Black Hole Engineering

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.

A diagram of the features of the Schwarzschild geometry, showing the event horizon (white circle) and central singularity.

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.

Energy is conserved, and mass is a manifestation of energy that is not moving. So when matter or radiation is swallowed by the hole, its energy is added to that of the hole and the mass of the hole increases by E = m c² to reflect this.

Charged and/or rotating black holes get more complicated:

A diagram of the features of the Reissner–Nordström geometry, showing the inner and outer event horizons (white solid circle), the location of the Schwarzschild event horizon for a black hole of equal mass but no charge (outer dashed circle), the location of the extremal horizon at half the Schwarzschild radius (inner dashed circle), and the central singularity.

Charged black holes

Charge is conserved. If electrically charged matter falls into a black hole, the hole itself will acquire the charge. The charge produces an electric field radiating away from the hole, much as the mass of the hole also creates a gravitational field.

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.

A charged black hole is described by the Reissner–Nordström geometry. 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, with a charge of

Q = M √[4 π ε₀ G] = M 8.61722×10^-11 C/kg.

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 diagram of the features of the Kerr geometry, showing the inner and outer event horizons (white ovals), outer boundary of the ergosphere (red oval), and ring singularity(dotted oval).

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.

Massive rotating bodies exhibit a process called frame dragging, and rotating black holes are no exception. Frame dragging is a gravitational analogue of magnetic induction from moving electric charges. It induces motion in space-time near the body co-rotating with the body and objects therein will be moved along with the space-time. Because space-time is dragged faster near the body than far from it, a stationary object in a free-fall orbit around the hole will appear to be rotating in the opposite direction to the hole to a distant observer even though it is in an inertial reference frame.

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 when the angular momentum J is

J = M² G / c = M² 2.22615×10^-19 m²/kg/s.

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. Here, frame dragging is so extreme that space-time is moving around the black hole faster than the speed of light. 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.

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. Extremal holes occur when

M² - (J c / (G M))² - (Q / √ [4 π ε₀ G])² = 0.

The new feature is the presence of a magnetic field whose magnetic axis is aligned with the spin axis. For a black hole with charge Q, angular momentum J, and mass M, the magnetic moment m (as measured in the far-field) is

m = Q J / M

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.

Caveats

All the above descriptions of black holes assumes a distribution of mass and charge that does not change with time. That is, it is static. It may be moving, as with the case of a rotating black hole, but the distribution of rotating stuff doesn't change. It may also be moving if you shift to a frame of reference where the hole is not at rest, but you can always find a frame of reference where the hole is at rest in the sense that it has no net linear momentum (and, in a more practical sense, isn't going anywhere. This also means that the occasionally encountered idea of "accelerate an object to such a high speed that it turns into a black hole" simply doesn't work and is not consistent with physics). If you have a static hole, it's properties are entirely defined by just the three quantities of its mass, charge, and angular momentum. Any two static black holes with these three quantities the same will be identical in every respect. To describe this, physicists use the somewhat odd terminology that "the black hole has no hair"; hair being things that do not directly derive from mass, spin, or charge.

Not all black holes need be static. At the moment of creation by the collision of two supermassive objects, for example, a black hole will momentarily have an event horizon that is elongated and wobbly. That is, it has "hair." However, it rapidly radiates gravitational waves until all its hair is shed and it settles down to a static state.

All of the above descriptions of different kinds of black holes assume that if you go far enough away from the black hole, space-time settles down into the ordinary mostly flat space-time where Newtonian gravity works and planets and satellites have regular orbits and geometry works like you would expect and things behave like we would otherwise naively expect them to. This is called asymptotic flatness, defined by the idea that if you go far enough away from the hole in any direction space-time will get as arbitrarily close to flat with increasing distance. Asymptotic flatness is a good approximation of our universe on scales up to and beyond galactic clusters. If you are only dealing with engineering projects within a single galactic cluster, you can generally assume that asymptotic flatness holds. There has been some work on black holes in universes that are not asymptotically flat, but we will not concern ourselves with that here as it is unlikely to be of relevance to engineering tasks.

