Black Hole Engineering

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Ah, black holes. Flaws in the fabric of the universe. Empty voids from which nothing can return. The ultimate unknowable mystery.

But what are they good for?

Basics

Lets start with a brief introduction to black holes.

Things like planets and stars and other massive bodies have gravitational fields around them that tend to draw things toward them and trap stuff on them. In order to get away from such a body, you need to shoot yourself off it with a speed higher than its escape velocity. If you don't have that much speed, you can't get away. When you pack enough mass into a small enough volume, its gravity gets so high that the escape velocity is higher than the speed of light. Because nothing can go faster than light, nothing can escape. This is a black hole.

That's the description motivated by Newtonian gravity, anyway. But when gravity gets really strong Newtonian gravity breaks down and you need to use general relativity instead. Curiously, the size and mass where light (and everything else) is trapped is the same as the Newtonian case. But instead of light and other things flying out, looping around, and coming back space-time gets strange. At the critical distance where light would be trapped you get a surface called an event horizon. Nothing that passes into an event horizon can ever get back out again. The gravity at and inside the event horizon is so strong that it rotates space and time enough that the direction inwards toward the center becomes your inevitable future. You can no more resist going toward the middle of the hole that you can avoid seeing what fate awaits you.

An uncharged and non-rotating black hole at rest is described by the Schwarzschild geometry. The radius of its event horizon is the Schwarzschild radius

r_S = 2 G M / c²

where M is the mass of the black hole, G is the gravitational constant, and c is the speed of light in vacuum. As an example, a black hole with a mass of 100 million metric tons would have a Schwarzschild radius of 1.48 × 10^-16 meters. This is slightly under one-fifth the radius of a proton.

At the center of a black hole lies a point at which our description of physics breaks down, called the singularity. While of immense scientific interest, it is irrelevant for engineering because it is inside the event horizon so it cannot possibly affect us or our environment.

Charged and/or rotating black holes get more complicated. Energy, momentum, angular momentum, and electric charge are all conserved; as you dump stuff into a black hole it acquires these conserved quantities from the things it eats. Energy takes the form of the mass M of the black hole, which we already discussed above. Momentum absorbed by the hole sets it moving, as usual for momentum applied to a massive object. A black hole that acquires angular momentum will begin rotating. And a black hole that eats charged things takes that electric charge on itself, with a radiating electric field corresponding to that charge.

Charged black holes

A charged black hole is described by the Reissner–Nordström geometry. The charge produces an electric field radiating away from the hole, much as the mass of the hole also creates a gravitational field. For the same mass, a net charge will cause the event horizon to shrink. A second horizon will form inside the first horizon that will grow with increasing charge, although for the purpose of black hole engineering this is not particularly relevant because anything going through the outer horizon is lost to our universe one way or the other.

As charge is added, the two horizons approach each other until they meet at a distance of half of the Schwarzschild radius calculated for an uncharged hole of the same mass. This forms one example of an extremal black hole. In this case the mass-energy of the charge, considered as a sphere of charge located in a thin shell at the event horizon, makes up the entirety of the mass of the black hole with no room left over for mass from any matter or other kinds of energy. It is thus easy to see that simply adding more and more charge to a black hole that is not yet extremal cannot actually form an extremal black hole. Likewise, adding charge to an already extremal black hole at most keeps it extremal as you add electrostatic mass-energy that keeps up with the increase in charge (and all physical charged particles also have their own mass, which would take it out of the extremal condition). Some theories suggest that it is impossible for extremal black holes to form by any physical process, although these theories have been disputed.

A charged black hole is not expected to last long in the real world. The charge will draw in particles of the same charge and repel particles of the opposite charge, tending to neutralize it in any environment where any matter exists (even tenuous space plasma). An engineer intending to work with charged black holes will need to ensure it exists in a high vacuum environment and perhaps add additional features to slow the rate of neutralization or methods to top off its charge by adding additional charged particles. As will be seen later, a charged black hole will also spontaneously shed particles to get rid of its charge, making keeping it charged even harder.

Rotating black holes

You get a rotating black hole when the hole devours things which have angular momentum and that angular momentum becomes a property of the hole. Black holes have no surface features so you can't actually see things on the hole going around. But the angular momentum manifests in other physically observable ways.

Most astrophysical processes that lead to the formation of black holes involve the collapse or collisions of rotating bodies with non-zero angular momentum. Hence it is expected that all naturally occurring black holes are born rotating. As we will see later, they may not remain rotating but large rotating holes are likely to remain rotating for long periods of time.

