Analytic solutions of the geodesic equation for ${\mathrm{U}(1)}^2$ dyonic rotating black holes

In this article, we derive the geodesic equations in the ${\mathrm{U}(1)}^2$ dyonic rotating black hole spacetime. We present their solutions in terms of the Kleinian $\sigma$ function and in special cases in terms of the Weierstraß $\wp$, $\sigma$, and $\zeta$ functions. To give a list of all possible orbits, we analyze the geodesic motion of test particles and light using parametric diagrams and effective potentials.

Figure 1

Parametric $g^2-{N_g}^2$ diagram for $a=0.4$, $e=0.38$, $v=1.3$, and $m=0.5$. The blue line represents $r=0$ as a zero, and the green line shows where double zeros appear.

Figure 2

Plots of the singularity given by $B-aA=0$; the dotted blue line represents $r\le 0$, and the red line represents $r\ge 0$.

Figure 3

Parametric $\tilde{L}-E^2$-diagrams for $\delta =0$, $\tilde{a}=0.4$, $\tilde{g}=0.1$, $\widetilde{N_g}=0.2$, $\tilde{e}=0.3$, $\tilde{v}=0.2$, and $\tilde{K}=0.1$. The continuous lines represent double zeros of $Y(\nu )$ in blue and $X(\tilde{r})$ in green. The dashed lines show zeros at $\tilde{r}=0$ and $\nu =0$.

Figure 4

Parametric $\tilde{L}-E^2$ diagrams for $\delta =0$, $\tilde{a}=0.4$, $\tilde{g}=0.55$, $\widetilde{N_g}=0.2$, $\tilde{e}=0.3$, $\tilde{v}=0.2$, and $\tilde{K}=0.1$. The continuous lines represent double zeros of $Y(\nu )$ in blue and $X(\tilde{r})$ in green. The dashed lines show zeros at $\tilde{r}=0$ and $\nu =0$.

Figure 5

Illustration of the effective potential $V_{\pm }$ and energies for examples of orbit types of Table 1. The blue line represents $V_{+}$, and the green line represents $V_{-}$. The red dots denote the turning points of the particle, and the grey area forbids geodesic motion due to $X\le 0$.

Figure 6

Orbits of test particles in the dyonic rotating black hole spacetime. The orbit is represented by the blue line, and the outer and inner horizons are denoted by gray and black ellipsoids.

Figure 7

Orbits of test particles in the dyonic rotating black hole spacetime. The orbit is represented by the blue line, and the outer and inner horizons are denoted by gray and black ellipsoids. The gray structure denotes the singularity.