The initial justification for nothing getting past the event horizon was that it would have to move faster than the speed of light, and nothing can move faster than light. But many science fiction works feature methods whereby information or objects (usually spacecraft) can go faster than light (FTL). Could a faster than light starship escape from inside the event horizon of a black hole? Possibly. It depends in the implementation, but under relativity FTL motion automatically implies time travel. And all of the results of relativity that inside a black hole the future is towards the center of the hole rather than forward in time would similarly be un-done by time traveling FTL. Likewise, your FTL spacecraft could likely go backwards around the ergosphere, if that's your thing. The article on wormholes covers some of the details for wormholes interacting with black holes, illustrating one way to get information out of a black hole's event horizon and the difficulty of implementing it. This could, in principle, allow access to the interior of black holes that we formerly ignored. Such as using rotating black holes as a time machine (but we can already do that if we can get there and out in the first place) or as wormholes to other universes.

Acquiring a black hole

If you want to do things with a black hole, first you need to get one. Here, we discuss various ways you might get your grubby little mitts on one of these monstrosities of physics.

Supermassive black holes

At the center of each galaxy resides a gigantic black hole with a mass ranging from tens of thousands to billions of times more massive than our sun. To acquire a supermassive black hole, you'll need to travel to the center of a galaxy. The mass of these black holes means that they can be difficult to take with you and you might need to do your work where you originally found the hole.

Stellar mass black holes

Stars do not readily form black holes, despite their immense gravity trying to pull them together. When you try to squish a star down to make a black hole, that squishing makes its temperature rise. A rising temperature makes the star hot, which increases its pressure, which pushes back against your squishing. This can be very annoying when trying to make a black hole. You need to wait for that thermal energy to radiate away. But even worse the hot, dense interior of the stuff you are squishing makes a great environment for thermonuclear fusion to occur. This fusion creates heat and you have to wait for that heat to radiate away, too, before you can get the stuff to contract down further.

But even after everything has fused, there can be limits to your squishing. As the stuff in the stars gets denser and denser, you get to a point where all the low energy places to park the electrons are all taken up. To make the star denser, you need to put the electrons in higher energy states. This takes energy to get the electrons there, which means even more pressure pushing back. This is a state of matter called electron degenerate matter, and the resulting object is called a white dwarf star. For stars with a mass of about 1.44 times the mass of our sun or less, the electron degeneracy pressure keeps the star from getting small enough to form a black hole. This threshold mass is called the Chandrasekhar limit.

Okay, so you get together a star with more mass than the Chandrasekhar limit. Now you're good to go, right? You have enough mass to just push past that annoying electron degeneracy pressure. Not so fast, buckaroo! Once the energy of the electrons gets high enough it becomes energetically favorable for them to combine with protons to form neutrons (this happens for energies of about 0.78 MeV for free protons). Now you get a dense ball of neutrons and have the same issue that you previously had with electrons, but worse. This mass of degenerate neutrons is called a neutron star. It takes a mass of a bit more than twice the mass of the sun to overcome the pressure of degenerate neutron matter (the Tolman–Oppenheimer–Volkoff limit). But once you do that, there is nothing preventing the remains of the star from squishing down into a black hole under its gravity.

All of this is to show that it can be hard to make a black hole from stars. And that's not even considering other complications, like how stars tend to shed a lot of their mass as they collapse so you need considerably more mass than the Tolman–Oppenheimer–Volkoff limit to make your black hole.

But do not fret! The universe has been kind enough to make black holes out of stars for you. There has been enough time for many of the more massive stars to burn through their fusion fuel and collapse to make black holes. Even those that remain as neutron stars sometimes run in to other neutron stars and form black holes.

Needless to say, a stellar mass black hole is going to be very heavy. If your civilization cannot move stars around, this will be a location you go to rather than a piece of equipment you carry around with you.

