In order to build a space elevator we’d first send a bundle of cable to geosync (or manufacture it in space) and then lower one end to the ground. Once it’s anchored, we’d get new cables into place by running them up the old cables like a flag pole.

The cost of sending stuff to space has dropped precipitously in the last few years, as people of staggering genius discovered not-throwing-your-rocket-away technology. But even with reusable equipment, rocket flight has a number of drawbacks that are unlikely to ever be fixed. Rockets are powered by rocket fuel, which means that there’s a whole extra process for supplying them energy. And since a rocket needs to carry all the fuel it burns, the first (and largest) stage of a rocket is almost entirely fuel used to lift other fuel. And then of course are all of the usual issues that crop up when you ride an explosion into space. It’s good that the lower stages are now being reused, but it’s bad that they exist at all.

Vehicles that climb up and down a space elevator can be as small as you like, run on electricity, and don’t suffer from the burning/exploding snafus native to rocketry. While the trip will be much slower and controlled, more like travel by train than roller coaster, the real advantage of space elevators is infinite free energy forever and a safe, convenient way to get to and from the surface of Earth and literally *anywhere* else in the solar system. Since the tether is anchored to the ground, the rotation of the Earth supplies sideways velocity to anything attached to the tether and (like anything that rotates) that sideways velocity adds up to a lot if you get far enough out.

The higher something is, the longer it takes to orbit. Near the surface (within a few hundred km) orbits around the Earth take about 90 minutes, while the Moon (385 km away) takes about a month. One Lunar month in fact. In between is “geosynchronous orbit”, where objects can orbit the Earth in exactly one day, allowing them to stay over a single point on the equator indefinitely. If you ever see a satellite dish that’s nailed to a wall, it’s pointing at a satellite in geosync, because only those satellites sit perfectly still in the sky with respect to the ground. If you run a cable from the equator straight up, below geosync it will want to fall, because it will be traveling slower than orbital speed (that’s why ropes don’t stand up on their own), but above geosync it will want to fall up, because it will be traveling faster than orbital speed. If you swing a rope around over your head it will point straight away from you and for exactly the same reason space elevators stay upright (as long as enough of its mass is above geosync). Basically, space elevators turn Earth into a giant space weed wacker.

The big hurdles to building a space elevator are: 1) it sounds like a physicist stoner fantasy and 2) for Earth there are only a few things you can build a space elevator out of and we can’t manufacture any of them on the scales we need. The first problem is an old one: electricity, radios, computers, satellites, and practically every other important advancement were deemed impossible and pointless right up until they became easy and essential. The second problem is a bit more fundamental.

A hanging rope has to hold up its own weight in addition to whatever is tied to it. There’s a limit to how far a rope can hang before it snaps under its own weight, called the “breaking length”. Perfect name. The breaking length is a function of both the density of the material and the tensile strength; it’s greatest for materials that are light and strong. Unfortunately, the breaking lengths for our usual building materials is a little short. For example, the breaking length for steel is about 6.4 km.

So why no just make the top of the cable thicker than the bottom? On the scale of sky scrapers that’s exactly what we do. But on the scale of the sky itself, it’s silly. Since the cable is pulled in both directions away from geosync, it needs to be thickest there; geosync is “the top of the hill”. By declaring that you’d like the tension to be constant along the entire cable (the best way to avoid snapping), you can derive how many times thicker it needs to be at geosync compared to its size at ground level (or wherever else you’d prefer). This is the “taper ratio”. Building a space elevator out of steel would entail a taper ratio of around 10^{175}, meaning that a steel cable that’s 1 cm on the ground would have a diameter of around 10^{60} times the size of the observable universe at geosync, which would be difficult to construct for… any number of reasons. Carbon nanotubes on the other hand have a breaking length of around 6,000 km and a taper ratio of 1.6 between the ground and geosync. Even if you make the ratio 100, to accommodate extra equipment, lots of climbers, and a generous safety margin, that’s still totally feasible. It might look a little weird if you looked at the geosync station and the ground anchor at the same time, but you can’t: they’re 36,000 km apart.

The problem is that carbon nanotubes don’t come in 40,000+ km lengths. The record so far is about half a meter. That’s *really* impressive for a single molecule, but it would be nice to have intact strands stretching the full length of the tether, rather than tying or gluing them together (which introduces weak points). That right there is the big stumbling block for actually building space elevators: carbon nanotubes are presently too short and expensive. But that’s an engineering issue. We know we *can* do it, we just need to work out the details and then get good at it. In the last 20 years new fabrication techniques have massively increased the yield and dropped the price of nanotubes by a factor of more than 100. Recently, nanotube “yarn”, made from lots of short bits of nanotube, has demonstrated that it can handle about a third of the theoretical maximum, which is more than good enough for a space elevator (the taper ratio would be, gasp… 4.5ish).

