Strong-field tidal distortions of rotating black holes: Formalism and results for circular, equatorial orbits

Tidal coupling between members of a compact binary system can have an interesting and important influence on that binary’s dynamical inspiral. Tidal coupling also distorts the binary’s members, changing them (at lowest order) from spheres to ellipsoids. At least in the limit of fluid bodies and Newtonian gravity, there are simple connections between the geometry of the distorted ellipsoid and the impact of tides on the orbit’s evolution. In this paper, we develop tools for investigating tidal distortions of rapidly rotating black holes using techniques that are good for strong-field, fast-motion binary orbits. We use black hole perturbation theory, so our results assume extreme mass ratios. We develop tools to compute the distortion to a black hole’s curvature for any spin parameter, and for tidal fields arising from any bound orbit, in the frequency domain. We also develop tools to visualize the horizon’s distortion for black hole spin $a/M\le \sqrt{3}/2$ (leaving the more complicated $a/M>\sqrt{3}/2$ case to a future analysis). We then study how a Kerr black hole’s event horizon is distorted by a small body in a circular, equatorial orbit. We find that the connection between the geometry of tidal distortion and the orbit’s evolution is not as simple as in the Newtonian limit.

Figure 1

Convergence of contributions to the horizon’s tidal distortion. We show $R_{\mathrm{H},lm}^{(1)}$ summed over $m$ for a given $l$, scaled by a factor $(r_{\mathrm{o}}^3/\mu )$ to account for the leading dependence on small body mass and orbital radius. The largest amplitude oscillation is for $l=2$ (red in color). The next largest is $l=3$ (green), followed by $l=4$ (blue), $l=5$ (magenta), with the smallest oscillations shown for $l=6$ (cyan). (Higher order contributions are omitted since their variations cannot be seen on the scale of this plot.) These curves are for a circular orbit at $r_{\mathrm{o}}=6M$, which has $u=0.41$, the largest value for the Schwarzschild cases we consider. As such, this case has the slowest convergence among Schwarzschild orbits. The falloff with $l$ is more rapid for all other cases.

Figure 2

Comparison of numerically computed scalar curvature perturbation $R_{\mathrm{H}}^{(1)}$ for Schwarzschild with the analytic expansion given in Eqs. (3.21)–(3.32). The four panels on the left compare numerical [red (dark gray)] and analytic [green (light gray)] results for an orbit at $r_{\mathrm{o}}=50M$. Panels on the right are for $r_{\mathrm{o}}=6M$. In both cases, we plot $(r_{\mathrm{o}}^3/\mu )R_{\mathrm{H}}^{(1)}$, scaling out the leading dependence on orbital radius and the orbiting body’s mass. We show contributions for $l=2$, $l=3$, and $l=4$, plus the sum of these modes. For $r_{\mathrm{o}}=50M$, we have $u=0.14$, and we see very good agreement between the numerical and analytic formulas. In several cases, the numerical data lie on top of the analytic curves. For $r_{\mathrm{o}}=6M$, $u=0.41$, and the agreement is not as good. Although the amplitudes disagree in the strong field (especially for large $l$), the two computations maintain good phase agreement well into the strong field.

Figure 3

Equatorial section of the embedding of a distorted Schwarzschild horizon. Each panel shows the distortion for a different orbital radius, varying from $r_{\mathrm{o}}=50M$ to $r_{\mathrm{o}}=6M$. The black circles are the undistorted black hole, and the red curves are the distorted horizons, embedded with Eq. (3.38). These plots are in a frame that corotates with the orbit, and are for a slice of constant ingoing time $v$. The green dashed line in each panel shows the angle at which the tidal distortion is largest; the black dotted line shows the orbit’s position. Notice that the bulge leads the orbit in all cases, with the lead angle growing as the orbit moves to smaller orbital radius. We have rescaled the horizon’s tidal distortion by a factor $\propto r_{\mathrm{o}}^3/\mu$ so that, at leading order, the magnitude of the distortion is the same in all plots.

