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Non-linear regime of the Generalized Minimal Massive Gravity in critical points

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Abstract

The Generalized Minimal Massive Gravity (GMMG) theory is realized by adding the CS deformation term, the higher derivative deformation term, and an extra term to pure Einstein gravity with a negative cosmological constant. In the present paper we obtain exact solutions to the GMMG field equations in the non-linear regime of the model. GMMG model about \(AdS_3\) space is conjectured to be dual to a 2-dimensional CFT. We study the theory in critical points corresponding to the central charges \(c_-=0\) or \(c_+=0\), in the non-linear regime. We show that \(AdS_3\) wave solutions are present, and have logarithmic form in critical points. Then we study the \(AdS_3\) non-linear deformation solution. Furthermore we obtain logarithmic deformation of extremal BTZ black hole. After that using Abbott–Deser–Tekin method we calculate the energy and angular momentum of these types of black hole solutions.

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Notes

  1. In our notation \(G=1\).

  2. Here we should mention that may be exist other solutions which are not investigated here. Boundary condition restrict the scope of solutions. So by some special boundary conditions some of solutions will be excluded. To clarifying the matter, we focus on the solution (41). If we demand following boundary conditions

    $$\begin{aligned} g_{\mu \nu }= \begin{pmatrix} \mathcal {O} (r^{2}) &{} \mathcal {O} (1) &{} \mathcal {O} (r^{2}) \\ &{} \mathcal {O} (\frac{1}{r^{4}}) &{} \mathcal {O} (1) \\ &{} &{} \mathcal {O} (r^{2}) \end{pmatrix} \end{aligned}$$

    \(d_{2}\) vanishes in (41) and we lost logarithmic term. In any case we must impose some boundary conditions, but they needs to satisfy finiteness of asymptotic charges. Due to this, here (38) and (41) are good solutions with finite asymptotic charges. In the other hand there are some interesting solutions, like (39), (40), etc, whose asymptotic charges can not be calculated by the ADT method.

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Acknowledgments

We would like to thank the anonymous referee for useful comments on this work.

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  1. Department of Science, University of Kurdistan, Sanandaj, Iran

    M. R. Setare & H. Adami

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  1. M. R. Setare
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Correspondence to M. R. Setare.

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Setare, M.R., Adami, H. Non-linear regime of the Generalized Minimal Massive Gravity in critical points. Gen Relativ Gravit 48, 36 (2016). https://doi.org/10.1007/s10714-016-2033-6

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