This report for NIAC (NASA Institute for Advanced Concepts and incomplete acronyms) goes through the problems, feasibility, and cost of a space elevator and comes up with a $40 billion price tag for the first elevator and a hell of a lot less for the second. Budget estimates like this are not the sort of thing that should be taken literally. “$40 billion” really means “somewhere between $4 and $400 billion”. The science of reading tea leaves and related sub-fields, like economics, are not great for exact predictions.

So what’s the point? In short, you personally will never go to the Moon or another planet by rocket. If you do go, it’ll be by elevator. Typically when you see a diagram of a space elevator, you see something like the image at the top of this article: Earth, geosynchronous orbit, and a space elevator extending just a little bit beyond it. But that tail above geosync is worth considering. Therein fun happens.

The higher you are the faster you’ll be moving and the less gravity you have to fight against. A space elevator to 45,000 km gives us access to Earth orbit. The savings from not needing rockets to put stuff in orbit means the elevator would pay for itself in short order, which is… fine. On the other hand, an elevator extending out to 200,000 km changes the course of human history. It gives us immediate and cheap access to and from *everything* in the solar system. Our world would go from dangerously small and crowded, to inconceivably vast and resourced in a single generation (unless there’s a reason to drag our feet). Just for some idea of where we might want to go, there’s likely to be more water on both Enceladus and Europa than on Earth and 16 Psyche is a brick of metallic iron the size of a state, which is especially appealing considering that Earth has *zero* natural metallic iron (it’s a long, polluting road from iron ore to iron). Easy access to effectively infinite resources along with plenty of room to grow is a plus.

If you want to know how to throw a rock that would land on Mars in the least time, using the least energy, the answer is a “Hohmann transfer orbit“. When you hear about the launch window for Mars only being open once every 780 days (once per synodic period), that’s the window for the Earth-to-Mars Hohmann transfer orbit. Any other trajectory typically takes longer and requires a lot more rocket power at both ends.

A space elevator is like a sling for space craft and cargo. By climbing to the appropriate height and letting go during the appropriate launch window, you can go anywhere in the solar system without the need to lift off from Earth or work to escape its gravity. And the gains in efficiency that come from that are not small. That whole “rockets need to use fuel to carry their fuel” thing gets out of hand really fast. Freedom from lift-off means freedom from most of your rocket.

The Hohmann transfer orbits we use to get to each other planet, are the same ones we’d use to return. So the “Mars port” is also the best place to catch incoming stuff from Mars. Asteroid miners wouldn’t have to worry about safely landing cargo on Earth or even slowing it down; they’d just need to aim at the Ceres port and lower it from there. Earth is about the worst place to build a space elevator, which is why we need nanotubes to do it. You can easily do the same trick with rotating asteroids to fling stuff back to Earth using cables made from steel and suddenly you’ve got a free ride back home as well.

When designing space elevators, one of the big stumbling blocks often seems to be powering the climber to go up and down. The consensus seems to be that lasers are a good way to get power to a single climber going in a single direction (electrical cables conduct about as well as wood over tens of thousands of miles). But that’s very 20th-century-space-thinking. Consider this: counter weights.

If you’ve ever had the opportunity to look inside an open elevator shaft, you may have noticed that it’s more complicated in there than a single cable attached to the top of the elevator car. That’s because of the counter weight (and probably safety or something). The cable coming out of an elevator car goes over a pulley and attaches to a counter weight, so that as the elevator car rises the counter weight falls. For every 1 kg going down, you get 1 kg going up for free (or nearly free, what with friction). The same idea can be applied to space elevators. It takes a lot of power to get a ship up to the Mars port, but if there’s also a ship returning, then everyone gets a much cheaper ride. If there’s more cargo coming in from the asteroid belt than ships leaving Earth, then suddenly “counter weight tourism” begin to make sense. Rather than intentionally shedding energy to bring cargo to the ground, you may as well send stuff up. People, spaceships, garbage, Pog collections, whatever you got.

Almost as good, at 150,000 km, coincidentally between the ports for inner and outer solar system destinations, is the “break even line”. If you imagine the elevator as “uphill” from the ground to geosync and “downhill” after that, then the break even line is where you’ve gone uphill as much as you’ve gone down hill, so (ignoring friction and assuming the judicious use of counter weights) it takes a total of zero energy to get to the break even line and if you want to go to the outer solar system, there’s a net *gain*.

