Structure and thermodynamics of charged nonrotating black holes in higher dimensions

We analyze the structural and thermodynamic properties of $D$-dimensional ($D\ge 4$), asymptotically flat or anti–de Sitter, electrically charged black hole solutions, resulting from the minimal coupling of general nonlinear electrodynamics to general relativity. This analysis deals with static spherically symmetric (elementary) configurations with spherical horizons. Our methods are based on the study of the behavior (in vacuum and on the boundary of their domain of definition) of the Lagrangian density functions characterizing the nonlinear electrodynamic models in flat spacetime. These functions are constrained by some admissibility conditions endorsing the physical consistency of the corresponding theories, which are classified in several families, some of them supporting elementary solutions in flat space that are nontopological solitons. This classification induces a similar one for the elementary black hole solutions of the associated gravitating nonlinear electrodynamics, whose geometrical structures are thoroughly explored. A consistent thermodynamic analysis can be developed for the subclass of families whose associated black hole solutions behave asymptotically as the Schwarzschild metric (in the absence of a cosmological term). In these cases we obtain the behavior of the main thermodynamic functions, as well as important finite relations among them. In particular, we find the general equation determining the set of extreme black holes for every model, and a general Smarr formula, valid for the set of elementary black hole solutions of such models. We also consider the one-parameter group of scale transformations, which are symmetries of the field equations of any nonlinear electrodynamics in flat spacetime. These symmetries are respected by the minimal coupling to gravitation and induce representations of the group in the spaces of solutions of the different models, characterized by their thermodynamic functions. Exploiting this fact we find the expression of the equation of state of the set of black hole solutions associated with any model. These results are generalized to asymptotically anti–de Sitter solutions.

Figure 1

Qualitative behavior of the admissible Lagrangian densities $\phi (X):0$ (i) around the vacuum [$X\sim 0$; $\phi (X\sim 0)\sim X^{{\gamma }_2}$], corresponding to the three B cases and IRD asymptotic behaviors of the ESS solutions, (ii) for large ESS fields [$X\to \infty$; $\phi (X\to \infty )\sim X^{{\gamma }_1}$], corresponding to the A1 and UVD central-field behaviors, and (iii) for finite maximum field-strength models ($X\le a^2=E_{\max }^2$), corresponding to the A2 central-field behavior. The ${\gamma }_i$ constants are related to the central and asymptotic behaviors of the ESS fields through Eqs. (25), (26), and (31)–(33). In the A2 cases [see Eq. (27)] the Lagrangian density exhibits a vertical asymptote at $X=a^2$ (if $\sigma \le D-2$) or takes a finite value with a divergent slope there (if $\sigma >D-2$). In the intermediate range of $X>0$ values, matching the central and asymptotic regions, $\phi (X)$ must be strictly monotonically increasing, for admissibility [see Eq. (20)]. This figure is qualitatively similar for any value of $D\ge 4$.

Figure 2

Qualitative $M-r_h$ diagram for the asymptotically Schwarzschild BH solutions of admissible G-NEDs belonging to families with central-field behaviors A1, A2, and UVD in $D>4$ dimensions. The curves correspond to fixed values of $Q$. There are always unique minima in these curves corresponding to extreme black holes. Naked singularities correspond to the configurations with a mass below the value of that of the extreme BH for a given charge and exist in all cases. Furthermore, all families support two-horizon BHs. The A1 and A2 families, supporting soliton solutions in flat space, exhibit also nonextremal single-horizon BH solutions for values of $M$ above the total electromagnetic energy of the configuration [$M>ϵ(Q,D)$]. The increasing parts of these curves correspond to the external event horizons, whose radii increase monotonically with the mass. The dashed curve, to which all the curves converge at large $r_h$, corresponds to the $M-r_h$ relation for the Schwarzschild BHs. The small frame displays the qualitative behavior for the A2 family in $D=4$ spacetime dimensions, where extreme and nonextreme black points arise for $Q=Q_c$ and $Q<Q_c$, respectively, $Q_c=(16\pi a{)}^{-1}$ being the critical value of the charge in these cases.

Figure 3

The metric function $g(r)$ for ESS black holes embedded in AdS spacetime. At short distances, $g(r\to 0)$ diverges to $\pm \infty$, depending on the family and the range of parameters (see the main text). At large $r$ the metric function reaches the parabola $1+r^2/l^2$ (asymptotically AdS behavior). In the intermediate region the different configurations (naked singularities, two-horizon BHs, extreme and nonextreme one-horizon BHs) follow from the cut points of the curves with the horizontal axis, which define the location of the horizons. The AF curve displays the large $r$ behavior of the asymptotically flat BHs.

