Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation

There is a well-known, intuitive geometric correspondence between high-frequency quasinormal modes of Schwarzschild black holes and null geodesics that reside on the light ring (often called spherical photon orbits): the real part of the mode’s frequency relates to the geodesic’s orbital frequency, and the imaginary part of the frequency corresponds to the Lyapunov exponent of the orbit. For slowly rotating black holes, the quasinormal mode’s real frequency is a linear combination of the orbit’s precessional and orbital frequencies, but the correspondence is otherwise unchanged. In this paper, we find a relationship between the quasinormal-mode frequencies of Kerr black holes of arbitrary (astrophysical) spins and general spherical photon orbits, which is analogous to the relationship for slowly rotating holes. To derive this result, we first use the Wentzel-Kramers-Brillouin approximation to compute accurate algebraic expressions for large-$l$ quasinormal-mode frequencies. Comparing our Wentzel-Kramers-Brillouin calculation to the leading-order, geometric-optics approximation to scalar-wave propagation in the Kerr spacetime, we then draw a correspondence between the real parts of the parameters of a quasinormal mode and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the quasinormal-mode parameters to coefficients that modify the amplitude of the scalar wave. With this correspondence, we find a geometric interpretation of two features of the quasinormal-mode spectrum of Kerr black holes: First, for Kerr holes rotating near the maximal rate, a large number of modes have nearly zero damping; we connect this characteristic to the fact that a large number of spherical photon orbits approach the horizon in this limit. Second, for black holes of any spins, the frequencies of specific sets of modes are degenerate; we find that this feature arises when the spherical photon orbits corresponding to these modes form closed (as opposed to ergodically winding) curves.

Figure 1

Low-overtone QNM spectrum of three Kerr black holes of different spins with approximate degeneracies in their spectra. From left to right, we plot the three lowest-overtone QNM excitations for (i) $a/M=0.69$ in which $(l,m)=(j,2)$ are black triangles and $(l^{\prime },m^{\prime })=(j+1,-2)$ are blue squares, where $j=3,\dots ,9$; (ii) $a/M=0.47$ in which $(l,m)=(j,3)$ are magenta dots and $(l^{\prime },m^{\prime })=(j+1,-3)$ are cyan cycles, where $j=3,\dots ,9$; (iii) $a/M=0.35$ in which $(l,m)=(j,4)$ are red diamonds and $(l^{\prime },m^{\prime })=(j+1,-4)$ are purple stars, where $j=5,\dots ,10$. For these spin parameters, the mode with positive values of $m$ and ${\omega }_R$ (a corotating mode) of index $l$ is approximately degenerate with the mode with $m^{\prime }=-m,$ and ${\omega }_R$ (a counter-rotating mode) of index $l^{\prime }=l+1$.

Figure 2

Difference in ${\Omega }_R(a,\mu )$ [Eq. (2.35)] that arises from using the approximate formula for $A_{lm}$ [Eq. (2.28a)] as opposed to the exact formula. Here $a/M=0.7$, 0.9, 0.95, and 0.99 correspond to black solid, red dashed, blue dotted, and purple long-dashed curves, respectively. The quantity plotted on the vertical axis has been scaled by ${10}^5$.

Figure 3

Difference in ${\Omega }_I(a,\mu )$ [Eq. (2.39)] from using the approximate formula for $A_{lm}$ [Eq. (2.28a)] rather than the exact formula. Here $a/M=0.7$, 0.9, 0.95, and 0.99 correspond to black solid, red dashed, blue dotted, and purple long-dashed curves, respectively. We scale the quantity plotted along the vertical axis by ${10}^4$ in this figure.

Figure 4

Real part of the QNM spectra from the WKB approximation. Black solid curves show $\Omega$ for $a/M=0$ (the flat curve) and $a/M=1$ (the curve that increases towards 0.5); red (light gray) dashed and dotted curves show $a/M=0.3$ and 0.5, while blue (dark gray) dotted and dashed curves show $a/M=0.9$ and 0.99.

Figure 5

Imaginary part of the QNM spectrum computed in the WKB approximation. Black solid curves show ${\Omega }_I$ for $a/M=0$ (again the flat curve) and $a/M=1$, the curve that decreases and heads to zero. The red (light gray) dashed curve shows $a/M=0.5$, while blue (dark gray) dotted and dashed curves show $a/M=0.9$ and 0.99, respectively. For $a/M=1$, modes with $\mu \gtrsim 0.74$ approach zero (modes that do not decay), while others still decay.

