Dynamics of test particles in the five-dimensional, charged, rotating Einstein-Maxwell-Chern-Simons spacetime

We derive the complete set of geodesic equations for massive and massless test particles of a five-dimensional, charged, rotating black hole solution of the Einstein-Maxwell-Chern-Simons field equations in five-dimensional minimal gauged supergravity and present their analytical solutions in terms of Weierstrass elliptic functions. We study the polar and radial motion, depending on the black hole and test particle parameters, and characterize the test particle motion qualitatively by means of effective potentials. We use the analytical solutions in order to visualize the test particle motion by two- and three-dimensional plots.

Figure 1

Plot of $\mathrm{\Delta }$ for $\mu =1$, $a=0.3$, $b=0.2$ and various values of $q$. The black solid line denotes the neutral case, dashed lines refer to positive charges $q$, and dotted lines refer to negative charges. The solid green line denotes the extremal case, where both horizons merge.

Figure 2

Plots of several $\theta$-potentials for $E\ge 0$. The grey-colored areas denote energy values, for which $\mathrm{\Xi }<0$; thus, they need to be excluded for physical test particle motion. Plots (a)–(d) emphasize the influence on the metric parameters $\delta$, $K$, $\mathrm{\Phi }$ and $a$. Plots (e) and (f) show two typical potentials with possible $\theta$-motion (dashed red line) between two turning points (red points) or a constant $\theta$-motion related to a minimum in the potential.

Figure 3

Effective potentials for massive (a)–(c) and massless (d)–(f) test particles. The grey areas are physically forbidden by the radial geodesic equation. Regions which are energetically restricted due to the conditions of the $\theta$-polynomial are hatched. Different orbit types (A)–(D) are given by the red lines of constant energy for massive test particles, and the corresponding massless potential is shown below.

Figure 4

Massive, two-dimensional escape orbit (blue) in the $X-Y$-plane for parameter values: $a=0.3$; $b=0.2$; $q=0.2$, $\mu =1$, $\mathrm{\Phi }=-2$, $E=1.42274248$. The grey dashed lines represent the horizons and the green dotted line denotes the static limit. The inner static limit is beyond the singularity, which is depicted by the grey disk, and therefore not shown.

Figure 5

Massive, two-dimensional many-world bound orbit in the $X-Y$ plane for the following parameter values: $a=0.3,b=0.2,q=0.2,\mu =1,\mathrm{\Phi }=-2,E=1.42274248$. Here, both values of the static limit (green dotted lines) and turnaround boundaries (red dotted lines) are shown. Both boundaries seem to merge outside the event horizon, while they are clearly distinct inside.

Figure 6

Massive, two-dimensional two-world escape orbit in the $X-Y$-plane for parameter values: $a=0.4,b=0.3,q=0.2,\mu =1,\mathrm{\Phi }=-1.5,E=3$.

Figure 7

Massless, two-dimensional two-world escape orbit in the $X-Y$ plane for the following parameter values: $a=0.3,b=0.2,q=0.2,\mu =1,\mathrm{\Phi }=-2,E=1.42274248$.

Figure 8

Massive escape orbit in the $X-Z$ plane for the following parameter values: $a=0.3,b=0.1,q=0.4,\mu =1,\mathrm{\Phi }=-1,\mathrm{\Psi }=-0.2,K=1.8$, $E=1.09$.

Figure 9

Massive many-world bound orbit in the $X-Z$ plane for the following parameter values: $a=0.3,b=0.1,q=0.4,\mu =1,\mathrm{\Phi }=1,\mathrm{\Psi }=-0.2,K=1.8,E=0.9$.

Figure 10

Massless many-world bound orbit in the $X-Z$ plane for the following parameter values: $a=0.3,b=0.1,q=0.4,\mu =1,\mathrm{\Phi }=-1,\mathrm{\Psi }=-0.2,K=1.8,E=0.5$.

Figure 11

Massive two-world escape orbit in the $X-Z$ plane for the following parameter values: $a=0.3,b=0.1,q=0.4,\mu =1,\mathrm{\Phi }=0.5,\mathrm{\Psi }=0.5,K=1.8,E=1.2$.

Figure 12

Massive, three-dimensional projection of an escape orbit in the $X-Y-Z$ space for the following parameter values: $a=0.3,b=0.2,q=0.2,\mu =1,\mathrm{\Phi }=-2,\mathrm{\Psi }=-0.2,K=5,E=1.5549783$.

Figure 13

Massive, three-dimensional projection of a many-world bound orbit in the $X-Y-Z$ space for the following parameter values: $a=0.4,b=0,q=-0.4,\mu =1,\mathrm{\Phi }=0.6,\mathrm{\Psi }=-0.2,K=1.8,E=0.5$.

Figure 14

Massive, three-dimensional projection of a two-world escape orbit in the $X-Y-Z$ space for the following parameter values: $a=0.3,b=0.1,q=0.4,\mu =1,\mathrm{\Phi }=-1,\mathrm{\Psi }=-0.2,K=1.8,E=3$.