appears, instead of the second order equation.

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Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.2012-R1A1A2A10040499).

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  1. Institute of Basic Sciences and Department of Computer Simulation, Inje University, Gimhae, 621-749, Korea

    Taeyoon Moon & Yun Soo Myung

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  1. Taeyoon Moon
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Correspondence to Yun Soo Myung.

Appendix: Inappropriateness of type I and II for parity-odd theory

Appendix: Inappropriateness of type I and II for parity-odd theory

We first substitute type I perturbation (2.18) into the TMG Eq. (3.5). It turns out that \((t,\phi )\) and \((r,\phi )\) components of (3.5) yield just \(\mu h_0(r)=0\) and \(\mu h_1(r)=0\), which implies that type I perturbation becomes null unless \(\mu =0\). Similarly, for type II perturbation (2.19), the components \((t,t), (r,r), (\phi ,r),\) and \((\phi ,\phi )\) of (3.5) are given by

$$\begin{aligned} \mu H_0(r)~=~0, \quad \mu H_2(r)~=~0,\quad \mu H_1(r)&= 0,\quad \mu H_3(r)~=~0, \end{aligned}$$

which leads to all null components for the type II perturbation of \(H_0=H_1=H_2=H_3=0\) unless \(\mu =0\).

For GMG, applying type I perturbation (2.18)–(3.12), we find that the corresponding solution to \((t,r)\) and \((r,r)\) components of Eq. (3.12) is given by \(h_0(r)=h_1(r)=0\). In type II case (2.19), we note that \(H_0(r), H_2(r)\), and \(H_3(r)\) can be expressed in terms of \(H_1(r)\) by considering the traceless condition, \((\phi ,t)\) component, and \((\phi ,r)\) component in (3.12), respectively:

$$\begin{aligned} \begin{aligned} H_0(r)&=\frac{\mathcal{M}-r^2/\ell ^2}{\omega r}\Big [(\mathcal{M}-3r^2/\ell ^2)H_1(r)+r(\mathcal{M}-r^2/\ell ^2)H_1^{\prime }(r)\Big ],\\ H_2(r)&=\frac{1}{\omega (\mathcal{M}-r^2/\ell ^2)}\Big [(\mathcal{M}-r^2/\ell ^2)H_1^{\prime }(r)-2rH_1(r)/\ell ^2\Big ], \\ H_3(r)&=-\frac{r(\mathcal{M}-r^2/\ell ^2)}{\omega }H_1(r). \end{aligned} \end{aligned}$$
(5.1)

Substituting (5.1) into \((t,r)\) and \( (t,\phi )\) components of (3.12), we find \(H_1(r)=0\). In this case, it yields \(H_0(r)=H_2(r)=H_3(r)=0\) when using (5.1) again. As a result type II perturbation becomes null for parity-odd gravity theories.

This proves the inappropriateness of type I and II for parity-odd theory.

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Moon, T., Myung, Y.S. Stability of the massive graviton around a BTZ black hole in three dimensions. Gen Relativ Gravit 45, 2493–2507 (2013). https://doi.org/10.1007/s10714-013-1600-3

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