Figure 6

Fractional error, $\delta {\omega }_R/{\omega }_R$, of the WKB approximation to the $s=2$, gravitational-wave, quasinormal-mode spectrum, multiplied by $L^2$. The four panels correspond to four different spins which (going clockwise from the top left) are $a/M=0.3$, 0.5, 0.95, and 0.9. Errors for $l=2$, 3, 4 are highlighted as red solid, brown dashed, and pink dotted lines, while the rest ($l=5,\dots ,14$) are shown in gray. This shows that the relative error approaches the $O(1/L^2)$ scaling quite quickly.

Figure 7

Fractional error, $\delta {\omega }_R/{\omega }_R$, of the WKB approximation to the $s=0$, scalar-wave, quasinormal-mode spectrum, again scaled by $L^2$. The four panels correspond to the same four spins in Fig. 6. The points shown in the four panels are for values of $l$ in the range $l=2,3,\dots ,14$. Because all values of $l$ nearly lie on the same curve, the relative error has converged at an order $O(1/L^2)$ even for very low $l$. The overall error is also significantly lower than that for the $s=2$ modes.

Figure 8

Fractional error, $\delta {\omega }_I/{\omega }_I$, of the WKB approximation to the $s=2$, gravitational-wave, quasinormal-mode spectrum, also scaled by $L^2$. The panels and the curves are plotted in the same way as in Fig. 6, and the error scales similarly.

Figure 9

Fractional error, $\delta {\omega }_I/{\omega }_I$, of the WKB approximation to the $s=0$, scalar-wave, quasinormal-mode spectrum, again multiplied by $L^2$. The four panels and the points are shown in the same way as in Fig. 7, and there is a similar rapid convergence of the error.

Figure 10

Schematic plot of trajectories in the $r\mathrm{\text{−}}\theta$ plane of homoclinic orbits outside of the peak of the potential (specifically for a black hole with spin $a/M=0.7$ and a photon orbit with radius $r_0/M=2.584$). The two horizontal grid lines mark the turning points, $\theta ={\theta }_{\pm }$; between these turning points, there are two homoclinic orbits passing through every point, while at turning points only one orbit passes through. Vertical grid lines indicate when the value of parameter $\lambda$ has changed along the orbit by (an arbitrarily chosen value) $\Delta \lambda =0.046M$. Near the spherical photon orbit, each homoclinic orbit undergoes an infinite number of periodic oscillations in $\theta$ while $r-r_0$ is growing exponentially as a function of $\lambda$.

Figure 11

The values of $r$ and $\cos {\theta }_{+}$ of spherical orbits, for $a/M=0$ (black, solid vertical line), 0.5 [red (light gray) dashed curve], 0.9 [blue (dark gray) dashed curve], and 0.99 999 (black, solid curve). Note that for $a=0$ all such orbits have $r=3M$, while for $a=M$ a significant fraction reside at $r=M$.

Figure 12

Radii of corotating spherical photon orbits as a function of $\mu$, for $a/M=0.9$ (black solid line), 0.99 (red dashed curve), 0.9999 (blue dotted line). For extremal Kerr black holes, a nonzero fraction of all spherical photon orbits are on the horizon.

Figure 13

Orbital frequency, ${\Omega }_{\theta }$, plotted against $\mu$, for $a/M=0.3$ [red (light gray) solid curve], 0.7 [blue (dark gray) solid curve], 0.9 (purple dashed line), and 1 (black dotted line). The orbital frequency vanishes for a significant range of $\mu$ for extremal black holes.

Figure 14

Precessional frequency, ${\Omega }_{\varphi }$, versus $\mu$ plotted identically to those curves in Fig. 13 representing the same black-hole spins. The precessional frequency approaches the horizon frequency, ${\Omega }_H$, for a range of values of $\mu$ for extremal black holes.

Figure 15

A diagram showing the spin parameters, $a$, and the ratios of the multipolar indices $m/L$, at which the orbital and precessional frequencies have a ratio of $p/q$. Although we only perform our numerical calculations at a discrete set of $m/L$ values (shown by the dots), in the eikonal limit, each set of points for a given ratio of $p/q$ approaches a continuous curve.

Figure 16

For black holes with spins $a/M=0.768$, 0.612, and 0.502, the spherical photon orbits with ${\omega }_{\mathrm{orb}}=2{\omega }_{\mathrm{prec}}$ on the left, ${\omega }_{\mathrm{orb}}=3{\omega }_{\mathrm{prec}}$ in the center, and ${\omega }_{\mathrm{orb}}=4{\omega }_{\mathrm{prec}}$ on the right, respectively. These orbits correspond to quasinormal modes in the eikonal limit with $m/L=0.5$. The top figures show the photon orbit, the red, solid curve, on its photon sphere (represented by a transparent sphere). The dashed black line is the equatorial ($\theta =\pi /2$) plane, which was inserted for reference. The bottom figures are the same photon orbits, but plotted in the $\varphi \mathrm{\text{−}}\theta$ plane, instead.