Black holes may not be uncommon in the universe, but they can be dark (it's in their name, after all). So stellar mass black holes can be hard to find. But there are ways. If the black hole has a stellar companion, it can siphon gas from the companion to produce a bright x-ray source. If a dark black hole passes in front of another star, it can make that star temporarily brighter through gravitational lensing. So you may be able to locate a stellar mass black hole – we have already located a great many of them. The problem of getting to said stellar mass black hole is still an unsolved problem, however.

Primordial black holes

There are no known natural processes to make black holes in our universe with a mass less than the Tolman–Oppenheimer–Volkoff limit. However, it is possible that our universe might have been born with small black holes already in place. These primordial black holes could potentially be significantly smaller than stellar mass black holes. Primordial black holes with initial masses of less than five hundred million (5×10⁸) tons will have evaporated by now (see below for why black holes evaporate). 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.

It is not necessary for primordial black holes to be small. They could have initially formed at any size. Indeed, there has been discussion among the scientific community that the seeds of supermassive black holes were primordial black holes which would necessarily have been of large size.

Surviving primordial black holes that are not supermassive black holes would contribute to the dark matter of the universe. Indeed, it is possible that most of the universe's dark matter consists of these primordial black holes. Ocasionally, a small primordial black hole might pass through a solar system and be detected by its minute gravitational effects on planetary orbits.

Artificial black holes

If you can't find a hole, maybe you can make one. If your culture is capable of assembling massive stars and you're willing to wait a few tens or hundreds of millions of years, this is something that can be done. However, if you're looking to make holes of sub-stellar size, no one today has even the faintest idea of how it could be done.

For quite a while, one of the favorite ideas was a method called a kugelblitz. Technically, this can be any arrangement of radiant energy or energy made of fields that surpasses the Schwarzschild critereon and forms a horizon, but since the development of the laser one of the favorite kugelblitzes has been to shine many enormously powerful laser pulses into a tiny spot. When the laser pulses simultaneously reach the focal spot, their combined energy is sufficient to form a black hole.

Unfortunately, it doesn't work. Before the light can get concentrated enough to self-gravitate into a black hole, it gets intense enough for light to start interacting with light. This scatters the light out of the beam, preventing the light from focusing tightly enough to form a black hole.

So that's the current state of the art. If there are ways to make small black holes, we haven't thought of them yet.

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. Gravitational waves will also be radiated away at all temperatures similarly to electromagnetic radiation. The fraction of radiation coming off as various particle types is shown in the table below for black holes large enough to have insignificant muon, pion, and heavier particle radiation.

Mass (tons) >> 1 × 1017 << 1 × 1017 & >> 2 × 1010 << 2 × 1010 & >> 1 × 108 Temperature (K) << 1200 >> 1200 & << 6 × 109 >> 6 × 109 & << 1.2 × 1012 Temperature (eV) << 0.1 >> 0.1 & << 500,000 >> 500,000 & << 100,000,000 Electromagnetic fraction 90% 17% 7.6% Gravitational fraction 10% 2% 0.9% Neutrino fraction 0 81% 55.7% Electron & Positron fraction 0 0 35.8%

Fraction of power emitted as different kinds of radiation as a function of mass for larger mass black holes. For black holes smaller than 1 × 108 tons, the radiation doesn't so neatly separate with many new kinds of radiation coming on-line without as obvious separations between them. Near the threshold masses, there is a gradual transition from one radiation scheme to another as the temperature gets high enough to occasionally excite the new particle type over the existence threshold.

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 100 million metric tons, for reasons that will become clear later. We have already found that this hole is only about a fifth the size of a proton. But that tiny speck of compact mass has a temperature of 1.23 × 10¹² kelvin. It puts out a radiated power of 1.4 × 10¹² watts (of which something like 7 × 10¹¹ watts is usable), which is a rate of mass loss of 15.6 micrograms per second. Or in somewhat more descriptive terms, the interacting radiation has about the energy released by the detonation of 170 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 67 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.