At just short of 200,000 km (just over halfway to the Moon) a space elevator can fling stuff into interstellar space, completely clear of the solar system. While that goes a long way toward traveling to other stars (we don’t have to worry about climbing out of the Sun’s gravity), on its own a space elevator won’t get us moving fast enough to go anywhere outside of the solar system on a even remotely reasonable time scale, if at all. For example, we recently had an interstellar interloper pass through the inner solar system, ‘Oumuamua. ‘Oumuamua is a rock traveling at a fairly typical clip of 26 km/s relative to the Sun when it’s in interstellar space. Unfortunately, to catch up with even this, the closest of all extra-solar objects, we’d need a tether that extends beyond the Moon (out to around 545,000 km). That presents two problems: first, the taper ratio jumps into the thousands even for nanotubes, and second, the Moon’ll smash the elevator. Still, “everywhere in the solar system for free” is better than a swift a kick in the pants.

To be clear, the space elevator described above is not crazy science fiction; it’s science fact that just doesn’t happen to exist right now. Even at the extreme end of the tether you’d experience a maximum of a tenth of Earth’s gravity, so you’re not exactly crushed by centrifugal forces, and the taper ratio never gets above about 2.5. This is something we can build and comfortably use. It would be a project on the scale of the Great Wall of China (21,196 km) or the Interstate Highway System (77,556 km), but it’s definitely doable.

**Answer Gravy**: It may have occurred to you that we live in 3 dimensional space. That means that places roughly in the plane of the equator are potential targets for a space-elevator-fling, but things above one of the poles are out of reach. The spin of the Earth keeps it oriented in space, like a gyroscope (well, not *like* a gyroscope) and that means that if you extend the equator out into space you’ll always hit the same line of stars across the sky, the “celestial equator”. Everything in the solar system orbits the Sun in roughly the same plane, and since the Earth is included in that plane it appears to us as another line across the sky, “the ecliptic”. With rare exceptions (mostly things orbiting way out there), everything in our solar system can always be found close to the ecliptic. These two planes are such important landmarks (spacemarks?) that their intersection is used to define 0° in celestial coordinates; kind of the “Greenwich of the sky”.

So an important question before building a space elevator for the purpose of flinging is “How close are the celestial equator and the ecliptic?”. It turns out that they’re at most 23.5° apart. Not coincidentally, this is the Earth’s axial tilt (the ecliptic is what the Earth is tilted with respect to).

If , , and , are the target, fling, and missing velocities and if is the angle between the target velocity and the celestial equator, then by the Pythagorean theorem and by definition and . The total energy the space craft needs is , and since , the space elevator covers of the total energy cost. That means that in the worst case scenario about of the energy you need to get where you’re going is free. Considering that liftoff is no longer a going concern, this is a great opportunity for ion drives (much more efficient, but too weak for liftoff) to pick up the slack.

If you’re wondering how we can tell which materials are worth trying, it comes down to the usual physics process: say something reasonable, say it again with math, watch helplessly as the math spirals out of control, and finally (hopefully) learn something useful. If the stress on any part of the cable is especially high, then that’s where it’ll snap. To avoid weak points, we just need the entire cable to be under the same stress.

The total force that a cable can support is the cross-sectional area times the tensile strength (the maximum stress). This is because two cables can supply twice the force, so the total force must be proportional to the number of cables or (imagine bundling them together) the cross-section. Every elevation of the cable, r, supports the weight of the entire cable below it. As you rise up a tiny distance, dr, the cable now has to support the same weight, plus the additional weight of the cable between r and r+dr. So the additional area, dA, just needs to support the additional weight.

Weight is force times mass. Figuring out the mass is easy: it’s , where is the density of whatever the cable is made of and is area times height, the volume of the “puck”. It may bother you that by ignoring the dA we’re pretending that the cable isn’t tapering. This is one of those big advantages of infinitesimals (and calculus in general). The volume we’re ignoring is . dA and dr are “infinitesimals”, meaning that (basically) they’re so small that things start to work out. As small as they each are, their product, , is a hell of a lot smaller. So ignore it. Things are working out.

“The additional area times the tensile strength is exactly enough to deal with the additional weight” written in mathspeak is

where T is the tensile strength, is the acceleration of gravity on Earth’s surface, is the Earth’s radius, and is the time it takes Earth to rotate once. We can simplify this a little by realizing that at the radius of geosync, , these two forces are equal

which means we can get rid of π (which is practically a variable, for all you can do with it) and t. Now: some algebra and calculus.

