The collisional Penrose process
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Abstract
Shortly after the discovery of the Kerr metric in 1963, it was realized that a region existed outside of the black hole’s event horizon where no timelike observer could remain stationary. In 1969, Roger Penrose showed that particles within this ergosphere region could possess negative energy, as measured by an observer at infinity. When captured by the horizon, these negative energy particles essentially extract mass and angular momentum from the black hole. While the decay of a single particle within the ergosphere is not a particularly efficient means of energy extraction, the collision of multiple particles can reach arbitrarily high centerofmass energy in the limit of extremal black hole spin. The resulting particles can escape with high efficiency, potentially serving as a probe of highenergy particle physics as well as general relativity. In this paper, we briefly review the history of the field and highlight a specific astrophysical application of the collisional Penrose process: the potential to enhance annihilation of dark matter particles in the vicinity of a supermassive black hole.
Keywords
Black holes Ergosphere Kerr metric1 Introduction
In 1969, Roger Penrose realized that reactions taking place within the ergosphere could result in particles with negative energy [3]. What does it mean for a particle to have negative energy? Locally, any observer with 4velocity \(u^\mu \) measures the energy of a particle with momentum \(p_\mu \) with a simple inner product: \(E_\mathrm{obs} = p_\mu u^\mu \). An observer at rest at infinity has \(u^\mu = [1,0,0,0]\) so we can define the “energy at infinity” of a particle to be \(E_\infty \equiv p_t\). For stationary spacetimes, \(p_t\) is an integral of motion, so a particle’s energy at infinity is conserved. Thus it is possible for a particle to have positive energy, as measured by an observer in the ergosphere, yet still have \(p_t >0\). Such a particle simply would not be able to escape the ergosphere, and would rapidly be captured by the black hole (all the while maintaining Christodoulou’s limits [4] on the black hole mass and spin).
Shortly after Penrose’s 1969 paper, some authors proposed that this remarkable feature of spinning black holes might be responsible for the highenergy radiation seen from some active galaxies. But a careful analysis by Bardeen et al. [5] and Wald [6] showed that it was impossible to attain relativistic energies due to the Penrose process alone. A particle would have to emit a daughter particle with energy a significant fraction of the original particle’s rest mass in order for the surviving particle to itself reach relativistic velocities. And in this case, while the escaping particle may in fact be moving near the speed of light, it can only do so by sacrificing its own rest mass energy.
Now, \(121\%\) efficiency in converting mass to energy is nothing to dismiss lightly. It far exceeds the efficiency of nuclear fusion, and even exceeds the radiative efficiency of quasars (assuming Novikov–Thorne thin accretion disks [8] and maximal spin, efficiency of \(40\%\) might be attained [9]). It is the ultimate “free lunch:” getting more energy out than you put in. Yet it still cannot explain the GeV or even TeV emission seen in blazars, or the largescale, relativistic jets seen emerging from radio galaxies with bulk Lorentz factors of 10 or more, and this was one of the outstanding mysteries that Penrose, Wald, and others at the time were trying to solve [3, 6].
Shortly after Wald published his limit of \(121\%\), Piran and collaborators discovered a way to attain an even higher efficiency from the Penrose process [7, 10]. The crucial point was to use more than one particle, in order to achieve a much higher centerofmass energy in the ergosphere. This approach is known as the Collisional Penrose Process.
The collisional Penrose process is a great deal richer than the simple decay problem considered by Wald, where it was clear from inspection what was the geometry required for maximal efficiency. For two incoming particles, the range of possible values for the total energy and momentum are vastly greater. Further expanding the population of colliding particles to those with \(E_0/m \ne 1\) and nonplanar orbits makes the problem virtually intractable for analytic methods. However, from symmetry arguments we can limit our focus on specialized planar orbits around extremal black holes when looking for the highest efficiency reactions.
Figure 2 shows these turning points as a function of the impact parameter \(b\equiv \ell /\varepsilon \) for both massless and massive particles, for maximal spin \(a_*=1\). For the massive particles we set \(\varepsilon =1\), corresponding to a particle at rest at infinity. One can visualize a massive particle incoming from the right with \(b < 2(1+\sqrt{2})\) or \(b > 2\), reflecting off the centrifugal potential barrier and returning back to infinity (yellow regions). Alternatively, if the impact parameter is small enough [i.e., \(2(1+\sqrt{2})< b < 2\)], the particle will get captured by the black hole. Due to framedragging, the cross section for capture is much greater for incoming particles with negative angular momentum [12].