A rotating black hole is described by the Kerr geometry. This has some similar behavior to the Reissner–Nordström geometry of charged black holes. You get the formation of an inner horizon that grows with increased rotation, and the outer horizon shrinks. Also similar to charged black holes, a hole that is spinning fast enough can become extremal such that the spin alone is providing the energy for its mass term. Different from charged holes is that the singularity at the center forms a ring rather than a point. None of this is of any interest to the engineer, as it is all hidden behind an event horizon and cannot affect our world.

Of more interest however, is that you get a region outside of the event horizon where it is impossible to stop moving. This region is called the ergosphere. Similar to how once you go past the event horizon time rotates so that your future is toward the center of the hole, in the ergosphere time rotates so that your future is in the direction of the hole's spin. You can no more come to a stop or go the other direction than you can go back in time.

Massive rotating bodies exhibit a process called frame dragging, and rotating black holes are no exception. A stationary object in a free-fall orbit around the hole will appear to be rotating to a distant observer even though it is in an inertial reference frame. Frame dragging is a gravitational analogue of magnetic induction from moving electric charges. Don't get this confused with what distinguishes the ergosphere – frame dragging will happen well outside of the ergosphere.

Charged and rotating black holes

A black hole with both charge and angular momentum behaves much like you would expect from the solutions for charged black holes and rotating black holes. You get an ergosphere, frame dragging, electric field, and the possibility of extremal black holes. The new feature is the presence of a magnetic field whose magnetic axis is aligned with the spin axis. This black hole is described by the Kerr-Newman geometry. The mathematics of this geometry allow for the event horizon to disappear and the ring singularity to be displayed to the world. However, to obtain this condition you need to go past the extremal case, which is generally thought to be physically impossible.

Energy

Hawking radiation

Famously, nothing that goes into a black hole can ever come back out again. But something comes out. For it turns out that black holes have a temperature and that, like everything with a temperature, they emit radiation. In fact, being perfectly black, they radiate as a perfect black body. This radiation is called Hawking radiation after its discoverer, physicist Stephen Hawking. For normal sized black holes, those the size of stars or galaxies, this temperature is very small and the radiation power is absolutely minuscule. But the smaller the hole, the hotter it gets and the more power it radiates. For a Schwarzschild black hole with mass M, the Hawking temperature T_H is

T_H = ℏ c³ / (8 π G k_b M)

where ℏ is Planck's constant, π is the circle constant, and k_b is Boltzmann's constant. Curiously, this means that the wavelengths around the peak emission of light in its spectrum is near the size of its event horizon. The power radiated by a hole of this temperature in the form of electromagnetic radiation is

P_H = ℏ c⁶ / (15360 π (G M)³).

However, there are additional forms of radiation beyond electromagnetic energy which will add to this radiated power. If the black hole's temperature (in units of energy, so multiply the temperature by the Boltzmann constant to get the units right) is of the same order or higher than the rest mass-energy of a type of particle, that type of particle will also be emitted. The lowest mass particles known that are not electromagnetic radiation are neutrinos. Neutrinos are slippery elusive little fellows and we still don't know their rest masses, but an upper bound on the rest mass of the lightest neutrino species is approximately 0.1 eV. This corresponds to a temperature of 1160 K and a black hole mass of about a hundred thousand trillion (10¹⁷) tons. Temperatures higher than this and masses lower than this will need to take neutrino radiation into account. A black hole with a mass of less than twenty billion (2×10¹⁰) tons at a temperature of 6 billion kelvin will be radiating electrons and positrons. As the mass continues to decrease additional particle types such as muons and pions will start to contribute to the radiation; at even higher temperatures quarks and gluons will be produced that decay into particle jets creating various hadrons. Although gravitions are massless particles predicted by many models of physics, they have never been observed. If they do exist, they would also be radiated away at all temperatures similarly to electromagnetic radiation. At temperatures below the limit for neutrino emission, a black hole can be expected to radiate about 90% electromagnetic radiation and 10% gravitational radiation. For temperatures where electron and positron emission is insignificant but neutrino emission is, about 81% of the energy will be radiated as neutrinos, 17% as electromagnetic radiation, and 2% as gravitational radiation. And if temperatures are hot enough for electron and positron production but not higher mass particles like muons or hadrons, you can expect 35.8% of the energy in the form of electrons and positrons, 55.7% as neutrinos, 7.6% as electromagnetic emissions, and 0.9% as gravitational waves.

