Black string in dRGT massive gravity
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We present a cylindrically symmetric solution, both charged and uncharged, which is known as a black string solution to the nonlinear ghost-free massive gravity found by de Rham, Gabadadze, and Tolley (dRGT). This “dRGT black string” can be thought of as a generalization of the black string solution found by Lemos. Moreover, the dRGT black string solution includes other classes of black string solution such as the monopole-black string ones since the graviton mass contributes to the global monopole term as well as the cosmological-constant term. To investigate the solution, we compute mass, temperature, and entropy of the dRGT black string. We found that the existence of the graviton mass drastically affects the thermodynamics of the black string. Furthermore, the Hawking–Page phase transition is found to be possible for the dRGT black string as well as the charged dRGT black string. The dRGT black string solution is thermodynamically stable for \(r>r_c\) with negative thermodynamical potential and positive heat capacity while it is unstable for \(r<r_c\) where the potential is positive.
In general relativity (GR), where the corresponding graviton is a massless spin-2 particle, is the current description of gravitation in physics and has importantly astrophysical implications. One of the alternating theory of gravity, which follows a very simple idea of generalization to GR, is known as massive gravity where interaction terms, corresponding to graviton mass, are added into GR. The result of such an introduction is that the model of gravity is modified significantly at a large scale so that, from the cosmological point of view, a system like our universe can enter the cosmic accelerating expansion phase. Not only that; the gravity at smaller scale is not altered much by such a modification to ensure the same predictions as GR makes.
However, adding generic mass terms for the graviton on given background usually brings about various instabilities and ghosts for the gravitational theories. To avoid the appearance of the ghost in massive gravity suggested by Boulware and Deser , the set of possible interaction terms is constructed accordingly by de Rham, Gabadadze and Tolley (dRGT) [2, 3]. In particular, the interaction terms are constructed in a specific way to ensure that the corresponding equations of motion are at most second order differential equations so that there is no ghost field. The allowed interaction terms for the four-dimensional theory consist of three kinds of combination; the quadratic, the cubic, and the quartic orders of the metric. Unfortunately, such nonlinearities involve complexities in the calculation as a price to pay for eliminating the ghost and usually make it cumbersome to find an exact solution. Nevertheless, there have been plenty interesting approaches to tackle the complexity to obtain spherical symmetric solutions in massive gravity theories [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In particular, it was found by Vegh  that a spherically symmetric black hole with a Ricci flat horizon is a solution to the four-dimensional massive gravity and then the solution was studied in the aspect of thermodynamical properties and phase transition structure [5, 6]. The spherically symmetric solutions for dRGT were also considered in [9, 10]. The extension of the solution in terms of electric charge was found in [11, 12]. Moreover, the bi-gravity extension of the solution was found in  which covers the previously known spherically symmetric solutions (see [14, 15, 16] for reviews on black holes in massive gravity). Recently, other extensions of the dRGT massive gravity solution were explored [27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
In astrophysics, it is commonly known that black holes can be created by the gravitational collapse of massive stars. However, the famous Thorne  hoop conjecture states that gravitational collapse of a massive star will yield a black hole only if the mass M is compressed to a region with circumference \( C <4 \pi M \) in all directions. If the hoop conjecture is true, cylindrical matter will not form a black hole. However, the hoop conjecture was given for a spacetime with a zero cosmological constant. When a negative cosmological constant is introduced, the spacetime will become an asymptotically anti-de Sitter spacetime. Indeed, Lemos  has shown a possibility of cylindrical black holes from the gravitational collapse with cylindrical matter distribution in an anti-de Sitter spacetime, violating in this way the hoop conjecture. This cylindrically static black hole solutions in an anti-de Sitter spacetime are called black strings. The pioneering work on the black string is due to Lemos [39, 40, 41], and soon its charged and rotating  counterpart were also obtained.
The main purpose of this paper is to present an exact black string solution, including its generalization to the charge case in dRGT massive gravity, and also to discuss their thermodynamical properties with a focus on thermodynamic stability. The possibility to obtain a Hawking–Page phase transition is explored. We use units which fix the speed of light and the gravitational constant via \(G = c = 1\), and use the metric signature (\(-,\;+,\;+,\;+\)).
2 dRGT massive gravity
3 Black string solution in dRGT massive gravity
One can see that the horizon structure depends on the sign of \(\alpha _m^2\). If \(\alpha _m^2>0\), corresponding to an anti-de Sitter-like solution, the maximum number of horizons is three. If \(\alpha _m^2<0\), corresponding to de Sitter-like solution, the maximum number of horizons is two. The generic behavior of the horizon structure is shown in Fig. 1. This solution modifies the black string solution in general relativity with cosmological constant  by the three parameters \(c_0, c_1\) and \(\alpha _m^2\).
3.1 Thermodynamics of dRGT black string
4 Charged dRGT black string
4.1 Grand canonical black string
4.2 Canonical black string
5 Concluding remarks
The dRGT massive gravity describes nonlinear interaction terms as a generalization of the Einstein–Hilbert action when the graviton is massive. It is believed that dRGT massive gravity may provide a possible explanation for the accelerating expansion of the universe that does not require any dark energy or cosmological constant. In this paper, we have presented a class of black strings, both charged and uncharged, in dRGT massive gravity where the effective cosmological constant is negative, and we studied the thermodynamics and phase structure of the solutions in both the grand canonical and the canonical ensembles. The black string obtained is immensely simplified due to the choice of the fiducial metric. As expected the solution contains the Lemos solutions  as a particular case. Our dRGT black string solution can be identified, e.g., as a monopole-black string of general relativity for a suitable choice of the parameters of the theory where the graviton mass in massive gravity naturally generates the cosmological constant and the global monopole term.
We have also analyzed the thermodynamical properties of the dRGT black string solution. The thermodynamic quantities have also been found to contain corrections from the graviton mass except for the black string entropy which is unaffected by massive gravity and still obeys the area law. By analyzing the thermodynamical properties of the black string solution, we found that it is possible to obtain the Hawking–Page phase transition in the dRGT black string case, while this is not possible for Lemos solution. The conditions to provide such a transition were explored. In fact, we determined the phase transition in both charged and non-charged case by analyzing the sign of the potential at \(r=r_c\), with the stable (unstable) branch for \(r > (<) r_c\).
The results presented here are a generalization of previous discussions of Lemos [39, 40, 41], on the black string, in a more general setting. The possibility of a further generalization of these results to higher dimensions is an interesting problem for future research. The rotating dRGT black string solution is also interesting, since most astronomical objects are rotating. We leave this investigation for further work.
This project is supported by the ICTP through Grant no. OEA-NT-01 and the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (no. 2016R1C1B1010107). PW is also supported by the Naresuan University Research Fund through Grant no. R2559C235. S.G.G. would like to thank SERB-DST Research Project Grant no. SB/S2/HEP-008/2014 and also IUCAA, Pune, where part of this work was done, for hospitality.
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