Slowly rotating black hole solution in the scalar-tensor theory with nonminimal derivative coupling and its thermodynamics

Abstract

We obtain a slowly rotating black hole solution in the scalar-tensor theory of gravity with nonminimal derivative coupling to the Einstein tensor. Properties of the obtained solution have been examined carefully. We also investigate the thermodynamics of the given black hole. To obtain thermodynamic functions, namely its entropy we use the Wald procedure which is suitable for quite general diffeomorphism-invariant theories. The applied approach allowed us to obtain the expression for entropy and the first law of black hole thermodynamics. Having introduced thermodynamic pressure which is related to the cosmological constant we have examined thermodynamics of the black hole in the so called extended phase space. The extended phase space and specifically chosen scalar "charge" allowed us not only to obtain the generalized first law but also derive the Smarr relation. The behaviour of black hole's temperature, heat capacity and Gibbs free energy shows a lot of similarities with the behaviour of the corresponding values for Schwarzschild-AdS black hole in standard General Relativity.

Static topological black hole with nonminimal derivative coupling and nonlinear electromagnetic field of Born-Infeld type

“…Since the metric function W (r) diverges on the horizon changing its sign while crossing the horizon, the same is true for the function (ϕ ′ ) 2 , unless the expression α + Λη − 2β 2 η + 2βηr 1−n q 2 + β 2 r 2(n−1) also changes it sign on the horizon. It leads to the consequence that in the inner domain ϕ ′ becomes purely imaginary (phantom-like behaviour), similar situation takes place in case of neutral and charged black holes [52,55]. We point out there that the kinetic energy of the scalar field K = ∇ µ ϕ∇ µ ϕ is finite at the horizon and is positive in the inner domain up to the moment when the expression α + Λη − 2β 2 η + 2βηr 1−n q 2 + β 2 r 2(n−1) changes it sign.…”

“…It should be stressed that the infinite sums in the written above relations ( 26) and ( 27) are convergent when r > d and r 2(n−1) > q 2 /β 2 (which also takes place for large distances), but there is no difficulty in writing the evident form for the metric function U (r) when r 2(n−1) > q 2 /β 2 and d < r or for other two possible options for the distance (small distances). As we have noted above, the condition d 2 > 0 is imposed after the integral form for the metric function U (r) is written, we point out here that solutions with d 2 < 0 might be studied, but similarly as it was shown for neutral black hole [52] or power-law field [55] the corresponding solutions do not represent a black hole.…”

“…Having supposed that the coupling parameters α and η are positive (and assuming that the expression 2αr 2 + εη(n − 1)(n − 2) is positive in the outer domain) one arrives at the conclusion that the expression α + Λη − 2β 2 η + 2βηr 1−n q 2 + β 2 r 2(n−1) should be negative outside the horizon, what can be achieved if the cosmological constant Λ is negative. We point out here that similar condition on the cosmological constant was imposed in case of neutral [52] and charged [55] black holes in Horndeski gravity. Since the metric function W (r) diverges on the horizon changing its sign while crossing the horizon, the same is true for the function (ϕ ′ ) 2 , unless the expression α + Λη − 2β 2 η + 2βηr 1−n q 2 + β 2 r 2(n−1) also changes it sign on the horizon.…”

“…in this case the dominant term for small distances is the same as for the nonminimally coupled theory without any electromagnetic field [52]. This fact can be explained by nonsingular behaviour of the electromagnetic potential at the origin of coordinates for the case n = 3 and what is not true for higher dimensions.…”

“…Then static black holes' solutions were obtained and investigated for various dimensions and in more general setup of Horndeski Gravity [42,43,44,45,46,47,48,49]. Some attention has been paid to the examination of slowly rotating neutron stars and black holes [50,51,52]. Some aspects of black hole thermodynamics were studied [53,54,55].…”