The publicly available abstract just states "In this study, we discuss whether building a Dyson sphere around a black hole is effective." but no statements as to what they arrive at.
https://arxiv.org/abs/2106.15181 is the preprint.
The authors' model rests on an asymptotically flat incoming Vaidya spacetime[1] equipped with a cosmic microwave background that dilutes away over time (like the real CMB) but without the worries of tracking the cosmological constant sufficiently near the black hole.
They unsurprisingly find that the energy budget from the background light that collides into the black hole in this model is an uninteresting part of the system's extractable energy budget. The real CMB has a low energy-density at all times when our universe has star-filled galaxies in it, and one would need a high boost to get significant power from it, or equivalently, one would have to gravitationally modify the energy-density locally. (That boost could be found by an observer hovering just outside their model non-rotating black hole, or in a different model where (for example) where a black hole's rotation can lead to an observer developing angular speeds approaching the speed of light; however there is no obvious way to transmit harvested power of this sort "up" to a Dyson sphere thanks to gravitational redshifting at least. The authors admit that calculating an energy budget which relies upon black hole rotation is too hard, and move on).
Skipping over the accretion disk section, which I'll return to, the authors also find that in the absence of angular momentum and, an incoming spherically symmetrical uncharged "rain" onto the black hole does generate significant outgoing radiation, which is also unsurprising. This is after all essentially what the CMB is, just with a higher "molecular weight" than light. Of course this particular rain is unphysical: our universe is populated by clouds of gas, and stars, which periodically -- not uniformly in time -- intersect black holes like the one in our galactic centre [2], causing occasional flares. They do not spend much time describing an integral approach around this. For what it's worth most principled uses of the Vaiyda metric consider outgoing radiation as an easy proxy for Hawking radiation (or even more often for modelling the cooling of white dwarfs). A massive incoming Vaiyda "anti-shine" has some interesting consequences or requirements for regions far from a black hole with lifetimes of more than a few million years!
The authors do not properly explain how they equip their model with a relativistic jet (they repeat an empirical luminosity relationship between astronomically observed jets accretion disks, but no causality is found in their model, and their model is different from "real" astrophysics) which is quite striking given their everywhere spherically-symmetric, exactly zero angular momentum, and exactly zero magnetic charge setting for for their CMB and Bondi accretion analyses. Their Chou 2020 reference [3] underpinning their relativistic jets purports to describe a spacetime where the authors break this no-angular-momentum condition. From that breaking they take spherical symmetry into axisymmetry, and with some addition of magnetic interactions, they arrive at an environment that might plausible host relativistic jets. But they also suddenly introduce significant consequences of angular momentum and other nongravitational charges near their model black hole that are not accounted for in their CMB or Hawking analyses. They do not explain how to reconcile their findings from two qualitatively different spacetimes, nor do they attempt an argument that the quantitative differences vanish "below" the altitude of the Dyson sphere. (Do you have to make an axisymmetric Dyson swarm, where the equatorial collectors have a high orbital angular velocity?)
I think that the authors would have written a more interesting (and challenging) paper if they had chosen to explore the components of the energy budget of their Dyson-around-a-black-hole system using a gravitational and matter background that does not change from subsection to subsection.
That they didn't raises two questions I think are large.
Firstly, can they get their §2.6 relativistic jets somehow without introducing both angular momentum and magnetic interactions? (The formation of relativistic jets is a live area of current research).
Alternatively, can they find the heavy calculation lifting in other published work for their energy estimates from CMB-infall and infall-onto-accretion-structure physics? Their Chou 2020 references at least shows that they are competent to do a wide search for papers which purport to do that kind of heavy lifting. Their §2.3 already refers to other work in accretion disc physics that reasonably covers both extremes of angular momentum, but there's a gap between the two which is likely interesting for relativistic jets that an be "mined" by their Dyson sphere, and closer to the spin parameters of known black hole candidates.
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[1] §2.4 could be much clearer about this, especially in light of the "Kerr-Vaiyda" metric they found in [4]; they consider a vanishingly small a parameter (sometimes implicitly) whenever they discuss their model black hole, and resile from the consequences of a significant nonzero spin, except briefly for the case of the maximum possible spin. Perhaps they found it easiest to import the maths from [4]'s results accepting them as good especially in the limit where a vanishes, rather than sticking to a more widely used model -- https://en.wikipedia.org/wiki/Vaidya_metric. After a brief glance at the edit history, I have stray thoughts about the timing of the introduction of [4] at the end of that wikipedia page.
[2] examples https://www.pbs.org/newshour/science/massive-gas-cloud-colli... https://phys.org/news/2021-07-milky-supermassive-black-hole.... and, in a not-too-distant galaxy, https://www.sciencealert.com/arp-299-supermassive-black-hole...
[3] Chou 2020 is https://doi.org/10.1016/j.heliyon.2020.e03336 (pay-for-publication open access by bottom-feeding Elsevier). I believe the author to be https://www.researchgate.net/profile/Yu-Ching-Chou where one finds a link to this reference, and that the publication record of the single-author relativity papers listed there speaks for itself.