2 Banados–Silk–West
This divergence for the extremal case is closely related to the curvature structure of the horizon. It is well known that the singularity of the Kerr metric at the horizon is only a coordinate singularity for Boyer–Lindquist coordinates [14]. A whole class of coordinates exists (e.g., Kerr–Schild, Doran) that do not blow up at the horizon, and are thus useful for calculating the trajectories of particles near or across the horizon. However, in the limit of extremal Kerr, the curvature singularity approaches the horizon at Boyer–Lindquist \(r=M\) (a superextremal black hole would have the singularity outside the horizon, and thus violate the cosmic censorship conjecture), and thus physical, coordinateindependent quantities such as the centerofmass energy can diverge.
In response to BSW, Lake [15] and Gau and Zhong [16] showed that the c.o.m. energy for collisions inside the horizon will generically diverge even for nonextremal black holes as the particles approach the inner Cauchy horizon, which is itself outside of the curvature singularity (they all coincide with \(r=M\) in the extremal limit). Along these lines, Ref. [17] showed that, for black holes with \(a_*>1\) (naked singularities), infinite c.o.m. energy collisions were quite generic.

Berti et al. [19] point out two practical problems: even in the limit of an initially extremal black hole, a single collision would deposit the mass and angular momentum of the debris particles, lowering the black hole spin far below the levels needed for Planckscale collisions. Additionally, the critical orbits required for diverging \(E_\mathrm{com}\) take an infinite amount of proper time to actually reach the horizon. During this time, a particle would orbit so many times that it would actually emit a significant amount of energy and angular momentum in gravitational radiation, in turn reducing the spin of the black hole. For the much more astrophysical spin limit of \(a_*=0.998\) [9], the peak energy would be a paltry 6.95 times the restmass energy [20].

Jacobson and Sotiriou [21] show that the scaling of \(E_\mathrm{com}\) is extremely weak with the spin. For nearmaximal \(a=1\epsilon \), they find the peak energy to be \(E_\mathrm{com} \sim 4.06 \epsilon ^{1/4}\) (see Fig. 4 above). Aside from this weak scaling restriction, they also show that any energy gained by colliding near the horizon will necessarily be lost by the redshift of escaping the black hole potential.

Harada and Kimura [20] demonstrated similar scaling for particles falling in from the innermost stable circular orbit (ISCO). For (nonplunging) particles on ISCO orbits colliding with generic particles falling in from infinity, the peak centerofmass energy has an even weaker scaling: \(E_\mathrm{com}\sim 5 \epsilon ^{1/6}\).

Bejger et al. [22] focus on the problem of the escaping particle’s energy. They agree that an arbitrary centerofmass energy can be achieved, but like [21], point out that the reaction products lose much of their energy on the way out from the horizon, ultimately limiting the efficiency of the process to \(129\%\) for equalmass particles falling in from infinity.

Harada et al. [23] carry out a more general calculation including nonequal mass particles and Compton scattering reactions, yet mistakenly calculate an even smaller upper limit of \(109\%\) for efficiency in the BSWtype reaction.

Ding et al. [24] are the first to introduce the additional limitations that will arise from a quantum theory of gravity. By including the effects of a noncommutative spacetime via a parameterized effective field theory, they show that the maximum centerofmass energy attainable is of the order of a few thousand times the particle rest mass, but depends on the black hole mass (in quantum gravity, black holes are no longer scale invariant).

Galajinsky [25] repeats the BSW calculation in both Boyer–Lindquist and NearHorizon Extremal Kerr (NHEK) coordinates, and surprisingly finds two different answers for the maximum c.o.m. energy, with it diverging in the classical Boyer–Lindquist approach, but remaining finite in NHEK. This apparent paradox is likely due to the order in which various diverging limits are taken, and which values are allowed for particle trajectories, with the consensus appearing that the B–L result is correct [26].

