Black hole binaries: Ergoregions, photon surfaces, wave scattering, and quasinormal modes

Closed photon orbits around isolated black holes are related to important aspects of black hole physics, such as strong lensing, absorption cross section of null particles and the way that black holes relax through quasinormal ringing. When two black holes are present—such as during the inspiral and merger events of interest for gravitational-wave detectors—the concept of closed photon orbits still exist, but its properties are basically unknown. With these applications in mind, we study here the closed photon orbits of two different static black hole binaries. The first one is the Majumdar-Papapetrou geometry describing two extremal, charged black holes in equilibrium, while the second one is the double sink solution of fluid dynamics, which describes (in a curved-spacetime language) two “dumb” holes. For the latter solution, we also characterize its dynamical response to external perturbations and study how it relates to the photon orbits. In addition, we compute the ergoregion of such spacetime and show that it does not coincide with the event horizon.

Figure 1

There are different closed orbits for light in a static binary. We find, generically, three different closed trajectories: one global outer geodesic that encircles both holes, an “8-shaped” trajectory and two “smaller” ones encircling only each of the BHs. These plots show the different “light rings” for a MP solution with $a=10M$. The “8-shaped” orbit (black curve) has a period $T=98.79M$ in coordinate time. The global outer orbit (blue curve) has a period $T=95.47M$. The period of the orbits around a single hole (red curves) is $T=26.39M$.

Figure 2

Same as Fig. 1, but for massive free particles instead of light (i.e., timelike instead of null geodesics). The “8-shaped” orbit (black curve) has energy $E=0.9159$ and period $T=291.64M$ in coordinate time. The global outer orbit (blue curve), has energy $E=0.9644$ and period $T=474.01M$. The orbits around a single hole (red curves) have energy $E=0.9159$ and period $T=38.84M$.

Figure 3

Null geodesics on the symmetry plane of a two-BH MP spacetime, characterized by $a=M/5$. The radii of the circular orbits do not depend on $L$ and are expressed in units of $a$. From left to right, the energy of the orbits are $E=0.0366882$, 0.0366939, 0.0367020. The red curve shows the portion of the motion between two consecutive apastrons, for which the accumulated angle is given by $\mathrm{\Delta }\varphi =2\pi q$. When $q$ is a rational number, the motion is periodic, and the geometric properties of the plot can be extracted from $q$. From left to right, $q$ is given by ($1+7/10$), ($1+33/47$), and ($1+45/64$). Notice that $q$ increases monotonically with $E$. Geometrically this means that the motion is divided into more “leaves”, which creates more vertices in the figures. The stable circular orbit has a radius ${\rho }_{\mathrm{in}}=1.167a$, and the unstable orbit is at ${\rho }_{\mathrm{out}}=9.740a$.

Figure 4

Lyapunov exponent as a function of the BHs masses for null geodesics in the $z=0$ plane of the MP spacetime. The black dots are the numerical results, while the red dashed line corresponds to the analytic expression.

Figure 5

Timelike geodesics on the symmetry plane of a two-BH MP spacetime, characterized by $a=M/5$. From left to right, the energy and angular momentum of the orbits are: ($E=0.0920047$, $L=0.1a$), ($E=0.456023$, $L=a$), and ($E=0.585419$, $L=10a$). From left to right, $q$ is given by $56/111$, $29/53$, and $27/28$. The radii of both circular orbits (stable and unstable) depend on $M$ and $L/a$ for timelike geodesics. From left to right, the inner circular orbit radii ${\rho }_{\mathrm{in}}/a$ are given by 0.0977,0.3093,0.8892. There are no unstable outer orbits for these combinations of $M$ and $L/a$.

Figure 6

Different “soundcurves” when the separation between the sinks is $a=2$. These curves were computed by numerically solving the null geodesic equations for the analogue metric (10) with the velocity field (9).

Figure 7

Acoustic horizon (solid line) surrounding both sinks for separation $a=1$. The horizon exists only if $a\le 1$. The ergocurve (dashed line) clearly does not coincide with the horizon, except at a few points.

Figure 8

Evolution of a quadrupole ($c_0=c_1=0$, $c_2=1$) Gaussian (with $x_0=y_0=0$, $R_0=5$, $\sigma =1/2$) profile around a single ($a=0$) acoustic hole. The scalar was extracted at $x=30$, $y=0$. A ringdown, exponentially decaying, phase is visible, and followed by a late-time power-law tail. This and subsequent plots show the absolute value of the scalar.

Figure 9

Scattering of a sound wave off a binary dumb-hole with $a=1$. The Gaussian, centered at the origin, has a width $\sigma =1/2$ and is localized at $R_0=5$. The field $\mathrm{\Phi }$ is extracted along the $x$-axis (black solid line) and along the $y$-axis (red, dashed line) at a distance of 30 from the origin. Top: Monopolar wave, with $c_0=1$. For monopolar waves the prompt signal quickly gives way to a power-law tail $\mathrm{\Phi }\sim t^{-1.1}$ at late times, in agreement with prediction (17). Bottom: Quadrupolar wave, with $c_2=1$. We find a ringdown $\omega =0.472-0.170i$ and a power law $\sim t^{-1.2}$.

Figure 10

Scattering of a sound wave off a binary dumb-hole with $a=5$. The Gaussian width is $\sigma =1/2$. The field $\mathrm{\Phi }$ is extracted along the $x$-axis (black solid line) and along the $y$-axis (red, dashed line) at a distance of 30 from the origin. Top: Quadrupolar wave, with $c_2=1$, $x_0=5$, $y_0=0$, $R_0=3$. First ringdown stage for $x$ matches well that of a one single, individual BH. Second ringdown is of lower frequency. Ratio is 1.2. Middle: Monopolar wave, with $c_0=1$, $x_0=y_0=0$, $R_0=10$. Bottom: Quadrupolar wave, with $c_2=1$, $x_0=y_0=0$, $R_0=10$.

Figure 11

Left: Null geodesic curves in two MP BHs, whose event horizon consists of two disconnected components. Depicted are two circular orbits around each of the disconnected components (blue lines) and a “8-shaped” orbit (green line) around the event horizon. Middle: Two BHs merge to form a single BH. The event horizon has only a single connected component and admits a “crease set” (depicted by thick red-line) consisting of the past end points of some of the null geodesic generators of the event horizon. Here “two black holes” is meant to be the number of cross sections of the event horizon on a particularly chosen Cauchy time slice as illustrated. Note however that since the event horizon has, as a part, the crease set, which is spacelike, the notion of the number of cross sections (as well as the topology of the cross section) is dependent on the choice of Cauchy time slice. Also because of the spacelike nature of the crease-set, null/timelike orbits cannot be of “8-shaped.” Inside the event horizon there are apparent horizons, for which null geodesic curves can possibly form circular orbits (blue lines). Right: Two ultracompact stars collide to form a single BH. The system can admit a “8-shaped” null orbit (green line) as well as circular null orbits (blue lines) around each of the ultracompact stars if they are sufficiently compact. Further study is needed to see whether these “8-shaped” and circular null orbits eventually can go around outside the event horizon (as illustrated here) or must go inside the event horizon, after the BH formation.