Moving black holes: Energy extraction, absorption cross section, and the ring of fire

We consider the interaction between a plane wave and a (countermoving) black hole. We show that energy is transferred from the black hole to the wave, giving rise to a negative absorption cross section. Moving black holes absorb radiation and deposit energy in external radiation. Due to this effect, a black hole hole of mass $M$ moving at relativistic speeds in a cold medium will appear to be surrounded by a bright “ring” of diameter $3\sqrt{3}GM/c^2$ and thickness $\sim GM/c^2$.

Figure 1

Energy gain of a (high-frequency) photon scattered off a moving BH. The photon has initial energy $E_i$, impact parameter $b$, and scatters off a BH moving with velocity $v$ in the opposite direction. The final energy is $E_f$. The peak of each curves agrees, to numerical precision, with Eq. (1). For impact parameter $b<3\sqrt{3}M$, the photon is absorbed by the BH.

Figure 2

Numerical values of the quantity $\frac{d\sigma }{db}$ obtained through (null geodesic) geometric optics, and wave scattering with $\omega M=17$. The two methods are in clear agreement for $b\gg M$. Note that the numerical values of the $v=0$ curves are extremely small and compatible with zero, up to numerical errors.

Figure 3

Appearance of a BH moving in a bath of cold (and countermoving) radiation. The BH is moving along the $z$-axis towards us at a speed $v=0.9$. The colors denote energy flux intensity on a screen placed a short distance away from the BH. The peak energy flux is 10 times larger than that of the environment. The bright ring has width $\sim M$ for all boost velocities $v$. For very large $v$ even a randomly moving gas of photons will leave a similar observational imprint, since countermoving photons will be redshifted away.