Penrose process

In a rotating black hole, anything entering the ergosphere gets pulled around the black hole by the spinning space-time. If you dive into the ergosphere and then shoot something backward against the direction you're being swirled in, this is a rocket and you get pushed forward just like any other rocket. But if you do the math, if you dive in deep enough (but still outside the event horizon!) when you come out of the ergosphere you can be going much faster than if you fired your rocket outside the black hole. What gives? How can you get more energy than you started with? Well, it turns out that the energy came from the black hole itself. You decreased both the black hole's mass-energy and its angular momentum when you did that, and got shot out with that extra energy and angular momentum.

This has obvious uses for getting energy. If you drop things into the black hole, and have them push stuff out backward to fall into the black hole, you can harvest the black hole's rotational energy by using the dropped things to do work when they come zipping back out.

For an uncharged extremal rotating black hole and a trajectory grazing the event horizon, up to 20.7% of the mass-energy of the ejected particle can be returned as kinetic energy by this process. However, for a charged rotating black hole there is no upper limit to the efficiency of the process. In fact, you can gain more energy from the Penrose process with a charged black hole than was in the mass-energy of the particle you ejected!

Penrose batteries

For an uncharged extremal rotating black hole, nearly 30% of the mass-energy of the black hole can be extracted via the Penrose process. This percentage can get even larger for a charged rotating black hole.

Of course, once you extract that energy, you can't use the black hole for the Penrose process any more. However, you could charge it up again by throwing matter into the hole with high angular momentum with respect to the hole. It is even better if the matter is highly charged. Assuming that the black hole is large enough that it can be fed efficiently (see below), you can re-use your black hole battery over and over again.

Superradiant scattering

An effect similar to the Penrose process with matter can be accomplished with radiation. Light is shone into the rotating black hole. A portion is absorbed by the black hole, but more energy than was lost is given to the light by the ergosphere, a process known as superradiant scattering. If this light is then reflected back into the black hole again and again, it can get amplified indefinitely – at least until the intensity of the light gets so high that it breaks your mirror. The idea of enclosing a rotating black hole with a mirrored shell is called a black hole bomb. All of this allows you to extract the energy of a rotating black hole using light and receiving energetic light in return. You no longer need worry about the energy coming out as extremely penetrating radiation of high energy particles.

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. If the infalling matter is charged, the black hole will aquire that charge. If the infalling matter is off-center or spinning, the black hole will acquire the angular momentum of the system once the matter is absorbed.

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.

In some cases, the circling debris may puff up into a shape more like a doughnut than a flat disk. These toruses are generally expected to be less efficient at radiating energy out of the infalling matter, with the radiation getting trapped in the torus and serving to puff it out rather than escaping.

The accretion disk process around a non-rotating, uncharged black hole can extract up to 5.7% of the mass energy of infalling matter into radiated energy and energy of the jets. The efficiency at radiation can increase to up to 42% for an extremal rotating black hole. If this radiated energy from the accretion disk 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.

The speed of sound in a solid makes a useful approximation for where inertial effects overcome material strength effects. Thus, the Bondi radius can serve as a useful approximation of how big of a channel will be ripped out of something that has a black hole pass through it.

If the Bondi-Hoyle accretion rate is too low, the black hole will be losing mass 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 Bondi radius for such a black hole will be about half a micrometer, or about 5000 atoms in radius, so the tunnel it will make falling through rock will be fairly small.

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?

Consuming protons and neutrons

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.

Discharging via Hawking radiation

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.

Propulsion

People often like to get from one place to another. A black hole gives you various options for moving things around.

Penrose launcher

If you have a large enough rapidly rotating black hole, you can drop an entire spacecraft in it. If you get deep enough into the ergosphere, you can use the Penrose process by firing your rockets at the point of closest approach. Now you can get yeeted out at ridiculous speeds. If you can survive the tidal forces that close to the event horizon, you can potentially get a machine for flinging you around the galaxy at relativistic speeds.

Black hole rockets

Taking a black hole with you has the advantage that you don't need to rely on any black hole based infrastructure at your destination. An obvious method of propelling yourself with a black hole is to use the energy emitted by a hole to energize your propellant, rather than using a chemical or nuclear reaction for your rocket thrust. Perhaps you can directly use the astrophysical jet as your rocket propellant. Or the radiant light or energy from Hawking radiation or a black hole bomb as a photon drive. All of these methods will require careful engineering to avoid very low accelerations from the high mass of the black hole while avoiding getting a black hole so small that it immediately evaporates in an explosion far larger than your spacecraft can survive.