Figure 4

Phase diagram in the $Q–M$ plane for typical A1 and A2 models (main frame) and UVD models (small frame) in $D\ge 4$. The $r_h=0$ curve in the main frame [corresponding to $M=ϵ(Q,D)=Q^{(D-1)/(D-2)}ϵ(Q=1,D)$] defines the vanishing inner horizon BH configurations. This curve separates the regions associated with the single horizon (ShBH) and two-horizons (2hBH) black hole configurations. Below the extreme BH curve (EBH) only naked singularity configurations are possible. Both curves meet at the origin, except for A2 models in $D=4$, for which the $r_h=0$ curve and the extreme BH curve (dashed line) meet when the charge takes the critical value $Q_c=(16\pi a{)}^{-1}$ ($a$ being the maximum field strength). In these cases, the piece of the $r_h=0$ curve for $0<Q\le Q_c$ corresponds to extreme ($Q=Q_c$) and nonextreme ($Q<Q_c$) black points, which are absent for all admissible models in $D>4$. The analysis of the diagram in the small frame (UVD cases) is similar, but now the $r_h=0$ curve does not exist and the charged single-horizon BH configurations are absent.

Figure 5

Qualitative behavior of the functions $\eta (r_h)$ obtained from Eq. (114) for asymptotically flat BH solutions associated with two different G-NEDs at constant $Q$ (top figures), and from Eq. (119) for two asymptotically AdS black hole solutions corresponding to two sets of values of the parameters $Q$, $D$, and $l$ of the same G-NED (bottom figures). The $\eta (r_h)$ curves are always monotonically increasing and exhibit a vertical asymptote at the origin (excepted for A2 models in $D=4$ dimensions, whose particular features have been extensively analyzed in Ref. [70]). In the asymptotically flat cases (top), they also exhibit a horizontal asymptote at the constant value $\eta =\mathrm{\Upsilon }(D)$, whereas they diverge parabolically in AdS cases (bottom). The slopes of the dashed straight lines connecting the origin with the points of the $\eta$ curves give the temperatures of the associated BH configurations, which are plotted in the small frames. The tangency points define local or absolute extrema of the $T-r_h$ curves and the $\eta =0$ points correspond to the ($T=0$) extreme BH configurations. The temperature vanishes at large $r_h$ for the asymptotically flat BHs and diverges linearly for the asymptotically AdS ones. The models used in obtaining the curves for the asymptotically flat cases are the Born-Infeld one (upper left) and the UVD-B2 model discussed in Ref. [70] [Eq. (73) and Fig. 12 of this reference], which exhibits a richer and more complex behavior of the temperature. For the asymptotically AdS cases the Euler-Heisenberg model has been used for two different sets of parameters, leading also to different qualitative behaviors of the temperature.

Figure 6

Qualitative behaviors of the curves of extreme BHs in the $Q–S$ plane for the three families of central-field behaviors. The UVD and A1 cases (upper frame) and the A2 cases (lower frame) are represented for several values of the spacetime dimension $D$. The curves are positive definite everywhere, except for the A2 family in $D=4$, where extreme and nonextreme black points are present [in ($Q=Q_c$, $S=0$) and ($Q<Q_c$, $S=0$), respectively]. The model used in calculating the curves of the upper frame is the Euler-Heisenberg one, belonging to the A1 family in $D=4$ and to the UVD family in $D>4$. The lower frame is obtained from the Born-Infeld model, as representative of the A2 family in any dimension.

Figure 7

Qualitative shapes of the EOS surfaces built by the method of the characteristics associated with the sets of BH solutions of different families of G-NED models. The top frame displays the typical behavior of the families A2 in $D=4$ and is obtained here using the Born-Infeld model. The bottom frame shows the qualitative behavior for the other families in any dimension (here the example used is the Euler-Heisenberg model in $D=4$).

Figure 8

Quantitative behavior of asymptotically AdS extreme BHs for several values of the cosmological constant in the $Q–S$ plane. The upper frame corresponds to the gravitating Euler-Heisenberg model in $D=5$, as a representative of the UVD models in higher dimensions. The lower frame comes from the gravitating Born-Infeld model, as a representative of the special case of A2 models in $D=4$. In these last cases, the continuation of the curves in the unphysical region ($S<0$) has been removed, except for the asymptotically flat curve $\mathrm{\Lambda }=0$ (dashed piece).