Charged rotating AdS black holes with Chern-Simons coupling

We obtain a perturbative solution for rotating charged black holes in five-dimensional Einstein-Maxwell-Chern-Simons theory with a negative cosmological constant. We start from a small undeformed Kerr-AdS solution and use the electric charge as a perturbative parameter to build up black holes with equal-magnitude angular momenta up to fourth order. These black hole solutions are described by three parameters, the charge, horizon radius and horizon angular velocity. We determine the physical quantities of these black holes and study their dependence on the parameters of black holes and arbitrary Chern-Simons coefficient. In particular, for values of the CS coupling constant beyond its supergravity value, due to a rotational instability, counterrotating black holes arise. Also the rotating solutions appear to have vanishing angular momenta and are not manifest uniquely by their global charges.

Figure 1

The scaled quantities $E_{SG}$ and ${J_a}_{SG}$ in terms of ${\mathrm{\Omega }}_{H}l$.

Figure 2

Energy (left) and entropy (right) versus horizon angular velocity for $Q/l^2=0.5$, $r_{+}/l=0.9$, $\alpha =0,\dots ,5$. When ${\mathrm{\Omega }}_H=0$ all curves intersect at $E/l^2=2.09082$ (a) and $A_H/4l^3=3.59747$ (b), which corresponds to the RN-AdS solution. For comparison, the $Q=0$, $\alpha =0$ KAdS solution is also illustrated. Curves for positive and negative charges are presented by solid and dashed lines, respectively.

Figure 3

The temperature versus the horizon angular velocity. We choose $Q=0.5l^2$, $r_{+}=0.9l$ and $\alpha =0,\cdots 5$. The intersection point at ${\mathrm{\Omega }}_H=0$ is at 0.395935, corresponding to the RN-AdS solution. For comparison the $Q=0$, $\alpha =0$ KAdS solution is also illustrated. Curves for positive and negative charges are presented by solid and dashed lines, respectively.

Figure 4

The angular momentum versus the horizon angular velocity. Curves for positive and negative charges are presented by solid and dashed lines, respectively.

Figure 5

Horizon area versus the angular momentum. Curves for positive and negative charges are presented by solid and dashed lines, respectively.

Figure 6

Energy versus angular momentum for charge $|Q|/l^2=0.5$ and $r_{+}/l=0.9$. The intersection with the energy axis at $E/l^2=2.09082$ for $J=0$ is the RN-AdS solution. For completeness we include the $Q=0$ KAdS solution. Curves for positive and negative charges are presented by solid and dashed lines, respectively. These curves all reflect about the $J=0$ axis for ${\mathrm{\Omega }}_H<0$; we have not drawn the associated graph.

Figure 7

Energy (upper left), entropy (upper right), angular momentum (lower left), and temperature (lower right) plotted as a function of charge, with the $J=0$ RN-AdS results plotted for comparison, with $r_{+}/l=0.9$ and ${\mathrm{\Omega }}_{H}l=0.717879$. In the upper-left figure the intersection point at $Q=0$ is at $E/l^2=2.29188$ corresponding to the KAdS solution for all values of $\alpha$, whereas the corresponding point for the RN-AdS solution is at $E/l^2=1.72721$. In the remaining figures, the $Q=0$ intersection points are at $A_H/4l^3=3.97521$ (with the RN-AdS intersection at $A_H/4l^3=3.59747$) $J/l^3=0.927443$, and $\kappa l/2\pi =0.360645$ (with the RN-AdS intersection at $\kappa l/2\pi =0.463318$).

Figure 8

The electrostatic potential versus the charge (left) for fixed ${\mathrm{\Omega }}_{H}l=0.717879$ and the horizon angular velocity (right) for fixed $|Q|/{\ell }^2=0.5$.

Figure 9

The energy (left) and the entropy (right) versus the CS coupling for $|Q|/l^2=0.5$, $r_{+}/l=0.9$, ${\mathrm{\Omega }}_{H}l=0.717879$. In (a) the intersection of two lines is given by $E/l^2=2.73574$. In (b) the joining point is given by $J/l^3=1.28512$. Curves for positive and negative charges are presented by solid and dashed lines, respectively.