Patil et al. [27] point out that for ultrahigh values for the c.o.m. energy, the critical particles must come in with such finelytuned values of angular momentum that they take a nearly infinite amount of coordinate time to reach the horizon, or even the radius of collision necessary for Planckscale energies. A potential way around this problem with multiple collisions was identified by Griv and Pavlov [28].

Like Berti et al. [19], McCaughey [29] also questions the possibility of an extremal black hole existing in nature. In particular, he focuses on the problem of Hawking radiation combined with the Penrose process of virtual particles in the ergosphere, which have a tendency of spinning down the black hole. Unfortunately, that paper does not include a quantitative estimate of the physical spindown rate as a function of black hole mass and spin (see below).

Most recently, Hod [30] raised yet another problem for reaching the highest c.o.m. energies, based on Thorne’s classic hoop conjecture [31]. Simply put, if you pack enough energy into a small enough area, you form a black hole. In the context of the BSW process, if this energy is in the form of the colliding particles, and they are close enough to the horizon, then the two black holes instantly merge, and the daughter particles cannot escape, regardless of their nominal energy and angular momentum.
For spin \(a_*= 1\epsilon \), the critical angular momentum (maximum c.o.m. energy) for incoming particles is \(b_\mathrm{crit} \approx 2(1+\epsilon ^{1/2})\). The critical radius for these collisions is at \(r_\mathrm{crit} \approx 1+2\epsilon ^{1/2}\), and the c.o.m. energy scales like \(E_\mathrm{COM} \approx (r1)^{1/2} \approx \epsilon ^{1/4}\) [19, 21]. So for an incoming particle with rest mass on the order of a GeV, in order to reach Planck energies (\(\sim 10^{19}\) GeV) the critical spin is \(1a_*\lesssim 10^{76}\). In other words, an extremal stellarmass black hole would spin down from Hawking radiation in under a second (however, see below in Sect. 3 for a less conservative limit on the critical spin value).
Focusing for now on the supermassive black holes, where Hawking radiation should not be important, there is however another, more astrophysical mechanism to spin down the black hole. All astrophysical black holes are surrounded by a bath of isotropic thermal radiation from the cosmic microwave background. At a present temperature of 2.73 K, this radiation is far more energetic than the Hawking radiation from any black hole larger than the mass of the moon (\(\sim 10^{8} M_\odot \)). Furthermore, it is isotropic, so a Kerr black hole will preferentially absorb photons with negative angular momentum, thereby accelerating the spindown process.
Clearly, the effect of CMB accretion dominates over Hawking radiation for any astrophysical black hole. Even for the smallest known black holes, an initially extremal black hole would spin down well below the critical BSW/Planck spin value in a tiny fraction of a second.
In addition to these many critiques and commentaries on BSW, there has been an even larger number of followon papers exploring analogous effects in nonKerr black holes. These papers were both within the limits of general relativity (e.g., Kerr–Newman metric), as well as alternative theories of gravity. However, since this review (and the entire Topical Collection) is specifically concerned with the Kerr metric, we consider these alternative approaches to be outside the scope of our present discussion.
3 SuperPenrose process
In Fig. 5 we show an example of this graphic for the classical Penrose process of a single particle decay. Following our approach in [36], each image corresponds to a specific choice of initial mass, energy, angular momentum, and distance from the black hole. The polar coordinates are defined in the particle’s frame (or centerofmass frame for collisional reactions), with the coordinate radial direction oriented to the right. The color represents the energyatinfinity of the daughter photons as a function of emission angle, and the radius of the disk represents whether or not that photon escapes (\(R=1\)), is captured by the black hole horizon (\(R=0.8\)), or has negative energy (\(R=0.6\)), in which case it will be also be captured by the black hole.
The middle frame of Fig. 5 corresponds to a decay at \(r=1.9999M\), just inside the ergosphere. At this point, we see the first genuine Penrose process reaction, with the forwardgoing particle having an energy just over unity, and the opposite particle has a very slightly negative energy (the tiny notch in the polar plot at \(\phi \approx 315^\circ \)). In the third frame, the reaction takes place deep in the ergosphere, and the Wald limit is reached with forwardpointing particles escaping with energy of \(E_3 = 1.21E_1\). It is interesting to note that, even this close to the event horizon, when the initial particle has the critical value for angular momentum, a majority (\(\approx 53\%\)) of the decay products are still able to escape the black hole.