The radiated energy comes from the black hole's mass-energy, so a black hole will shrink over time as its mass is radiated away. As the mass decreases, the temperature goes up and so does the power output. So you get a runaway process of the hole getting hotter and hotter and radiating more and more power until POOF! It's gone in a flash of light and radiation. If you only consider the radiated electromagnetic energy the lifetime remaining of any black hole, assuming more mass doesn't fall into it, is

t_H = 5120 π G² M³ / (ℏ c⁴).

As this does not take into account radiation of other particle types, it is an upper bound to the lifetime; the radiation of other kinds of particles will also carry away energy making the black hole lose mass faster. Details for including the emission of other kinds of particles can be found in reference . As an estimate, you can divide the electromagnetic lifetime by the ratio of the total radiated power to the electromagnetic power; although this does not take into account the variation in this ratio as the black hole changes mass you might expect most of its lifetime to be in a range where the types of particles emitted are not changing dramatically and in such a case this approximation applies.

This is a neat result. It allows perfect conversion of mass-energy into radiant energy (although the neutrino and gravitational radiation will be rather inconvenient to capture). However, the actual implementation can get a bit inconvenient.

Let's skip for the moment the details of how you get a black hole. We'll assume that you have a magic black hole making box that can pop out whatever size of hole you need. Now let's say you want a megawatt of usable power (so we ignore the gravitational waves and the neutrinos). What size of hole do you need? It turns out to be a cool 38 billion metric tons. A hole that size is rather hard to carry around with you. And its temperature will be 3.2 billion kelvin. At that temperature its usable radiation is primarily electrons and positrons, with a good dose of hard x-rays and gamma rays for good measure. On the plus side, it's about 2000 times smaller in radius than a typical atom. So you could slip it into your pocket; just don't expect it to stay there.

Here we see one of the issues on trying to utilize Hawking power from black holes. Usable amounts of power generally come with horrendous power to mass ratios with the energy released as highly penetrating ionizing radiation. And if you start getting to masses that are more practical to deal with, you've got more of a bomb than a reactor – a 1000 ton black hole will release all of its 20,000 gigatons TNT equivalent in under a second.

Let's take an example of a black hole with a mass of 50 million metric tons, for reasons that will become clear later. We have already found that this hole is only about a tenth the size of a proton. But that tiny speck of compact mass has a temperature of 2.45 × 10¹² kelvin. It puts out a radiated power of 7.64 × 10¹² watts (of which something like 4 × 10¹² watts is usable), which is a rate of mass loss of 85 micrograms per second. Or in somewhat more descriptive terms, about the energy released by the detonation of 940 tons of TNT every second. Left to its own devices, it will slowly get brighter and brighter, losing mass faster and faster, until it eventually radiates itself away in about 6 million years.

The description of Hawking radiation so far has assumed a black hole without charge or angular momentum. These properties will change the amount of radiation emitted for a given amount of mass. In particular, an extremal black hole of any kind has a temperature of zero and emits no Hawking radiation. A rotating black hole preferentially emits particles with spin and orbital angular momentum aligned with its own; a charged black hole preferentially emits particles with a charge the same as its own. Consequently, Hawking radiation will tend to discharge charged black holes and spin down rotating black holes. As angular momentum is emitted at a higher rate than mass-energy, rotating black holes will spin down to black holes with negligible rotation over timescales where loss of mass is appreciable. Similarly, charged black holes will rapidly discharge from hawking radiation on time scales far faster than their rate of mass loss.

Hawking radiation will have caused naturally occurring primordial black holes with initial masses of less than five hundred million (5×10⁸) tons will have evaporated by now. There are no known methods that nature can make new black holes with this small of mass, although it is possible that some primordial black holes with masses slightly above this limit will survive to the present day with their masses since reduced to below this limit by the intervening evaporation. However, it does mean that black holes with mass smaller than this are going to be quite rare the wild.

Penrose process

Penrose batteries

How much energy can you get out? If you start with a Schwarzschild hole, can you turn it in to a Kerr hole and extract more energy out of it than the original Schwarzschild hole? (Probably not, Hawking's area theorem and all).

Super-radiant scattering

(Black hole bombs)

Feeding a black hole

If you are extracting energy from a black hole, you might want to eventually put that energy back in to avoid using up your black hole too soon. You can do this by letting mass or other forms of energy fall into the hole, passing through its event horizon to get trapped forever.