Figure 4

Comparison of selected modes for the numerically computed scalar curvature perturbation $R_{\mathrm{H},lm}^{(1)}$ with the analytic expansion given in Eqs. (4.21)–(4.32). The four panels on the left are for orbits of a black hole with $a=0.1M$ at $r_{\mathrm{o}}=50M$; those on the right are for orbits of a black hole with $a=0.2M$ at $r_{\mathrm{o}}=10M$. The mode shown is indicated by $(l,m)$ in the upper right corner of each panel [we actually show the contributions from $(l,m)$ and $(l,-m)$]. In all cases, we plot $(r_{\mathrm{o}}^3/\mu )R_{\mathrm{H}}^{(1)}$, scaling out the leading dependence on orbital radius and the orbiting body’s mass. Curves in light gray (green) are the analytic results, those in dark gray (red) are our numerical results data. Agreement for the large radius, low spin cases is extremely good, especially for small $l$ where the numerical data lies practically on top of the analytic predictions. As we increase $q$ and decrease $r_{\mathrm{o}}$, the amplitude agreement becomes less good, though the analytic formulas still are within several to several tens of percent of the numerical data. The phase agreement is outstanding in all of these cases.

Figure 5

Equatorial section of the embedding of distorted Kerr black hole event horizons, $a=0.1M$ and $a=0.4M$. Each panel represents the distortion for a different radius of the orbiting body, varying from $r_{\mathrm{o}}=50M$ to the innermost stable circular orbit ($r_{\mathrm{o}}=5.669M$ for $a=0.1M$, $r_{\mathrm{o}}=4.614M$ for $a=0.4M$). As in Fig. 3, the green dashed line shows the angle at which the tidal distortion is largest, and the black dotted line shows the position of the orbit. As in Fig. 3, we have rescaled by a factor $\propto r_{\mathrm{o}}^3/\mu$ to account for the leading dependence of the tide on mass and orbital separation. In contrast to the Schwarzschild results, the bulge does not lead the orbit in all cases here. The amount by which the bulge leads the orbit grows as the orbit moves to small orbital radius (in some cases, changing from a lag to a lead as part of this trend).

Figure 6

Equatorial section of the embedding of distorted Kerr black hole event horizons, $a=0.7M$ and $a=0.866M$. Each panel represents the distortion for a different radius of the orbiting body, varying from $r_{\mathrm{o}}=50M$ to the innermost stable circular orbit ($r_{\mathrm{o}}=3.393M$ for $a=0.7M$, $r_{\mathrm{o}}=2.537M$ for $a=0.866M$), with the green dashed and black dotted lines labeling the locations of maximal distortion and position of the orbit, respectively, and with the distortion rescaled by a factor $\propto r_{\mathrm{o}}^3/\mu$. The bulge lags the orbit in most cases we show here, with the lag angle getting smaller and converting to a small lead as the orbit moves to smaller and smaller orbital radius.

Figure 7

Two example embeddings of the tidally distorted horizon’s surface. Both panels show the 3-dimensional Euclidean embedding surface, $r_{\mathrm{E}}(\theta ,\psi )$; the shading (or color scale) indicates the horizon’s distortion relative to an isolated Kerr black hole. The hole is stretched (i.e., $r_{\text{E}}$ increased by the tides relative to an isolated hole; red or dark gray) at the end near to and opposite from the orbiting body. It is squeezed ($r_{\text{E}}$ decreased by tides; blue or light gray) in a band between these two ends. As in other figures illustrating the embedded distorted horizon, we have rescaled the distortion by a factor $\propto r_{\mathrm{o}}^3/\mu$. On the left, we show a relatively gentle deformation around a moderately spinning black hole: $a=0.3M$, $r_{\mathrm{o}}=20M$. The distortion here is dominated by a quadrupolar deformation of the horizon (lagging the orbiting body, whose angular position is indicated by the small blue ball). On the right, we show a rather extreme case: $a=0.866M$, $r_{\mathrm{o}}=1.75M$. The deformation here is much more complicated, as many multipoles beyond $l=2$ contribute to the shape of the horizon.

Figure 8

Embedding of distorted Kerr black hole event horizons for a corotating orbit—i.e., an orbit for which ${\mathrm{\Omega }}_{\text{orb}}={\mathrm{\Omega }}_{\mathrm{H}}$. As in Figs. 3, 5, and 6, the green dashed line points along the direction of greatest horizon distortion, and the black dotted line points to the orbiting body; the distortions are all scaled by a factor $\propto r_{\mathrm{o}}^3/\mu$. At very small spins (for which the corotating orbital radius is very large), the bulge lags the orbit slightly, but the bulge leads for all other spins.