Charged dilatonic black rings and black saturns and their thermodynamics

In this paper charged black rings and black saturns are constructed in Einstein-Maxwell-dilaton theory in five dimensions. The neutral black ring and black saturn solutions are embedded in six dimensions and boosted with respect to the time coordinate and the added sixth dimension. Then the charged solutions are obtained by a Kaluza-Klein reduction. The influence of the charge is studied by analyzing the physical properties and the phase diagram. The different dilatonic solutions are compared and their thermodynamic stability is considered.

Figure 1

Parameter space of the rotating charged dilatonic black ring. (a) Sign of $\delta$. The solution contains a conical deficit (blue area) or conical excess (green area). The balanced solutions ($\beta =0$) are on the black curve in between the blue and green area. (b) Sign of the mechanical moment of inertia. $I>0$ in the green and $I<0$ in the blue area. On the dashed curve the black ring is balanced ($\beta =0$).

Figure 2

Phase diagram of the charged dilatonic black ring. (a) $a_H$ versus $j^2$ for different values of $\beta$. (b) $a_H$ versus $j^2$ versus $q$.

Figure 3

$j\text{−}q$ diagrams and $a_H\text{−}q$ diagrams of the charged dilatonic black ring. (a) $j\text{−}q$-diagram for different values of the parameter $\nu$. (b) $a_H\text{−}q$-diagram for different values of the parameter $\nu$.

Figure 4

Temperature diagrams of the charged dilatonic black ring. (a) Temperature versus $\nu$ for different values of $\beta$. (b) Temperature versus $a_H$ versus $q$.

Figure 5

Sign of the specific heat at constant angular momentum of the rotating charged dilatonic black ring in the fixed $Q$ ensemble. $C_J$ is positive in the green area and negative in the blue area. The balanced solutions can be found along the dashed curve. (a) $\beta =0$, (b) $\beta =1$, (c) $\beta =5$.

Figure 6

Sign of the isothermal moment of inertia of the rotating charged dilatonic black ring in the fixed $Q$ ensemble. $\epsilon$ is positive in the green area and negative in the blue area. The balanced solutions can be found along the dashed curve. (a) $\beta =0$, (b) $\beta =0.8$, (c) $\beta =1.5$.

Figure 7

Parameter space of the charged dilatonic black saturn. In the colored region the solution is in thermodynamical equilibrium. (a) Sign of $\delta$. The solution contains a conical deficit (blue area) or conical excess (green area). The balanced solutions ($\delta =0$) are on the black curve in between the blue and green area. (b) Sign of the mechanical moment of inertia. $I>0$ in the green and $I<0$ in then blue area. On the dashed curve the black saturn is balanced ($\delta =0$).

Figure 8

Phase diagram of the charged dilatonic black saturn. (a) $a_H$ versus $j^2$ for different values of $\beta$. (b) $a_H$ versus $j^2$ versus $q$.

Figure 9

$j\text{−}q$ diagrams and $a_H\text{−}q$ diagrams of the charged dilatonic black saturn and a comparison to the charged dilatonic black ring. (a) $j\text{−}q$-diagram for different values of the parameter ${\kappa }_2$. (b) $a_H\text{−}q$-diagram for different values of the parameter ${\kappa }_2$. (c) Comparison of the $a_H\text{−}q$-diagrams. Case ${\kappa }_2\approx 0.657549$ for the charged dilatonic black saturn and $\nu =0.5$ for the charged dilatonic black ring: For a given $q$, $a_H$ has its maximal value (and $j^2$ its minimal value).

Figure 10

Temperature diagrams of charged the dilatonic black ring. (a) Temperature versus ${\kappa }_2$ for different values of $\beta$ (b) Temperature versus $a_H$ versus $q$.

Figure 11

Sign of the specific heat at constant angular momentum of the charged dilatonic black saturn in the fixed $Q$ ensemble. $C_J$ is positive in the green area and negative in the blue area. The balanced solutions can be found along the dashed curve. (a) $\beta =0$, (b) $\beta =1$, (c) $\beta =5$.

Figure 12

Sign of the isothermal moment of inertia of the charged dilatonic black saturn in the fixed $Q$ ensemble. $\epsilon$ is positive in the green area and negative in the blue area. The balanced solutions can be found along the dashed curve. (a) $\beta =0$, (b) $\beta =0.5$, (c) $\beta =1$.

Figure 13

Comparison of the blackfold approximation of the charged black ring with the analytic solution of the charged black ring and black saturn. In the ultraspinning limit the blackfold approximation of the black ring agrees well with the analytic black ring solution and also with the black saturn solution. (a) Scaled electromagnetic term of the Smarr formula as a function of the scaled charge. The red dots correspond to the blackfold approximation of a charged black ring. The blue line corresponds to the analytic solution of a charged black ring/saturn. (b) Scaled horizon area term of the Smarr formula as a function of the scaled charge. The red dots correspond to the black-fold approximation of a charged black ring. The blue lines correspond to the analytic solution of a charged black ring. The ultraspinning limit is reached for $\nu \to 0$. (c) Scaled angular momentum term of the Smarr formula as a function of the scaled charge. The red dots correspond to the blackfold approximation of a charged black ring. The blue lines correspond to the analytic solution of a charged black ring. The ultraspinning limit is reached for $\nu \to 0$. (d) Scaled horizon area term of the Smarr formula as a function of the scaled charge. The red dots correspond to the blackfold approximation of a charged black ring. The blue lines corre- spond to the analytic solution of a charged black saturn. The ultraspinning limit is reached for ${\kappa }_2\to 0$. (e) Scaled angular momentum term of the Smarr formula as a function of the scaled charge. The red dots correspond to the blackfold approximation of a charged black ring. The blue lines correspond to the analytic solution of a charged black saturn. The ultraspinning limit is reached for ${\kappa }_2\to 0$.