Making Holes in Things

Sometimes, you need to put a hole in something. Not in the sense of putting a black hole inside of something, but drilling a cylindrical hole through something. Perhaps you are interested in machining part out of difficult to work materials. Perhaps you want to build a weapon that perforates your enemies. In either case, if you have a black hole available you could imagine sending the black hole through the target object and leaving a hole ... or at least a region of gravitationally disrupted material ... behind.

For its frontal surface area, a black hole has an enormous mass. It's sectional density and the pressures it exerts on the material it passes through will be so high that it will essentially ignore the material in its way. After passing through enough material, it will eventually be slowed down both by accumulating mass and through drag forces, but that will occur over distances well beyond what we are concerned with here. For practical purposes, the black hole will just punch through without being impeded in any way by the object in its path. Our goal is to figure out what happens to that object.

Direct absorption

Obviously, anything which directly encounters the event horizon will be lost forever. This gives us a lower bound on the size of the hole left as the black hole diameter of twice the Schwarzschild radius 2 r_S.

Gravitational disruption

A more significant effect is how the black hole will gravitationally accrete the material it passes through and eventually consume it. We have already looked at Bondi-Hoyle accretion. The choked flow treatment takes as a cutoff where the infalling fluid transitions from subsonic to supersonic speeds at the speed of sound. But the speed of sound is also a reasonable estimate of where inertial effects overcome material strength effects. Motion due to gravity is fundamentally inertial, so we can take the Bondi radius r_B as a rough estimate of the distance where the black hole's gravity is able to rip material apart. If the black hole is moving slowly compared to the speed of sound, this material will be consumed; if it is moving much faster than the speed of sound it merely leaves a gravitationally disrupted trail behind it. In either case we are left with a region of diameter 2 r_B where the target object is torn apart.

Vapor explosions

The black hole will emit radiation into the target object as it passes, either from Hawking radiation or from the radiation coming from its accretion disk. In practice, much of the Hawking radiation from small black holes will be in the form of highly penetrating radiation. But if we make the assumption that the radiation is absorbed locally (a reasonable assumption for larger black holes where the temperature is on the order of 10 keV or less) we can find the energy deposited per distance traveled by a black hole moving with speed v as dE/dx = P_H/v. Any neutrinos or gravitational waves emitted will be far too penetrating to affect this calculation; consider only the Hawking power from interacting particles (and even then, the muons, pions, hadronic showers, and weak vector bosons that you get from the smaller black holes all put a significant fraction of their decay energy into neutrinos, so only part of their energy can be used).

The radiation from the accretion disk is likely to be more amenable to local absorption. Find the rate of accretion, multiply by the square of the speed of light to find the mass-energy accretion rate, and then by the efficiency ε of turning accretion disk mass energy into radiation that was discussed earlier. Then divide by the speed to find the energy deposited per distance traveled to get dE/dx = ṁ_BH c² ε / v. Add this to the Hawking energy deposition to get the total dE/dx. If the accretion is Eddington limited, the accretion rate cannot bring the energy deposition above L_E/v.

Under the assumption that this energy is absorbed locally, it will heat a cylinder of material to a high pressure vapor. This vapor will then expand, pushing surrounding material violently away. The radius of the resulting cavity can be found if you know the cavity strength of the material K_c. This can be found from the compressive strength K and the shear modulus G, both of which can usually be looked up for many common materials:

K_c = (2/3) K + (1 + ln(2 G/K))

The volume of a cavity blown out by an energetic event will be K_c times the energy release. This gives a radius of the cylinder exploded out of the target object of

r_v = √[(dE/dx) / (π K_c )]

The diameter of the exploded hole will be twice the radius.

