Geodesic equations in the static and rotating dilaton black holes: Analytical solutions and applications

In this paper, we consider the timelike and null geodesics around the static (GMGHS, magnetically charged GMGHS, electrically charged GMGHS) and the rotating (Kerr-Sen dilaton-axion) dilaton black holes. The geodesic equations are solved in terms of Weierstrass elliptic functions. To classify the trajectories around the black holes, we use the analytical solution and effective potential techniques and then characterize the different types of the resulting orbits in terms of the conserved energy and angular momentum. Also, using the obtained results we study astrophysical applications.

Figure 1

Regions of different types of geodesic motion for test particles, in GMGHS, magnetically charged and electrically charged GMGHS spacetimes. $\tilde{Q}=0.25$, $\epsilon =1$, and $\epsilon =0$, for (a) and (b), respectively. The numbers of positive real zeros in these regions are as follows: $\mathrm{I}=4$, $\mathrm{II}=3$, $\mathrm{III}=1$, and $\mathrm{IV}=2$, for GMGHS and for magnetically charged GMGHS spacetimes; and $\mathrm{I}=3$, $\mathrm{II}=2$, $\mathrm{III}=0$, and $\mathrm{IV}=1$, for electrically charged GMGHS spacetime.

Figure 2

Plots of the effective potential together with examples of energies for the different orbit types. (a), (b) correspond to Table 1, and (c), (d) correspond to Table 2. The green curves represent the effective potential. The red dashed lines correspond to energies. The red dots mark the zeros of the polynomial $R$, which are the turning points of the orbits. In the khaki area no motion is possible since $\tilde{R}<0$. The vertical black dashed lines show the position of the horizons. $\epsilon =1$, $\tilde{Q}=0.25$, and $\tilde{L}=0.072$ for (a), (c) and $\tilde{L}=0.025$ for (b), (d).

Figure 3

Four examples of possible orbits in the spacetime of GMGHS and magnetically charged GMGHS black holes. A bound orbit (a), a two-world escape orbit (c), an escape orbit and a Mani-world bound orbit (d), with parameters $\epsilon =1$, $\tilde{Q}=0.25$ and $L=0.072$, $E=\sqrt{0.93}$ for (a), (d); $L=0.025$, $E=\sqrt{1.94}$ for (b); and $L=0.025$, $E=\sqrt{1.5}$ for (c). The blue lines show the path of the orbits, and the circles represent the inner and outer horizons.

Figure 4

Three examples of possible orbits in the spacetime of an electrically charged GMGHS black hole. A bound orbit (a), a terminating orbit (b), and an escape orbit (c), with parameters $\epsilon =1$, $\tilde{Q}=0.25$ and $L=0.072$, $E=\sqrt{0.93}$ for (a); $L=0.025$, $E=\sqrt{1.94}$ for (b); and $L=0.025$, $E=\sqrt{1.5}$ for (c). The blue lines show the path of the orbits and the circle represents the event horizon.

Figure 5

$\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$: Parametric $\tilde{L}\text{−}{E}^2$ diagram for the function $\tilde{\mathrm{\Theta }}$. $\tilde{\mathrm{\Theta }}$ possesses one zero in region b and two zeros in region c. In the a and d areas geodesic motion is not possible.

Figure 6

$\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$: Parametric $\tilde{L}\text{−}{E}^2$ diagram of the $\tilde{r}$ motion. The polynomial $\tilde{R}$ has 0 zero in region I, 2 zeros in region II, 2 positive zeros in region III, 4 positive zeros in region IV, and 3 positive and 1 negative zeros in region V.

Figure 7

$\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$. Combined $\tilde{L}\text{−}{E}^2$ diagrams of the $\tilde{r}$ motion (green lines) and $\theta$ motion (blue lines). In regions marked with the letter b, the orbits cross $\theta =\frac{\pi }{2}$, but not $\tilde{r}=0$. In regions marked with the letter c, $\tilde{r}=0$ can be crossed but $\theta =\frac{\pi }{2}$ is never crossed.

Figure 8

Plots of the effective potential together with examples of energies for the different orbit types of Table 4. The blue and green curves represent the two branches of the effective potential. The red dashed lines correspond to energies. The red dots mark the zeros of the polynomial $R$, which are the turning points of the orbits. In the khaki area no motion is possible since $\tilde{R}<0$. The vertical black dashed lines show the position of the horizons. $\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$, ${\tilde{r}}_{\alpha }=0.35$, and $\tilde{L}=0.35$, $\tilde{L}=0.45$, $\tilde{L}=0.5$, $\tilde{L}=2.5$, for (a), (b), (c), and (e), respectively.

Figure 9

Two examples of possible orbits in the spacetime of a rotating Kerr-Sen dilaton-axion black hole. An escape orbit (a) and a two-world escape orbit (b), with parameters $\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$, ${\tilde{r}}_{\alpha }=0.35$, $L=0.35$, $E=2.5$. The blue lines show the path of the orbits and the circles represent the event horizons.

Figure 10

Two examples of possible orbits in the spacetime of a rotating Kerr-Sen dilaton-axion black hole. A crossover two-world escape orbit (a) and a Mani-world escape orbit (b), with parameters $\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$, ${\tilde{r}}_{\alpha }=0.35$, $L=0.35$, $E=7.5$, and $L=0.5$, $E=0.5$, respectively. The blue lines show the path of the orbits, and the circles represent the inner and outer horizons.

Figure 11

Two examples of possible orbits in the spacetime of a rotating Kerr-Sen dilaton-axion black hole. A bound orbit (a) and a many-world bound orbit (b), with parameters $\epsilon =1$, $\tilde{a}=0.8$, $\tilde{K}=12$, ${\tilde{r}}_{\alpha }=0.35$, $L=0.5$, and $E=0.96$. The blue lines show the path of the orbits and the circles represent the inner and outer horizons.