Tidal disruption

If you have something smaller in size than a black hole's event horizon and you drop it straight in, it should enter the hole without any particular complications. But as the object approaches the hole, the hole's changing gravity will affect different parts of the object differently. Gravity drops off with distance, so the parts of the object nearest the hole will be getting pulled harder than those furthest away. This means that once you account for the average force on the object accelerating it toward the hole, you have an additional force acting on the body to tear it apart along the direction to the hole. Meanwhile the direction of gravity is toward the center of the hole, pointing radially inward. Again, after accounting for the average force on the object this means that the parts furthest to the left are experience a residual force pointing to the right and vice versa. So the net result is that tidal forces stretch an object along the direction towards the center of the hole and squish it together in the directions transverse to that direction. This is called "spaghettification".

Tidal forces fall off faster than the average force of gravity on an object. Whereas gravity falls off with the square of the distance, tides fall off with the cube of the distance. So far out from a black hole, you might be falling comfortably but as you get closer the tides get strong quickly. Very large black holes, like the supermassive black holes at the center of galaxies, might not generate any noticeable tides even as you fall though the event horizon. Smaller holes, on the scale of stellar mass black holes, do generate enough tides to spaghettify any astronaut unlucky enough to fall into them.

Accretion disks and astrophysical jets

If the thing you drop into a black hole isn't dropping straight in – maybe it has a bit of transverse velocity as it gets sucked down – it is likely to miss the event horizon and slingshot around on an orbit. However, even as it misses the all-devouring beast at the center tidal disruption is still pulling the object apart. A close enough approach will have the tides rip apart the object and smear it out into a smudge of debris. The inner parts of the debris cloud will be orbiting faster than the outer parts, leading to shear flow and friction and drag. This leads to heating of the debris, coming from the object's kinetic energy. After enough passes the former object will get spread out into a ring around the hole, called an accretion disk. The closer the debris is to the hole, the faster the difference in speed between adjacent streamlines and the more heating will occur. So you can get the inner parts of the ring glowing brightly with radiated heat.

Most physical process that can feed matter into a black hole start with the infalling matter having some angular momentum. Because the angular momentum is conserved it naturally results in accretion disks forming as the matter falls in.

As the inner part of the disk radiates heat, it loses kinetic energy and gets a little bit closer to the event horizon. As it gets closer it gains heat at a greater rate and its temperature increases. When it gets hot enough, the matter turns into a plasma. To a good approximation, plasmas cannot cross magnetic field lines. A strong field with a diffuse plasma will have the plasma move along the field line direction. A dense, fast plasma, on the other hand, can bully through weak field lines, stretching out the field so that it moves with the plasma. In a turbulent plasma, or, in this case, a circulating plasma, the field gets stretched out enough that it can come back and meet itself, getting stronger and stronger. This dynamo effect will amplify even very weak fields within the accretion disk, forming a strong magnetic field near the black hole.

And this is where things get a bit weird. Something happens – we're still not entirely sure what – and the interaction of the strong field with the energetic plasma right near the event horizon creates jets of fast moving plasma, high energy particles, and electromagnetic radiation shooting out along the axis of the accretion disk, usually in both directions.

The accretion process can extract somewhere between 10% and 50% of the mass energy of infalling matter into radiated energy and energy of the jets. If this energy can be collected, it can provide an additional source of energy beyond what you can get from Hawking radiation and its somewhat inconvenient limits. So now we must see what limits the rate of accretion to see how much energy we can get out of it and also how fast we can recharge our hole for the extraction of Hawking and Penrose energy.

Mass collection rates

Suppose you have a black hole inside of some material. This might be a rock, or a star-hot plasma, or the diffuse gas of interstellar space.

If you are at rest with respect to the surrounding material, you'll get that material falling toward you. It will pile up as it crams together trying to get to the hole, until you reach a point where the flow turns super-sonic and the material free-falls the rest of the way into the hole. Finding the feeding rate is thus a choked flow problem.

If the hole is moving through the material faster than the speed of sound, material passing close to the hole will get deflected by the hole's gravity to converge in a wake behind it. Where it collides with other gas coming in from all directions in the wake, the gas comes to a halt and from there it can freely fall into the hole from behind.

