Penrose process, superradiance, and ergoregion instabilities

Superradiant scattering is a radiation enhancement process that takes place in many contexts, and which has recently found exciting applications in astrophysics and particle physics. In the framework of curved spacetime physics, it has been associated with the classical Penrose process for particles. Superradiance is usually also associated with bosonic fields around geometries with ergoregions and horizons. These notions are in clear tension however: the Penrose process occurs for horizonless geometries, and particles are composed of fermions. Here, we resolve the tension in its different aspects, by showing that (i) superradiance occurs for self-interacting fermions on flat spacetime. (ii) Superradiance occurs also for horizonless geometries, where it leads to an ergoregion instability. Ultracompact, horizonless geometries will usually respond with echoes of growing amplitude, until rotational (or electrostatic) energy is extracted from the object. (iii) The Fourier-domain analysis leads to absence of superradiance when horizons are not present. We elucidate why this analysis fails to give meaningful results. (iv) Finally, we show that superradiant, ergoregion instabilities have a particle analog of similar growth timescales and which can power the formation of a structure outside a compact, rotating star.

Figure 1

Amplification factor for a monochromatic scalar wave of frequency $\omega$ scattered off a step potential (79).

Figure 2

Wave packet solution at $t=0$, 160 (left and right panel, respectively) for a Gaussian wave packet scattered off a step potential (79), characterized by $q{\mathrm{\Phi }}_0=10$. The initial data corresponds to a Gaussian centered at ${\omega }_0=2$, with ${\sigma }_{\omega }=0.1$, $z_0=-100$. At $t=160$ the initial pulse scattered off the barrier, a fraction was reflected and parts penetrated the barrier. It is apparent that the process is superradiant: the right-moving pulse at $t=0$ was converted into a higher-amplitude, left-moving pulse at $t=160$. As a reference, for a perfectly monochromatic pulse with frequency $\omega =2$ incident on such a potential barrier, the amplification factor is $Z\simeq 2.778$.

Figure 3

Amplification factor for a Gaussian wave packet described by (82) as function of ${\sigma }_{\omega }$, for a coupling $q{\mathrm{\Phi }}_0=10$ and ${\omega }_0=2$. For this scattering problem, monochromatic waves of frequency $\omega =2$ have $Z\simeq 2.778$.

Figure 4

Amplification factor for a monochromatic scalar wave scattering on the potential (90) (left, $L=0.001$) and on a RN BH (right, BH mass and charge are $M=1$, $Q=0.2$, respectively).

Figure 5

Wave packet solution at $t=0$, 40, for the left and right panel, respectively. The initial data is given by Eqs. (88) and (89) with ${\omega }_0=2$, ${\sigma }_z=2.24$ and $z_0=-20$. The potential has the form (90) with $q{\mathrm{\Phi }}_0=10$ and $L=0.001$. The figure on the left shows the form of the incident pulse before the scattering on the barrier. At $t=40$ the initial pulse scattered off the barrier, a fraction was reflected and a part penetrated inwards.

Figure 6

Wave packet solution at $t=0$, 30, 115, 155, for the left and right panel, respectively. The initial data is given by Eqs. (88) and (89) with ${\omega }_0=2$, ${\sigma }_{\omega }=2.24$ and $z_0=-20$. The potential has the form (90) with $q{\mathrm{\Phi }}_0=10$ and $L=0.001$. There is now a mirror (implemented with a Dirichlet boundary condition) at $z=20$. This figure shows the form of the incident pulse before the scattering on the barrier. At $t=30$ the initial pulse scattered off the barrier, a fraction was reflected and a part penetrated inwards (just like in Fig. 5). At latter times it is possible to see an analogue ergoregion instability (due to the successive superradiant amplification of the pulse confined in $0<z<20$). This instability is characterized by the emission of echoes of increasing amplitude.

Figure 7

Penrose process cascade in the neighborhood of a very compact (rotating) star. A particle decays in two inside the ergoregion, at a radius $\tilde{r}$ (the dashed circumference). The ingoing particle is made to reflect off the surface of the star and decay again at the same radius. The process continues indefinitely and gives rise to an ergoregion (exponential) instability.

Figure 8

Exponential growth of the energy of first 20 emitted (positive energy) particles for different values of $\tilde{r}/M$, with $x=0.2$ and $\delta ={10}^{-4}$.

Figure 9

Exponential growth of the energy of first 20 emitted (positive energy) particles for different values of $x$, with $\tilde{r}/M=1.6$ and $\delta ={10}^{-4}$.