Third Order Lovelock Gravity In The Black Hole

In this paper, we explore the third order Lovelock black holes and present the black hole solutions with negative and positive Gauss-Bonnet coefficients in flat and anti-de Sitter spacetimes, respectively, and then compute the mass, temperature and entropy of black holes. By considering the coefficients, two special black hole solutions are taken for coefficientsα2< 0 andα2> 0. Moreover, we perform the thermodynamically stable analysis of the black holes with different horizon structures. Firstly, we focus on the asymptotically flat third order Lovelock black holes. Based on a Hamilton-Jacobi approach beyond the semiclassical approximation, the corrected temperature and entropy of the asymptotically flat black holes are calculated in seven dimensional spacetimes. When k=1, we find that, in seven, eight and nine dimensional spacetimes, there exist an intermediate stable phase forα2>0.While no black hole exists in a dimension greater than nine. The solution with a2< 0 present black holes with two inner and outer event horizons, extreme black holes, or naked singularities provided the parameter of the solution are chosen suitable. It is worth to mention that there exist an intermediate stable phase for D≥7. For negative constant hypersurface event horizon, the solution f(r) does not present any black hole forα2> 0 andα2< 0 in seven and higher dimensional spacetimes.With regard to third order Lovelock black holes in anti-de Sitter space, there exist three kinds of horizon structure k=0,±1.When k=-1, the black holes with a2<0 are thermodynamically stable for the whole range rH, While there exist an intermediate thermodynamically unstable phase for black holes withα2> 0. For positive constant hypersurface event horizon, black holes withα2< 0 demonstrate an intermediate unstable phase for D=7. In eight dimensional spacetimes, a new phase of thermodynamically unstable small black holes appears if |α2| is under a critical value. For D≥9, black holes have similar the distributions of thermodynamically stable regions to the case where the coefficient is under a critical value for D=8. As toα2>0, the spherical,7-dimensional black holes for small values of Lovelock coefficient have an intermediate unstable phase, they are stable for large values of coefficient. We also find that there exists an intermediate unstable phase for these black holes in higher dimensions. Besides, all the thermodynamic and conserved quantities of black holes with flat horizon don't depend on the Lovelock coefficients and are the same as those of black holes in general gravity.It is of interest to generalize these static and spherically symmetric black hole solutions by including the effects of rotation. While, since the equations of motion of Lovelock gravity are highly nonlinear, it is rather difficult to obtain the explicit rotating black hole solutions. Through a small angular momentum as a perturbation was introduced into a non-rotating system and keeping track of how equations of motion is altered, we find that there exist off diagonal tφcomponent of equations of motion which is concerned with function g(r) and c(r). Besides, the off diagonal component of the stress-tensor of electromagnetic field is not independent of c(r). Notably, the action is in the absence of parameter a if substituting metric into the action. And then, the explicit charged Gauss-Bonnet black hole solution can be easily obtained. Furthermore, the approach is successfully adopted in discussing the slowly rotating third order Lovelock black hole in uncharged and charged cases. The angular momentum, magnetic dipole moment, and the gyromagnetic ratio of the Gauss-Bonnet and third order Lovelock black holes are calculated and it turns out that these two higher derivative curvature terms don't affect to the gyromagnetic ratio. It is worth mentioning that, taking into account all the relevant terms of the Lovelock action, then obtaining slowly rotating black hole solutions by solving the field equations for general space-times in higher dimensions, is a very complicated task. However, by working directly in the action, we obtain the slowly rotating Gauss-Bonnet black hole solutions in flat spacetimes. When the rotation of black hole is not very slowly, the event horizon and static limit surface will superpose no longer.