Completely independent, and largely ignorant, of the flurry of papers surrounding BSW, at that time we were working on calculating the phase space distribution and annihilation rates of dark matter particles around a spinning black hole [38]. Adhering to the wellknown strategy of “if your only tool is a hammer, everything looks like a nail,” we developed a version of the Pandurata raytracing code [35] to calculate fully 3dimensional trajectories of massive test particles coming in from rest at infinity. A sample of these particles will annihilate, and then we follow the photon trajectories either to the horizon or escape to infinity.
We showed in [11] that the absolute maximum efficiency is achieved for Comptonlike scattering between an outgoing photon with \(b_1=2\) and an infalling massive particle with \(b_2=2(1+\sqrt{2})\). The postscatter products are an ingoing photon with \(b_3=2\) and an ingoing massive particle with negative energy and angular momentum. Figure 11 shows the energy and escape distributions for these Compton scattering reactions, for a range of photon energy \(E_1\). In the limit of \(E_1>> E_2\) and \(r\rightarrow r_+\), the absolute maximum efficiency for Compton scattering is given by \(\eta = (2+\sqrt{3})^2\approx 1392\%\) [37]!
The most remarkable feature of this particular configuration is that, after the scattering event, the photon—now boosted by in energy by a factor of \(\gtrsim 10\)—is on an ingoing trajectory, reflects off the black hole’s centrifugal barrier, and then becomes an outgoing photon. At this point, it can scatter off a new infalling massive particle. In this way, the panels of Fig. 11 can be considered as three consecutive scattering events, each photon getting boosted to a higher energy, while the massive particles all have the same basic energy.
We reproduce some of the results of [39] in Fig. 12 for hypothetical massive particles with \(b_1=b_2=0\) annihilating in the ergosphere, with particle 1 on an outgoing trajectory. With this selection of deus ex machina particles, it is easy to reach very high values for \(E_\mathrm{com}\), \(\eta _\mathrm{max}\), and also the escape fraction. While it is impossible for these outgoing trajectories to originate from initially infalling particles, Berti et al. [39] proposes an alternative source: the products of earlier scattering reactions. However, for the simple cases they explore, the infalling rest mass must be sufficiently large to produce the appropriate outgoing trajectories, so that in the end, the net efficiency is no greater than the single rebound configuration we originally proposed in [11].
Before we move on to the next section, covering more generic numerical calculations of the Penrose process, it is valuable to discuss in more detail the analytic results of Leiderschneider and Piran [37]. Unlike the vast majority of papers cited thus far, Leiderschneider and Piran [37] extends their analysis to also include nonplanar trajectories. The reactions still take place in the equatorial plane, but the reactant particles themselves are allowed to move out of the plane (for the case of reactions outside of the plane, see [41]). In doing so, they were able to dispell one of the popular assumptions made in many previous works: due to symmetry, the highest energies must come from purely planar trajectories. For example, the “standard” BSW case of massive, infalling planar particles annihilating into photons just outside the horizon gives a peak efficiency of \(130\%\) [22]. By relaxing only the condition on the location of the collision, slightly larger values of \(b_1\) are allowed, and the resulting efficiency increases significantly: \(\eta _\mathrm{max} \approx 2.63\) [37] (see also [42] who correctly identify the important problem of taking the r, b limits in the proper order, but appear to make an arithmetic error and obtain a slightly smaller efficiency). This actually makes perfect sense: by selecting the largest possible value for \(b_1\) for a given radius, we are in effect setting the radial velocity to zero, because that radius corresponds to a turning point for that impact parameter. Therefore the efficiency naturally lies somewhere between the ingoing and outgoing results.
