# Are composite bosons superradiant scattering candidates?

Massive bosons can experience superradiant scattering in an ergosphere. This can in principle be used as a power source. However, elementary massive bosons ($$H^0,\,W^\pm,\,Z^0$$) are short-lived. Are stable but composite bosons such as deuterons candidates?

## 1 Answer

The onset of superradiant instability from massive bosons is governed by the dimensionless parameter $$\alpha=\mu M/ m_\text{P}^2$$, where $$\mu$$ is the mass of boson particles, $$M$$ is the black hole mass and $$m_\text{P}$$ is the Planck mass. For the efficient energy extraction from the black hole the parameter must satisfy the condition $$\alpha \lesssim 1$$. This condition could not be satisfied with astrophysical black holes and composite bosons of the Standard Model. For example for deuteron and black hole of $$1 M_\odot$$ we have $$\alpha \approx 1.4\times 10^{19}$$. For such large values of $$\alpha$$, the timescales of evolution of superradiant modes would be enormous (on the order of $$M \exp( C \alpha)$$ in GR units, with $$C$$ a constant of order 1 ) much greater than the lifetime of the Universe, and even much greater than expected lifetime of such a black hole due to Hawking radiation.

If there are very small black holes (primordial) that could provide the necessary small value of $$\alpha$$, then still superradiance from massive composite bosons would be impossible for a variety of reasons:

• Since $$\alpha < 1$$ means that Compton wavelength of a particle is larger than Schwarzschild radius of the black hole, tidal forces near the horizon of the black hole would be exceeding forces (e.g. nuclear) holding the particle together. The approximation of an effective field of the particle would not hold under those circumstances and one has to consider the quantum fields of constituent particles.

• For such small black hole there would be considerable Hawking radiation interacting with our composite bosons, making it impossible to consider them as free fields/particles interacting only gravitationally.

For overview of black hole (and other types of) superradiance I would recommend a book:

1. Brito, R., Cardoso, V., & Pani, P. (2015). Superradiance. Lect. Notes Phys, 906(1), 1501-06570, arXiv:1501.06570.
• Thank you. The timescale is exponential in $\alpha$ viz. Eq. (5.30). Where is it explained why $\alpha=0$ and $\alpha\gg 1$ respectively lead to no extraction and inefficient extraction? – J.G. Mar 29 at 6:57
• The case $\alpha\ll 1$ corresponds to non-relativistic quantum Kepler problem plus small corrections, smaller $\alpha$ leads to smaller corrections. In eq. 4.27 of the book, two terms of $\omega_I$ correspond to two effects: dampening through absorption by BH ($\sim r_+$) and amplification by superradiance. See also Fig. 4.9, for the dependence on $\alpha$ of $\omega_I$, Individual modes have a characteristic “tooth” shape, but as $\alpha$ increases different teeth/modes produce max values, and for large $\alpha$ the teeth aggregate along the curve of eq. 4.30. – A.V.S. Mar 29 at 18:17
• I'm afraid I don't see how $L^{(0)}=\frac{\sqrt{2Mr_0\Delta}-2aM}{r_0-2M},\,\mathcal{E}^{j}=\frac{\mu_0}{2}\left(1\mp(-1)^j\sqrt{\frac{2M(1-4\mu_\text{fin}^2/\mu_0^2)}{r_0}}\right)$ address these issues, probably due to my not understanding how their notation relates to your own. – J.G. Mar 29 at 18:49
• I see, I was referring to numbering in the book, in e-print this is eq. 5.27 and figure 28, respectively – A.V.S. Mar 30 at 5:13