Those C’s are constants of integration and, being unspecified numbers, you can do some pretty sloppy math with them: C ± C = C, e^{C} = C, that sort of thing. If you set that last C to , then you’ve declared that the cable has a maximum cross-section at geosync of 1 (units TBD). You can find values for the density and tensile strength of different materials here and you can graph them here, if you want to consider their feasibility for yourself.

Evangelista Torricelli, the first to measure the weight of the air above us, once famously wrote that “Noi viviamo sommersi nel fondo d’un pelago d’aria.” (“We live submerged at the bottom of an ocean of air.”). That’s a really good way to think about the atmosphere. It’s held in place around the Earth in exactly the same way that the oceans are held where they are; air has weight and gravity holds it down. Gravity doesn’t suddenly go away above the atmosphere. In low Earth orbit (where most satellites can be found) gravity is almost as strong as it is on the surface. So if you pump air or CO2 or any kind of matter above the Kármán Line (the generally agreed upon, but arbitrary, boundary of space), it will still be subject to gravity and will fall. You’ll have yourself a CO2 fountain.

If you build your pipe straight up from one of the poles, then it’ll be fountain-like even when it’s millions of miles tall. More importantly, there are no materials that can support such a pipe (it’s important, when building a CO2 pump to the stars, to be realistic). However, a pipe built up from the equator would swing as the Earth turns, and that changes things a lot. Shorter pipes (a mere several thousand km tall) will still act like fountains, but eventually that swinging motion becomes important.

Space elevators operate on exactly this idea; combating gravity using the Earth’s rotation. So figuring out what happens to the CO2 flying out of the top of a very tall pipe boils down to figuring out what would happen if you stepped out of a space elevator at the same height. And since it is entirely possible (but difficult) to build space elevators, this plan could work. In fact, since that elevator would need to be made from carbon nanotubes (no other material is known to be strong enough), we’d be using carbon to get rid of carbon. That’s almost like being efficient!

Douglas Adams once sagely observed that “*There is an art to flying, or rather a knack. The knack lies in learning how to throw yourself at the ground and miss. … Clearly, it is this second part, the missing, that presents the difficulties.*” Ordinarily this is difficult, but if the tower you throw yourself off is at least around 30,000 km from the center of the Earth, then by the time you get to the ground it won’t be there. Sweeping out a larger circle every day than a point on the ground means the platform you jump from will be moving sideways faster. Once it’s fast enough, when you throw yourself at the ground you’ll literally miss and find yourself in an orbit that skims just above the surface of the Earth at perigee (at the lowest) and returns up to the height of the platform at apogee (at the highest). Lacking an official name, I do hereby declare this to be the “Adams Line”.

Below the Adams Line, all of the CO2 pumped out of the pipe will fall back to Earth. Above the Adams line the released CO2 will find itself in orbit, and we’ll have given Earth a ring system. This wouldn’t be a permanent solution. All of the material in a ring system needs to “stay in its own lane” and orbit in a near-perfect circle (hence: “ring”) or things will run into each other, lose energy, dropping into lower orbits and eventually returning to Earth. A whole series of overlapping elliptical orbits swerving from the height of the pipe to as low as Earth’s surface would be a very messy, very unstable ring system.

But the higher the pipe, the more circular the orbit. Once the top of the pipe reaches geosync, it’s in orbit. If you were to step off of a platform you wouldn’t fall, you’d just drift. Gas released at this height wouldn’t swerve between altitudes as it orbited Earth, it would more or less stay where it is and instead would slowly fill up the thin ring that forms geosync. The most expensive and sought-after orbits around Earth would get dirtied up and be far less useful for satellites, but it’s a small price to pay.

We actually have examples of “pipes” releasing gas into their own orbits, so we know more or less what it would look like. Several of the moons in our solar system are geologically active and, having much lower gravity, eruptions from their surfaces don’t necessarily fall back to the ground.

So between the Adams Line and geosynchronous orbit we’d have a temporary ring system, perhaps giving us time to think up and even crazier plan, and above geosync we’d have a *fairly* stable ring system (no ring systems are entirely stable). But if we want to get rid of Earth’s CO2 forever, and not just orbitally sequester it, we need it to escape from Earth’s gravity entirely.