Summary of results from Ref. [37], the most complete and exact work to date on maximizing energy of escaping particles (\(E_\mathrm{max}\)) and efficiency (\(\eta _\mathrm{max}\)) for the collisional Penrose process. The labeling convention for the particles is XYZsgn, with X, Y, and Z describing the properties of particles 1, 2, and 3, respectively, and ‘sgn’ describing the direction of the radial velocity for particle 1
\(E_\mathrm{max}\)  \(\eta _\mathrm{max}\)  \(M_{3,\mathrm{max}}\)  

MMP−  \(2(2+\sqrt{3})\)  \(2+\sqrt{3}\)  \(\approx 3.73\)  
MMP\(+\)  \((2+\sqrt{3})(2+\sqrt{2})\)  \((2+\sqrt{3})(2+\sqrt{2})/2\)  6.37  
PmP−  \(2(2+\sqrt{3})E_1\)  \(2(2+\sqrt{3})\)  7.46  
PmP\(+\)  \((2+\sqrt{3})^2E_1\)  \((2+\sqrt{3})^2\)  13.92  
MpP−  \(2(2+\sqrt{3})\)  \(2(2+\sqrt{3})\)  7.46  
MpP\(+\)  \((2+\sqrt{3})(2+\sqrt{2})\)  \((2+\sqrt{3})(2+\sqrt{2})\)  12.74  
MMM−  \(4+\sqrt{11}\)  \((4+\sqrt{11})/2\)  3.66  \(2\sqrt{3}\)  \(\approx 3.46\) 
MMM\(+\)  \(7+4\sqrt{2}\)  \((7+4\sqrt{2})/2\)  6.32  \(\sqrt{3}(2+\sqrt{2})\)  5.91 
PmM−  \(4E_1+\sqrt{(12E_1^21)}\)  \(2(2+\sqrt{3})\)  7.46  \(2\sqrt{3}E_1\)  \(5.91E_1\) 
PmM\(+\)  \(2(2+\sqrt{3})E_1+\)  \((2+\sqrt{3})^2\)  13.92  \(\sqrt{3}(2+\sqrt{3})E_1\)  \(6.46E_1\) 
\(\sqrt{3(2+\sqrt{3})^2E_1^21}\)  
MpM−  \(4+\sqrt{11}\)  \(4+\sqrt{11}\)  7.32  \(2\sqrt{3}\)  3.46 
MpM\(+\)  \(7+4\sqrt{2}\)  \(7+4\sqrt{2}\)  12.66  \(\sqrt{3}(2+\sqrt{2})\)  5.91 
In Table 1 we reproduce a summary of the analytic results for peak efficiency from Ref. [37], combining their Tables 1 and 2. We follow their notation describing the parameters of collisions as massive particles ‘M’ and ‘m’, photons ‘P’ and ‘p’, and the direction of particle 1 (positive or negative radial velocity). When particle 2 has mass ‘m’, this means that one should take the limit of \(E_1>> m\). Similarly, when particle 2 is a photon of energy ‘p’, this corresponds to \(M_1>> p\) (these results for “heavy” massive particles were derived independently by Zaslavskii [44]). For massive products, we also list the peak rest mass attainable for particle 3, which does not necessarily correspond to the same trajectories used to achieve peak efficiency [37].
As can be seen from the results in Table 1, the absolute maximum efficiency is still that discovered in [11]. But we also see that generally, highefficiency collisions can be realized for a wide variety of generic reactions. The unifying theme appears to be the critical angular momentum for particle 1, along with nearhorizon collisions around extremal black holes.
4 Numerical calculations
The first such attempt at a numerical calculation was done in the seminal paper by Piran and Shaham [7], where they did a Monte Carlo simulation of the particles produced by elastic scattering of infalling protons with identical particles on stable circular planar orbits. While extremely impressive for the time, computational limitations restricted their calculations to a few thousand protons, enough to get a qualitative feel for the spectral properties and Penrose process rates, but hardly enough to fully sample the phase space. One representative example is shown in Fig. 13. This shows the outgoing energy spectrum for particle 3, also massive in this example. The black hole spin is \(a_*=0.998\), particle 2 is on a bound circular orbit at the ISCO, and particle 1 falls in from infinity with zero angular momentum. Of the 2000 protons participating in the 1000 collisions included in their calculation, only 23 escape with energy greater than \(E_1+E_2=1.674M_pc^2\).
