Rational orbits around charged black holes

We show that all eccentric timelike orbits in Reissner-Nordström spacetime can be classified using a taxonomy that draws upon an isomorphism between periodic orbits and the set of rational numbers. By virtue of the fact that the rationals are dense, the taxonomy can be used to approximate aperiodic orbits with periodic orbits. This may help reduce computational overhead for calculations in gravitational wave astronomy. Our dynamical systems approach enables us to study orbits for both charged and uncharged particles in spite of the fact that charged particle orbits around a charged black hole do not admit a simple one-dimensional effective potential description. Finally, we show that comparing periodic orbits in the Reissner-Nordström and Schwarzschild geometries enables us to distinguish charged and uncharged spacetimes by looking only at the orbital dynamics.

Figure 1

$V_{\mathrm{eff}}$ for $Q=0$, 0.4, 0.8, and 1, from bottom to top.

Figure 2

The RN and Schwarzschild potentials with horizons for each.

Figure 3

Regions in which the potential has three extrema (horizontal hatching) and those in which the smallest extremum is outside the horizon (vertical hatching).

Figure 4

$V_{\mathrm{eff}}$ for $L=3.1$, 3.4, 3.8, and 4.0, from bottom to top, with $Q$ fixed.

Figure 5

$L_c^2$ and $E_c^2$ as functions of $r$ for $Q=1$, $Q=0.5$, and $Q=0$.

Figure 6

$r_{\mathrm{IBCO}}$ (solid line) and $r_{\mathrm{ISCO}}$ (dotted line) as a function of $Q$.

Figure 7

Regions in which $L^2>L_{\mathrm{ISCO}}^2$ (horizontal hatching) and where the discriminant $D>0$ (vertical hatching). When we impose the bound $-1<Q<1$ the two regions are equivalent, which demonstrates that only when $L^2>L_{\mathrm{ISCO}}^2$ does the potential have three extrema.

Figure 8

A homoclinic orbit for $Q=1.0$, $Q_{*}=0$, $L=3.3$.

Figure 9

$V_{\mathrm{eff}}$ for $Q_{*}=\{0,0.1,0.2,0.3,1.0\}$ from bottom to top, with ${\mathcal{E}}_{\mathrm{eff}}=\frac{-1+E^2}{2}=0.99$, $Q=1$, $L=3.5$.

Figure 10

Regions of positive discriminant for $L=\{3.2,3.5,3.8\}$; hatching: (diagonal, vertical, horizontal); $E=0.98$.

Figure 11

Regions in which the potential has three extrema that are all $>r_{+}$, for $L=\{3.2,3.5,3.8\}$; hatching: (diagonal, vertical, horizontal); $E=0.98$.

Figure 12

$Q=1.0$; $L=3.6$. Each point in the parameter plot defines an effective potential and ${\mathcal{E}}_{\mathrm{eff}}$. The regions shown are those where the resulting orbits’ periastra are at a radius outside the horizon (horizontal hatching), where the periastra are inside the potential’s smaller minimum (vertical hatching), and where the discriminant is positive (diagonal hatching). The coincidence of the three regions gives us the subset of parameter space in which we can find bounded orbits in the first of the potential’s two minima.

Figure 13

Periodic orbits with $z=1$, 2, 3, and 4 for $L=3.8$, $Q=0.5$, $Q_{*}=0$.

Figure 14

The $(w,v,z)=(w,0,1)$ orbits for $L=3.5$, $Q=0.5$. From left to right, these are (0, 0, 1), (1, 0, 1), and (2, 0, 1). Orbits with $w>3$ are indistinguishable at this scale because the additional whirls are densely packed in $r$.

Figure 15

Two $z=4$, $w=1$ orbits with different $v$ values.

Figure 16

A periodic table of orbits. In each column, the lower orbit may be approximated by the corresponding orbit in the first row.

Figure 17

All extremal RN $z=1$, 2, 3 orbits with $w=0$ for the first column, $w=1$ for the middle column, and $w=2$ for the last column. For all orbits, $L=3.3$ and $Q=1.0$. Orbits increase in energy from top to bottom and left to right. Note that the first and second entries in the first column are blank because the $q=0+\frac{0}{1}$ and $q=0+\frac{1}{3}$ orbits are inaccessible.

Figure 18

All Schwarzschild $z=1$, 2, 3 orbits with $w=0$ for the first column, $w=1$ for the middle column, and $w=2$ for the last column. Orbits increase in energy from top to bottom and left to right.

Figure 19

All RN $z=1$, 2, 3 orbits with $w=0$ for the first column, $w=1$ for the middle column, and $w=2$ for the last column. Orbits increase in energy from top to bottom and left to right.

Figure 20

Energy levels for Schwarzschild orbits (first row) with $L=3.8$ and RN orbits (second row) with $L=3.3$, $Q=1.0$, $Q_{*}=0$. The second and fourth figures show the detail of the first and third figures, respectively, indicating the rationals within the $w=3$ band. These diagrams show lines for all orbits with $w\in (2,3,4,5,6)$, $z\in (1,2,3,4,5)$ with $v<z$ and $v$ and $z$ coprime.

Figure 21

Differences in energies at which we find orbits of various $q$ in the RN (marked with plus signs) and Schwarzschild (marked with dots) spacetimes.