Tidal effects around higher-dimensional black holes

In four-dimensional spacetime, moons around black holes generate low-amplitude tides, and the energy extracted from the hole’s rotation is always smaller than the gravitational radiation lost to infinity. Thus, moons orbiting a black hole inspiral and eventually merge. However, it has been conjectured that in higher-dimensional spacetimes orbiting bodies generate much stronger tides, which backreact by tidally accelerating the body outward. This effect, analogous to the tidal acceleration experienced by the Earth–Moon system, would determine the evolution of the binary. Here, we put this conjecture to the test, by studying matter coupled to a massless scalar field in orbit around a singly spinning rotating black hole in higher dimensions. We show that in dimensions larger than five the energy extracted from the black hole through superradiance is larger than the energy carried out to infinity. Our numerical results are in excellent agreement with analytic approximations and lend strong support to the conjecture that tidal acceleration is the rule, rather than the exception, in higher dimensions. Superradiance dominates the energy budget and moons “outspiral”; for some particular orbital frequency, the energy extracted at the horizon equals the energy emitted to infinity and “floating orbits” generically occur. We give an interpretation of this phenomenon in terms of the membrane paradigm and of tidal acceleration due to energy dissipation across the horizon.

Figure 1

Comparison between the flux ratio $\rho ={\dot{E}}_H^{\mathrm{Tot}}/{\dot{E}}_{\infty }^{\mathrm{Tot}}$ (in absolute value) calculated analytically and numerically, as a function of the particle velocity $v$ for $n=0$, 1, 2, 3, 4 ($D=4$, 5, 6, 7, 8) and $a=0.2M^{1/(1+n)}$. The straight curves correspond to the analytical formula with $l=1$, and the dots are the numerical results discussed in Sec. 5. In the slowly rotating regime, numerical results are in very good agreement with the analytical formula.

Figure 2

The flux ratio $\rho ={\dot{E}}_H^{\mathrm{Tot}}/{\dot{E}}_{\infty }^{\mathrm{Tot}}$ (in absolute value) as a function of the particle velocity $v$ defined in Eq. (11) for $n=0$, 1, 2, 3, 4 ($D=4$, 5, 6, 7, 8) and $a=0.99M^{1/(1+n)}$.

Figure 3

The ratio $\rho ={\dot{E}}_H^{\mathrm{Tot}}/{\dot{E}}_{\infty }^{\mathrm{Tot}}$ as a function of the orbital velocity defined in Eq. (11) for several values of $a$. Left panel: when $D=5$, the ratio is constant in the small $v$ region, and it approaches unity in the extremal limit, $a\to \sqrt{2M}$. Right panel: when $D=6$, the flux at the horizon can exceed the flux at infinity. For each curve, the intersection with the horizontal line corresponds to a floating orbit, $-\rho =1$. Note that, at large orbital velocity, the superradiant condition is not met and ${\dot{E}}_H>0$.