Reference gives one attempt to estimate the effects of a micro black hole passing through the human body. Here, they assume that the black hole has a speed on the order of the dark matter velocity dispersion of around 200 km/s, and find a minimum mass for serious injury or death to a human victim of 1.4×10¹⁴ kg. That work used different assumptions than are used here. If we take a black hole of that mass and speed passing through the human body (taking water as the primary constituent such that density 1 gram/cubic centimeter, A = 18, Z = 10, and a speed of sound of 1500 m/s) the Bondi accretion limit is 0.14 g/s (far less than the Eddington limit, so we are Bondi limited rather than Eddington limited). The Bondi radius is 8.3 mm, so we can assume that the gravitationally disrupted tissue alone is equivalent to the effect of a 16.6 mm bullet. If we assume a 5% efficiency at turning the mass-energy of the accretion disk into radiation, we get an accretion power of 616 GW, leading to a linear energy deposition of 3.08 MJ/m. The Hawking radiation is negligible compared to this, so we ignore it. The cavity strength can be crudely approximated as 1.2 MPa, which gives results roughly consistent with ballistics gelatin results. Crunching through the calculations, we find that the vapor explosion blows out a hole 90 cm in radius (180 cm in diameter), which is enough to explosively disassemble the entire person into splattered gibbets. We therefore see that the vapor explosion is the most significant factor and that the given 1.4×10¹⁴ kg is a significant overestimate of the minimum dangerous mass of a black hole.

Gravity Generation

People are healthiest when living in gravity. If you want to go out in space, there is no gravity. Even on worlds, if the world is small enough there might not be enough gravity for good health.

There are many proposals to address this, and they mostly involve spinning things around in centrifuges. Which, to be perfectly honest, is probably always going to be a better approach to making gravity than black holes. But we're not here for practicality, so lets look at using black holes as a gravity source.

The source of gravity we are most familiar with here on Earth is gravity from mass. You need a lot of mass to generate just a little bit of gravity, so it seems rather inefficient. However, the closer you can get to your mass the more gravity you get, following Newton's law of universal gravitation

g = G M / r²

where lower case g is the acceleration due to gravity, upper case G = 6.67430×10⁻¹¹ m³/kg/s² is the gravitational constant, M is the mass making the gravity, and r is the distance between the center of the mass and the place where you are measuring the gravitational acceleration. Technically, this is only for point masses or spherically symmetric masses, but we will be dealing with planets and black holes which are generally pretty close to spherical in most cases so we're okay. Given this, we can get the same gravity the closer we can get to the source of our mass without going inside of it which in turn argues for using the densest source of mass we can find. Which is black holes.

Gravity on Earth has a value of g_⊕ = 9.8 m/s². If we know the mass of our black hole, we can plug this in to the law of universal gravitation to find how far away we need to be to get a comfortable gravity. However, there is another consideration. Your head and your feet will be at different distances from the center of the hole, so if you are standing up your feet will experience more gravity than your head. The average person is somewhere around 1.5 to 2 meters tall, so if you need to be 10 cm from the black hole for 1 g_⊕ at your feet your head will nearly be in freefall. So we also want the distance for 1 g_⊕ to be significantly larger than a human height.

Let's take, for example, a case where we have 1 g_⊕ at a distance of 10 meters. Plugging this in to the law of universal gravitation, we find that we need a mass of 14.7 billion tons. Given that we need to pack all of this into a sphere with a radius of 10 meters or less, we require a density of more than 3.5 million grams per cubic centimeter. The densest material known is osmium, which is 22.6 grams per cubic centimeter. As we need a density five orders of magnitude more than this, normal materials will not cut it. Electron degenerate matter can approach these densities, but electron degenerate matter cannot hold itself together and will spontaneously explode under environmental conditions suitable for human life (specifically, if the gravity is only 1 g_⊕) so we can rule that out. Neutron degenerate matter has the same issue. Which leaves black holes as our only option.