The analysis of these two limits may be combined to give the Bondi-Hoyle accrection rate

ṁ_BH = 4 π ρ G² M²/ (c_s² + v²)^3/2

where ρ is the density of the stuff the hole is in, c_s is the speed of sound in the medium, and v is the speed of the hole through the medium. The distance at which the in-falling material goes from subsonic choked flow to supersonic free-fall is the Bondi radius

r_B = 2 G M / c_s³.

If the Bondi-Hoyle accretion rate is too low, the black hole will be losing matter faster to Hawking radiation than it will be gaining mass to accretion. This depends on the variables described above, but let's look at what happens if we drop it into solid rock. Assuming a typical density of rock of 2.7 grams per square centimeter and a sound speed in rock of about 5 kilometers per second, we find that holes that are larger than 105 million metric tons are able to absorb a net gain in mass while those below this limit lose more mass to Hawking radiation than they gain by eating the rock. If you want to feed your hole with rock, you'll need it to be bigger than 105 million metric tons.

The best material for feeding your black hole, according to the Bondi-Hoyle accretion rate, is the heavy metal thallium. If you drop your hole into a blob of thallium, it can achieve a net mass gain at a mass of only 22 million metric tons. For black hole masses below this, you cannot feed a black hole on normal matter at room temperature and pressure (whether it can feed at the crazy high pressures at the cores of planets or stars is a subject not explored here).

Radiation pressure

Both the Hawking radiation and the radiation from the accretion disk will be shining out of an accreting black hole. This radiation will encounter material from the accretion disk. The radiated light can scatter off electrons in the disk material; on average, this will push them outward. The electrons will then drag any assorted atomic nuclei in the disk material with them. This puts a limit on how much material can flow into the black hole – if it is too bright, it will push everything away. If the hole gets brighter than this limit, it can no longer feed.

This is often referenced in terms of the Eddington luminosity

L_E = 4 π G M (A/Z) m_p c / σ_T

where A is the average atomic weight of the plasma, Z is the average atomic number, m_p = 1.672622 × 10^-27 kg is the mass of a proton, and σ_T = 6.65246 × 10^-29 m² is the Thompson cross section for scattering light off an electron. If something is shining with the Eddington luminosity, it will keep matter from falling in. Strictly speaking, this assumes hydrostatic equilibrium; for problems that are time varying or with steady-state flows the Eddington limit does not necessarily apply. However, it is often a good first guess to figure out when the radiation chokes off the inflow in accretion disks. There are some configurations of accretion disks that can support luminosity higher than the Eddington limit, but most are at or below this limit.

If we assume that our black hole's accretion disk is Eddington limited, we can find out how big it needs to be in order to accrete any matter at all, or to achieve net mass gain after its Hawking radiation losses are accounted for. In hydrogen gas, with A/Z = 1, we find that a hole must have a mass of at least about 104 million metric tons for any matter to fall in past the Hawking radiation pressure. The hole's mass has to be in the 109 to 125 million metric ton range to gain mass via accretion faster than it is lost to Hawking radiation, depending on the efficiency at which matter in the accretion disk is converted into radiation. If you drop the hole into rock or other light elements you'll have an A/Z ratio of 2 or very slightly higher. Setting A/Z = 2, we find that you can't get any accretion for masses under 85 million metric tons and, again depending on the radiative efficiency of the accretion disk, you need somewhere in the range of 90 to 103 million metric tons to reach breakeven in terms of mass loss versus mass gain. Even for very heavy elements like lead or uranium, with an A/Z ratio of approximately 2.5, you need at least 80 million metric tons to accrete matter at all and somewhere between 84 and 97 million metric tons to break even.

In other words, if you want to be able to add mass to your black hole by having it gobble up surrounding matter, you'll want it bigger than many tens of millions of metric tons.

Interestingly, the limit for net mass gain for the Eddington limit is very similar to that of the Bondi_Hoyle limit. In order to get a black hole that gains mass, you're pretty much going to need at least a mass somewhere near the 100 million metric ton range.

Reaction rates at sub-atomic sizes

We now know the rate at which matter can fall on to a black hole, getting past both the radiation coming from the hole and its inner accretion disk and for getting past the choked flow of the material getting in its own way. But what about when it reaches the hole? Obviously, if the hole is bigger than the size of an atom any atoms it touches will immediately get sucked in. But a lot of holes of engineering interest are much smaller than this. A black hole with a mass of 100 million tons would have a Schwarzschild radius of about 5.7 times smaller than that of a proton. If a hydrogen atom fell into the hole, it would end up sitting there with the black hole inside of the proton. How quickly could the hole slurp up that proton and its companion electron?