Nearly 20 years later, with significant advances in computing power, a more comprehensive study was carried out in [45], covering a wider range of collisional cases, including pair production, Compton scattering, and gammarayproton pair production (\(\gamma +p \rightarrow p+e^+e^+\)). Again, the focus was on astrophysical applications, so the canonical spin of \(a_*=0.998\) was used. This work was further expanded in [46], exploring the range of angles and energy for outgoing photons.
While these earlier works were able to explore a much wider range of parameter space for the collision products, they were still generally limited to a relatively small number of specific initial conditions for the reactants. In [38] this author attempted to expand on this approach and carry out a numerical calculation of the full 6D distribution of both reactants and products for annihilation events around spinning black holes. Using the radiation transport code Pandurata to integrate geodesic trajectories, we populate the phase space by launching a large number of test particles around the black hole. At each time step along the trajectory, a weighted contribution is added to the 6dimensional phase space (in practice, “only” 5dimensional, because of the azimuthal spatial symmetry of the Kerr metric).
The main results of this calculation are shown in Fig. 14, reproduced from [38]. The contour plots show 2D cuts in the \((r,\theta )\) plane of the density distribution for bound and unbound populations, for spin parameters of \(a_*=0\) and \(a_*=1\). Because of the numerical nature of this approach, any spin can be used, but these obviously span the range of astronomical possibilities. The density distribution of the bound population agrees closely with the analytic results of [48] for nonspinning black holes, and [49] for the Kerr case. It is interesting to note that for the spinning case, the unbound density distribution is almost perfectly uniform in \(\theta \), rising steadily towards the horizon, despite the fact that many of these particles spend a large amount of coordinate time orbiting near the midplane before finally plunging. On the other hand, the bound population shows a clear break in symmetry, due to the increased stability of prograde, planar orbits. These orbits contribute to a density spike in the form of a thick torus, peaking around radius \(r=4M\) [38].
Despite the fact that the spatial density distribution for unbound particles appears quite uniform in \(\theta \) in Fig. 14, we see that the velocity distribution near the black hole is not at all isotropic. There is essentially a bimodal distribution of velocity, with retrograde particles plunging with large negative values of \(v^r\), and prograde particles corating with the black hole spin, peaked around \(v^r=0\) and \(\gamma v^\phi =c\).
As can be seen in Fig. 16, the bound population is much more isotropic. This is hardly surprising, as there are no stable retrograde orbits at \(r=2M\), and even the prograde orbits are almost perfectly planar and circular, spanning a narrow range of velocity as seen by a nearly stationary, LNRF observer.
5 Dark matter applications
Despite the wide variety of fundamental and fascinating results described in the previous sections, by most accounts the collisional Penrose process is unlikely to play a significant role in astrophysical processes. Even the highest efficiency reactions can only provide energy boosts on the order of a factor of ten or so, far below the ultrarelativistic particles seen from gammaray bursts or active galactic nuclei.^{1} And in any case, even those moderately highefficiency events require such fine tuning of initial conditions, they are probably impossible to realize in a natural setting.
One potential (although admittedly speculative) exception is the annihilation of dark matter (DM) particles in the ergosphere around a Kerr black hole. Numerous authors have pointed out the important role that supermassive black holes might play in shaping the DM density profile around galactic nuclei [47, 51, 52, 53, 54]. However, in almost all these cases, the enhanced DM density—and thus annihilation signal—is a purely Newtonian effect, and is therefore not within the scope of this review.
In Fig. 18 we show the spectra corresponding to this annihilation scenario, for a range of black hole spins, for both the unbound and bound populations. In order to highlight the effects of the black hole spin, we focus on reactions coming from close to the black hole. For the unbound population, this means using an energy threshold for the annihilation cross section of \(E_\mathrm{com} > 3m_\chi c^2\). For the bound population, no threshold is needed, as the density peak near the black hole naturally leads to the annihilation signal being dominated by photons coming from small r. Note the qualitatively different spin dependences in the two cases: for unbound DM particles, the low energy part of the spectrum is independent of spin, as all these photons come from plunging particles near the horizon, and experience significant gravitational redshift. At the high energy end, we see the clear importance of spin in generating highefficiency, extreme Penrose process reactions. For the bound population, on the other hand, the stable orbits do not intersect with very large c.o.m. energies, so even for very high spins we do not see much influence from the Penrose process. Yet the spin does play an important role in shaping the lowenergy end of the spectrum, as higher spins allow stable orbits closer to the horizon, and thus more extreme gravitational redshift, just as in the case of the red tail of the iron fluorescent lines seen around black holes of all sizes [56].