Such a hole would be smaller than an atom, although substantially larger than an atomic nucleus. It will produce about 20 MW of hard radiation but most of that is neutrinos; only a bit over 8 MW is going to interact with normal matter – mainly several hundred keV gamma rays, positrons, and electrons which are all easy enough to shield against. The black hole will last much longer than the current age of the universe and if you need to feed it the Eddington limited rate is a few grams per second while the Bondi limit is about a quarter kg/s for rock, a few kg/s for water, or a couple hundred kg/s for thallium. As far as the gravity, if your feet are at 1 g_⊕, then (assuming you are 1.7 m tall) your head will experience about 3/4 g_⊕. This is probably both healthy and comfortable, the black hole is relatively benign, and so this presents one option for artificial gravity.

Computation

A black hole's event horizon has a temperature. This implies, via thermodynamics, that it has an entropy. In information theory, the entropy of a system is a measure of its information content, and thus the Hawking radiation coming out of the black hole is the rate at which information is returned to the outside world. This brings up the idea of, what if you could input information via coded messages into the black hole, have the black hole process that information, and then return that information as patterns and correlations in its Hawking radiation?

If this all sounds very hand-wavy, that's because it is. You could apply the same argument to the glow coming off of a bar of hot iron. But one work has looked into this concept and found ways, at least in principle, to make black holes Turing complete so that they can be used, again in principle, as a computer. This raises the possibility of arbitrarily advanced civilizations with near omniscient abilities to measure radiation using black holes as the ultimate computation device.

Containment

There were a dozen other questions that Duncan was longing to ask. How were these tiny yet immensely massive objects handled? Now that Sirius was in free fall, the node would remain floating where it was--but what kept it from shooting out of the drive tube as soon as acceleration started? He assumed that some combination of powerful electric and magnetic fields held it in place, and transmitted its thrust to the ship.

Arthur C. Clarke, Imperial Earth

So, you have a black hole. And let's say you want to use it for a mobile application. This means you need to move it around. As you are likely dealing with something that has a mass of millions of tons or more, it will take a lot of force to accelerate it just a little bit. If you are going to use it for thrust for your spacecraft, or even if you need to move it around somewhere using a spacecraft, you're going to want to make sure it doesn't get left behind when your spacecraft moves. As you can see from the quote above, even some of the foremost minds in science fiction simply hand-waved this detail away.

This can get particularly bothersome if you are on a planet. A basic 100 million ton black hole weighs, well, 100 million tons. Or about a trillion newtons of force. It's smaller than the nucleus of an atom. Any chemical bond will fail with a force of about 0.010 μN; the black hole will exert something like fourteen orders of magnitude more force than is needed to break any known force holding it to other atoms in matter. The pressure of all the force concentrated into such a tiny area means that nothing material could keep it from simply falling down. After which it will end up orbiting through the planet, mostly ignoring the matter in the way but gradually slowing down over geological time spans. If this happens and you wanted to do something other than geoengineering with your black hole, you're probably out of luck.

So how can you exert a force on a black hole?

By Newton's third law of motion, anything that gets gravitationally attracted to a black hole also exerts the same force back on a black hole. A black hole near something else massive will be tugged toward the massive thing as the massive thing pulls the black hole. So if that massive thing is made out of matter, you can pull the thing which can pull the black hole. Unfortunately, the resulting force is probably going to be really weak. If you had a 200 meter diameter ball of osmium (the densest material known) it would have a mass of 95 million tons. At the surface of the ball, it would attract a black hole with a gravitational acceleration of 0.63 mm/s²; about 1/15,500 that of Earth's gravity. The acceleration is pitiful, and you're going to have to be carrying around a lot of extra mass (whether it is a significant amount of extra mass compared to your black hole is another matter). But you can apply the acceleration continuously over long periods of time. If you use this to couple your black hole rocket to your spacecraft you can accelerate at 54 m/s per day; or a km/s every 20 days. Perhaps surprisingly, this is not entirely unworkable.

Note that this method does not provide overall propulsion. Conservation of momentum dictates that you still must use some kind of thruster than expels or exchanges momentum with the outside environment. Rather, this gives you the limits at which your black hole can be accelerated by whatever method you are using to move your spacecraft and the hole without the hole falling away.

