Stable photon orbits in stationary axisymmetric electrovacuum spacetimes

We investigate the existence and phenomenology of stable photon orbits (SPOs) in stationary axisymmetric electrovacuum spacetimes in four dimensions. First, we review the classification of equatorial circular photon orbits on Kerr-Newman spacetimes in the charge-spin plane. Second, using a Hamiltonian formulation, we show that Reissner-Nordström diholes (a family encompassing the Majumdar-Papapetrou and Weyl-Bach special cases) admit SPOs, in a certain parameter regime that we investigate. Third, we explore the transition from order to chaos for typical SPOs bounded within a toroidal region around a dihole, via a selection of Poincaré sections. Finally, for general axisymmetric stationary spacetimes, we show that the Einstein-Maxwell field equations allow for the existence of SPOs in electro vacuum, but not in pure vacuum.

Figure 1

Phase diagram for equatorial circular photon orbits of positive radius ($r>0$) in Kerr-Newman spacetimes (cf. Fig. 2.1 in [33], Fig. 6 in [32] and Fig. 4 in [34]). All orbits are unstable except where noted to the contrary. Region $I$: two exterior ($r>r_{h+}$) orbits. $II$: two interior ($0<r<r_{h-}$) and two exterior orbits; the innermost orbit is stable. $III$ and $IV$: two orbits; the inner orbit is stable. $V$: no orbits. $VI$ and $VII$: four orbits; the inner pair is stable. Cases $III$ and $VI$ admit a counter-rotating orbit; $IV$ and $VII$ do not. For $a^2=0$, $0\le q<1$ (Reissner-Nordström and Schwarzschild), one exterior orbit. For $a^2=0$, $1<q^2<9/8$, two orbits; the inner one is stable. For $q^2=0$, $0<a^2<1$ (Kerr) there are two exterior and one interior orbit for $0<a^2<1$. In the extremal case $a^2+q^2=1$ there are three orbits: for $0<a<1/2$ one stable horizon orbit ($r=M$) and two exterior orbits; and for $1/2<a<1$ one stable interior orbit ($r<M$), one horizon orbit, and one exterior orbit [36, 37]. At $a=1/2$, $q=\sqrt{3}/2$, where regions $I$–$III$ and $VI$ meet, there is a marginally stable (${\mathcal{R}}^{\prime \prime }=0$) horizon orbit and one exterior orbit at $r=3M$.

Figure 2

Contour plots of the height function $h=\rho f^{-1}$ for Majumdar-Papapetrou diholes separated by coordinate distance $d$. Circles indicate stationary points: saddles and one local maximum. Filled circles represent the black hole horizons. The first three plots show equal-mass cases; the lower right plot shows an unequal mass case, $\eta =\frac{M_{+}-M_{-}}{M_{+}+M_{-}}=0.05$.

Figure 3

Existence regions for stable photon orbits around Reissner-Nordström diholes. Left: Majumdar-Papapetrou dihole ($Q_{\pm }=M_{\pm }$) with mass ratio $\eta$ and (dimensionless) coordinate separation $\tilde{d}=2d/(M_{+}+M_{-})$. Right: Equal-mass Reissner-Nordström black holes with equal charge-to-mass ratios $q=Q_{\pm }/M_{\pm }$ held in static equilibrium by a Weyl strut (see the text). The vertices shown are at $(0,\sqrt{16/27})$, $(0,\sqrt{32/27})$ [28, 42], and $(1,2/\phi )$ [51].

Figure 4

Poincaré sections (taken in the $z=0$ plane) and SPOs [viewed in the $(\rho ,z)$-plane with $\varphi$ suppressed] for the equal-mass Majumdar-Papapetrou dihole in the special case $d=M$.

Figure 5

Poincaré sections in the $z=0$ plane for SPOs on MP dihole spacetimes with coordinate separations $d=0.95M$ (left) and $d=1.05M$ (right). See also Fig. 2.