The overall vertical axes in Fig. 18 are arbitrary, because we still don’t know very much about the overall density scaling of DM distributions around black holes. Even more uncertain is the amplitude of the annihilation cross section, much less its energy dependence. We hope that in the future, as gammaray telescopes improve in angular and energy resolution, we will be able to use quiescent black holes in galactic nuclei to probe the properties of the DM particle, measure black hole spins, and explore the exotic physics that describe the ergosphere. Perhaps one day we might even discover advanced civilizations that have successfully harnessed the black hole spin as an energy source, as imagined by Penrose in his original paper [3]!
6 Discussion
We have provided a broad overview of some of the recent work on the collisional Penrose process, with particular focus on collisions around extremal Kerr black holes. Despite the numerous astrophysical limitations, since the publication of BSW [13], there has been a great deal of interest in determining the highest attainable collision efficiencies. These highefficiency reactions require both large centerofmass energies and also fine tuning of the reaction product trajectories in order to assure they can escape from the black hole. While nonKerr (or even nonGR) black holes could more generally lead to diverging centerofmass energies, we have restricted this review to classical, if extremal, Kerr black holes.
For more general astrophysical observations, dark matter annihilation appears to be one of the more promising applications of the Penrose process. One reason for this is that DM particles are most likely to travel along perfect geodesics, even in the presence of the diffuse gas and strong magnetic fields typically found around astrophysical black holes. Additionally, the DM density distribution is expected to peak near galactic nuclei, which also contain supermassive black holes. Lastly, the extreme gravitational field of the black hole is a promising mechanism to enhance annihilation, both through increasing the relative collision energy, and also through gravitational focusing that increases the DM density.
As with the question of peak efficiency for Penrose collisions, an important factor in the observability of DM annihilation around black holes is the question of the escape fraction for the resulting reaction products. We showed above in Sect. 3 the planar escape distribution for a selection of specially chosen collisions. More general calculations of the escape probability have been carried out in [57, 58, 59]. In short, the escape fraction decreases as the distance from the black hole decreases (and thus the centerofmass energy increases). This is true for particles plunging in from infinity. But for particles on stable, bound orbits, the escape fraction can actually be quite large, on the order of \(90\%\) or more [36, 38].
In most previous work on the subject, and in our own discussion above, the DM population is generally divided into bound, and unbound. However, when including selfinteractions (e.g., [60]), these two populations can mix, giving rise to new phenomenology and potentially greater enhancements of the annihilation signal [52, 53, 54]. Exotic DM particle models with energydependent annihilation cross sections (e.g., [55, 61, 62]) promise to make this field one of active research in the years to come.
Aside from DM annihilation, astrophysical applications of the classical, collisional Penrose process are limited. In particular, from everything we have seen in the review, extremely fine tuning and multiple collisions would be required to get anywhere close to the highenergy gammarays (or even cosmic rays) seen from many active galactic nuclei. On the other hand, the highenergy emission that is observed is likely indirectly related to the Penrose process, by general coupling matter to the spin of the black hole. This can be done far more efficiently when employing largescale fields, either in the form of superradiance [63, 64], or coupling the particles directly to electromagnetic fields that in turn penetrate the black hole horizon [50, 65]. Unfortunately, the high efficiency of these mechanisms at creating gammarays only serves to confuse and complicate any prospects of direct detection of DM annihilation around otherwise quiescent galactic nuclei. We look forward to the next generation of highenergy observatories that will be able to circumvent these confusion sources with greater sensitivity, and improved spatial and energy resolution.
Footnotes
Notes
Acknowledgements
We thank Alessandra Buonanno, Francesc Ferrer, Ted Jacobson, Henric Krawczynski, Tzvi Piran, Laleh Sadeghian, and Joe Silk for helpful comments and discussion. A special thanks to the editor of this Topical Collection, Emanuele Berti, for his encouragement and patience.
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