It is easy enough to get an estimate of how fast a proton or neutron will get eaten once a black hole is inside of it. Both protons and neutrons have a radius of about 8.4 × 10^-16 meters. Both are made up of three quarks. This gives a quark density of about 1.21 × 10⁴⁵ / m³ inside of the proton or neutron. Because the binding energy of the quarks is much larger than the mass-energies of the quarks, we can assume that they are highly relativistic and are moving at about light speed. Multiply the density by the speed to get the flux (particles passing through per area per time) of about 3.62 × 10⁵³ quarks / m² / s. Then multiply by the surface area of the hole to get the absorption rate of the quarks. Once one quark is eaten, color confinement ensures that the rest of the quarks cannot leave and the particle is stuck to the black hole until the rest of it is eaten, which time we can guestimate by the time needed to eat three quarks. For our 100 million ton black hole, this shakes out to about 3 × 10^-23 seconds to eat a proton or neutron, or 3.3 × 10²² protons and neutrons eaten per second. If we multiply by the mass of a proton or neutron, we find that the 100 megaton black hole can eat protons and neutrons at a rate of about 5.6 × 10^-5 kg/s if it has a constant supply of protons and neutrons ready to immediately fall into the hole once the previous one was eaten. Which is comfortably higher than the loss to Hawking radiation of 1.56 × 10^-5 kg/s.

This is okay for neutrons (if you can somehow find a supply of free neutrons), but for protons there is a problem. For every proton the hole eats, it gains one unit of elementary charge (that is, the charge that the proton had gets added to the charge of the hole). If it eats enough protons, it will gain enough charge to repel away any other proton (or atomic nucleus) that comes near enough to it that the electrons around the atom can no longer screen the electric charge of the proton or nucleus. The potential energy of a proton or nucleus bound to the black hole by their mutual gravitational attraction is

U_G = -m_p A M G / r

and the potential energy of the repulsion between the proton or nucleus and a charged hole that has absorbed Y other protons is

U_E = [Y Z q² / (4 π ε₀)] / r.

Here, Z is the number of protons in the nucleus under consideration (Z = 1 for a single proton), A is the number of protons + neutrons in the nucleus (A = 1 for a single proton), q = 1.602176487 × 10^-19 C is one unit of elementary charge, ε₀ = 8.854187817620 × 10^-12 C² / J / m is the permittivity of free space, m_p = 1.67262192369 × 10^-27 kg is the mass of a proton, and r is the distance between the black hole and the proton or nucleus. If the sum U_G + U_E is negative, the hole still attracts the proton or nucleus and matter free-falling into the hole can collide with the hole without issue. If the sum is positive the force is repulsive and the proton or nucleus cannot approach the hole. We see that this happens when

Y = 4 π ε₀ m_p (A/Z) M G / q².

For our 100 megaton black hole eating hydrogen (which has only protons as a nucleus), the hole can charge up to a maximum of Y = 49. For heavier nuclei with a mass to charge (A/Z) ratio of 2, the hole can charge up to Y = 97. Whatever the case, if the hole cannot get rid of this charge fast enough, the hole will get too much charge to freely eat everything falling into it.

There are many ways that the hole can shed its charge. It's gravitational field and positive electric charge pulls negatively charged electrons in to a high density, it can simply eat these electrons to reduce its charge. Alternately, the electrons densely packed around the protons might get captured by the protons to form neutrons that can fall into the hole and keep feeding it. For this case, however, the most efficient means of reducing the hole's charge is from its Hawking radiation.

The hole will have a chemical potential for electrons of

μ = q² Y / (4 π ε₀ r_s),

which is the potential energy to bring an electron from far away to the event horizon. If the chemical potential is significantly larger than the Hawking temperature (in energy units) and if the Hawking temperature (in energy units) is significantly larger than the mass energy of an electron

μ > k_B T_H > m_e c²

then the rate of positron emission from the hole is approximately μ/ℏ. Our 100 million ton hole with Y > 10 meets both these criteria. For Y = 11 the rate of positron emission is 1.6 × 10²³, a full order of magnitude larger than the rate at which protons can be absorbed, and only increases as the charge goes up. This discharges the hole faster than it is charged by gobbling up protons. We thus see that nothing prevents matter from falling into the hole at the macroscopic accretion rates.