Radiation from particles with arbitrary energy falling into higher-dimensional black holes

We consider point particles with arbitrary energy per unit mass $E$ that fall radially into a higher-dimensional, nonrotating, asymptotically flat black hole. We compute the energy and linear momentum radiated in this process as functions of $E$ and of the spacetime dimensionality $D=n+2$ for $n=2,\dots ,9$ (in some cases we go up to 11). We find that the total energy radiated increases with $n$ for particles falling from rest ($E=1$). For fixed particle energies $1<E\le 2$ we show explicitly that the radiation has a local minimum at some critical value of $n$, and then it increases with $n$. We conjecture that such a minimum exists also for higher particle energies. The present point-particle calculation breaks down when $n=11$, because then the radiated energy becomes larger than the particle mass. Quite interestingly, for $n=11$ the radiated energy predicted by our calculation would also violate Hawking’s area bound. This hints at a qualitative change in gravitational radiation emission for $n\gtrsim 11$. Our results are in very good agreement with numerical simulations of low-energy, unequal-mass black hole collisions in $D=5$ (that will be reported elsewhere) and they are a useful benchmark for future nonlinear evolutions of the higher-dimensional Einstein equations.

Figure 1

Energy and momentum radiated plotted as functions of $n$ for selected values of $E$.

Figure 2

The scalar potential of Eq. (6) as a function of the tortoise coordinate $r_{*}$ for $l=2$ and selected values of $n$. We use units $r_h=1$.

Figure 3

Energy spectra for $n=3$ and $n=6$ (in units $r_h=1$).

Figure 4

Multipolar distribution for $n=3,\dots ,9$.