You can also electrically charge the black hole. This will give it an electric field. If the black hole is also spinning, the combination of spin and charge will give it a magnetic field. You can then push or pull on the black hole with beefy capacitor plates or electromagnets. However, it can be challenging to give a black hole a large charge, or to have it keep its charge for long.

One problem is the electrical potential of the hole. The potential 𝒱, in volts, for a black hole with a charge Q in coulombs, is

𝒱 = (1/(4 π ε₀)) Q / r_S

where ε₀ = 8.8541878188×10⁻¹² F/m is the vacuum permittivity. A black hole will have a capacitance of

C = 4 π ε₀ r_S

and the energy to charge the black hole up is

W = (1/2) C 𝒱².

Generally, the charge you can achieve is limited by the voltage (or energy per particle, expressed in eV) you can get with your particle accelerator. For a given 𝒱, this means the most charge you can put on your hole is

Q = C 𝒱.

With modern accelerators, we might get electrons up to an energy of 1 TeV (1×10¹² eV), for a potential of 𝒱 = 1×10¹² V. For our example 100 million ton black hole, this gives a charge of Q = 1.65×10^-14 C with a negligible charging energy. We can put this next to a highly charged capacitor plate to accelerate it. You can generate fields as high as the vacuum breakdown limit for the materials used to make your plate, which is typically about 𝓔 ~= 10⁸ V/m. The force is F = Q 𝓔, or about (very roughly) 1 μN. Using F = M a, the acceleration a produced is a rather pathetic a ~= 10^-17 m/s², or about 10^-18 g_⊕. This is not going to get anyone anywhere in a reasonable time! But you can at least see the math needed to figure out how to move the hole so you can work other examples for yourself.

If you have a charged rotating black hole, as described earlier it will have a magnetic moment. If you put a magnetic moment in a magnetic field gradient dB/dx the magnetic moment will experience a force F = m dB/dx. If we take our 100 million ton black hole charged up to a trillion volts from above, and give it enough spin that it becomes extremal, you will have an angular momentum of J = 2.2×10^-8 kg m²/s. This gives it a magnetic dipole moment of m = 3.7×10^-33 A m². The highest magnetic field gradients we have managed to achieve have been about a GT/m. Thus, we have a force of approximately 3.7×10^-21 N and an acceleration of about 3.7×10^-32 m/s², which is many orders of magnitude worse than the already pathetic electric field case. But again, using these tools you can work out for yourself the best way to move your black hole if your black hole is not 100 million tons or is charged to a different potential.

But there is another issue to consider. If e 𝒱 / (T_H k_B), for e the fundamental charge, is not much less than 1, you will get significant discharging from the hawking radiation emitting unbalanced numbers of electrons and positrons. For e 𝒱 / (T_H k_B) much larger than 1 and for T_H k_B / (m_e c²) much larger than 1, the discharge rate is approximately e² 𝒱 / ℏ. In our previous example with a 100 million ton black hole, e 𝒱 / (T_H k_B) is about 10,000 and T_H k_B / (m_e c²) is about 200. Because these are much larger than 1 we can use our discharging estimate to find a discharge current of I = 24 million A. In a tiny fraction of a second, our charged black hole would be neutral again. Keeping it charged requires a power of P = I 𝒱 = 24 million terawatts from our particle accelerator.

But we have one more lever left to pull here. Momentum is conserved, so if we can get our black hole to consume matter moving at high speed the momentum of the matter the black hole eats will be transferred to the black hole. With a little bit of calculus you can find that for a Bondi-limited black hole, the optimum speed to shoot your mass stream at the black hole is √2 c_s. The force on the black hole is v ṁ_BH.

Again for our example 100 million ton black hole, if we shoot it with a jet of thallium at 1157 m/s (the optimum for thallium's speed of sound) the black hole will experience a force of 2.7 N and an acceleration of 2.7×10^-11 m/s². This is still much less than the gravity tractor that was the first suggestion we floated for pulling a black hole; but at least it is much better than using electric or magnetic fields! Again, this is just one example. Black holes with different masses will get different results.

Credit

Author: Luke Campbell