Investigations of strong cosmic censorship in 3-dimensional black strings

Published for SISSA by Springer Received: March 2, 2022 Revised: June 22, 2022 Accepted: June 25, 2022 Published: August 1, 2022 Jeongwon Ho,a Wontae Kima,b and Bum-Hoon Leea,b a Center for Quantum Spacetime, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107, Republic of Korea b Department of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107, Republic of Korea E-mail: , , Abstract: Investigating the quasinormal modes of a massive scalar field on the 3dimensional black string (3dBS), we study the strong cosmic censorship (SCC) conjecture for the 3dBS in the T-dual relationship with the 3-dimensional rotating anti-de-Sitter (BTZ) black hole. It is shown that even though geometries of the two spacetimes are quite different, such as asymptotically AdS for the BTZ black hole and asymptotically flat for the 3dBS, the BTZ black hole and the 3dBS share similar properties for the SCC. Concretely speaking, the SCC conjecture can be violated even for asymptotically flat spacetime, i.e. the 3dBS. These observations lead us to an assumption that the T-dual transformation preserves spacetime symmetries, at least, which are relevant to the SCC. In addition, we find a new feature of the quasinormal mode at the Cauchy horizon: in the case of in the 3dBS, the spectral gap, αBS at the Cauchy horizon is not determined by the ‘ω-frequency mode’, but the ‘m-frequency mode’. Keywords: Spacetime Singularities, Black Holes, Black Holes in String Theory, String Duality ArXiv ePrint: 2202.12561 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2022)018 JHEP08(2022)018 Investigations of strong cosmic censorship in 3-dimensional black strings Contents 1 2 Three dimensional black strings 3 3 Quasinormal modes of massive scalar fields on 3dBS 6 4 Strong cosmic censorship conjecture in 3dBS 9 5 Discussions 13 1 Introduction In general relativity, we believe that, given suitable initial data, we can uniquely determine the geometry by solving Einstein’s equations. However, in the case of rotating or charged black holes, which have an inner horizon serving as a boundary of future Cauchy development, the story is not so simple. Since the spacetime region beyond the Cauchy horizon (CH) is not uniquely determined by given initial data, general relativity loses its predictive power. About 40 years ago, Penrose [1, 2] proposed a conjecture that perturbations incoming from outside the event horizon are infinitely blueshifted at the CH and backreaction makes the CH singular, so that beyond the CH the Einstein equation ceases to make sense and the general relativity theory recovers its predictive power. It is called the strong cosmic censorship (SCC) conjecture [3–5]. Indeed, it has been shown that the SCC conjecture is viable for the Reissner-Nordström (RN) and Kerr black holes which are asymptotically flat [6–11]. On the other hand, it has been known that perturbations outside the event horizon decay with an inverse power law in an asymptotically flat spacetime [12–15]. Accordingly, it seems that the competition between the decay rate of perturbations outside the event horizon and the amplification of blueshifted perturbation determines the viability of the SCC conjecture. In other words, these perturbations are infinitely blueshifted at the CH, because in the case of the asymptotically flat spacetime background they do not decay fast enough outside the event horizon, so that the SCC is respected. Such an observation that the validity of the SCC conjecture depends on the asymptotic geometry of black hole naturally leads to studies on the SCC for the de-Sitter (dS) and the anti-de-Sitter (AdS) black holes on which outside the event horizons perturbations exhibit an exponential decay [16, 17] and an inverse logarithmic decay [18, 19], respectively. Indeed, it has been shown that according to the decay rates of perturbations, the SCC conjecture is strengthened for RN-AdS [20, 21] and Kerr-AdS black holes [22] and is weakened for RN-dS black holes [23–25] (See also [26, 27]). Specifically, the SCC conjecture is violated for the near extremal RN-dS black hole [23–25]. –1– JHEP08(2022)018 1 Introduction –2– JHEP08(2022)018 From this, it can be seen that together with asymptotic geometries, the near extremal condition plays an essential role in determining the fate of the SCC conjecture. Such a result could be understood as follows: taking the near extremal limit is effectively similar with applying the near horizon limit, which leads to an enhanced spacetime symmetry [28]. Since the enhanced symmetry gets the decay of perturbations faster outside the event horizon, the amplification of blueshifted perturbations would not be big enough for the perturbations to be singular at the CH. Thus, we may say that an enhancement in the spacetime symmetry has made the SCC conjecture fail. From the same footing, it is expected that conversely, a reduction in the symmetry makes the SCC to be realized. Indeed, it has been shown that the photon sphere quasinormal mode, which cannot be found in the spherically symmetric RN-dS black hole background, exists in the axisymmetric Kerr-dS black hole background and decays sufficiently slowly to ensure that the SCC is respected for any nonextremal value of the black hole parameters [29]. This story continues with the BTZ black hole [30]. The logarithmic decay of perturbations is originated from the stable trapping phenomenon for 4-dimensional asymptotically AdS black holes [19]. However, since the 3-dimensional general theory of relativity does not have gravitational dynamics, the stable trapping phenomenon does not appear in the rotating BTZ black hole, which is a 3-dimensional AdS black hole. Since the factor causing the logarithmic decay of perturbations outside the event horizon disappears, it is expected that the SCC conjecture could be broken even though the BTZ black hole is asymptotically AdS. In fact, Dias et al. [31] have shown that the near extremal rotating BTZ black hole badly violates the SCC conjecture (see also [32–35]). In summary, it has been found that asymptotic geometries (signs of cosmological constants), rotations of black holes, near extremal limits for black hole parameters, and the number of spacetime dimensions play an important role in the SCC conjecture and they are more or less associated with the background spacetime symmetry. Thus, even though we do not have a unified description for the relation between the spacetime symmetry and the SCC, we can say that the background spacetime symmetry is essential for examining the SCC. On the other hand, it is believed that in string theory, the T-dual transformation significantly changes string background geometries, but leaves unchanged the physics of the theory, i.e. all observable quantities in one description are identified with quantities in the dual description [36]. It is well known that in the context of the low energy str (...truncated)