At geosync objects orbit Earth once per day. Below that they orbit faster and above that they orbit slower (for example, the Moon, which is *way* beyond geosync, takes a month to orbit). So if the pipe is taller than about 42,000 km, the CO2 it releases will actually fly upward before falling into orbit, and the taller the pipe higher that orbit will be. Beyond about 53,000 km, the top of the pipe will be moving at escape velocity; the CO2 that comes out will fly away from Earth to find its way into orbit around the Sun.

Interesting fun-fact: as measured from the center of any planet, the height of the “Fling Line” is always the cube root of 2 (≈1.26) times the height of geosynchronous orbits. The Fling Line is a really good goal for anyone planning to build a space elevator, because it’s a free ticket to *everywhere*.

You might worry about some small amount returning to Earth, but you should be equally worried about someone in Antarctica having bad breath; there’s a lot of room for that problem to diffuse.

So that there is the answer. If you want to solve global warming, you can just gather up and pump CO2 straight up about 53,000 km. Easy peasy.

The weakness of brilliant plans like this isn’t feasibility (this could *technically* be done), it’s the naysayers who point out that the cure might be worse than the disease. After all, carbon is an essential part of the Earth’s biological and chemical systems. We’ve added about half-again as much CO2 into the atmosphere in the last few centuries, taking us from below 300 ppm to about 400 ppm today. There’s been 400 ppm of CO2 in the Earth’s atmosphere before now, most recently in the Micene 10 million years ago. Way back then life on Earth was doing alright (we got grasslands and kelp forests out of it, which isn’t terrible). Having a bunch of CO2 around isn’t necessarily bad; like highway driving, the danger doesn’t come from any particular situation so much as sudden changes in situation. Given that, suddenly and irreversibly chucking a third of Earth’s CO2 into space has a fair chance of being a *really* bad idea.

**Answer Gravy**: If you’re wondering where the various heights came from, you’ll be thrilled to know that you can figure out the height of both the Adams Line and Fling Line (not the official names), with nothing fancier than algebra.

A Keplerian orbit (a regular elliptical or hyperbolic orbit) can be described as:

where r is the distance from the center of the Earth (or whatever you happen to be orbiting), e is the eccentricity (e=0 for a circle, 0<e<1 for an ellipse), is the angle between the lowest point in the orbit and the present position, and p is (basically) a convenient place holder that determines the overall size of the orbit.

If and are the initial tangential (sideways) and radial (up/down) velocities, h is the initial radius, and is the initial angular position, then you can solve for the orbit using:

Normally you don’t know what the initial angular position is, you just know how high you are and how you’re moving.

This is a slightly streamlined version of the solution that can be found here. Jumping off of a platform that’s h above the center of the Earth and attached to said Earth means that , where t is the length of one sidereal day (the time it takes Earth to fully rotate once). This is just the distance (the circumference of a large circle, 2πh) over time (one day, t). Since you’re stepping off of a platform, your initial radial velocity is zero, . Finally, since the question is about the minimum height you could step off such that you would miss the Earth, you’re stepping off at the highest point in the orbit and then dropping, so . Normally you need both the second and third equations, but knowing you’re staring at the highest point means you already know what is. One less equation to worry about!

Use the second equation to solve for e:

and then plugging that into the first equation gives us the orbit:

If you plug in , you find that the height of the platform is in fact the height of the platform:

It’s good to double check. But to actually figure out what h is (this was the whole point) you want the lowest point in the orbit to be the Earth’s radius, R=6371 km (any lower and you hit the ground).

This is a fourth degree polynomial in h, and while it is *technically* possible to solve this by hand, you never want to solve this by hand. So plugging in the gravitational constant, , and all of the details about Earth, , , , and then giving this to a computer to solve for you, you find that one of the four solutions makes sense: h=29,825 km. So if you have to jump into an empty swimming pool, that’s a good height to do it.

The height of the Fling Line (not an official name), f, is a bit easier. You don’t have to mess around with orbits, you just have to set the speed your platform is moving at a given height, , equal to the escape velocity at that height, :

You can calculate the height of geosync, s, by setting the centrifugal force equal to the gravitational force for an object orbiting exactly once per (sidereal) day, t:

So that’s a cute, probably useless fact; for any planet, the Fling Line is always at times the radius of geosyncronous orbits.

]]>The picture that you’ve already seen is of M87*, the supermassive black hole (SMB) in the middle of the galaxy M87. Damn near every galaxy appears to have a SMB in its core. This isn’t a coincidence, they’re instrumental in the formation of galaxies. A galaxy’s SMB is almost always dead center and generally has a mass proportionate to the mass of the galaxy it calls home (or more likely, calls “mine”). Our home galaxy’s supermassive black hole, Sagittarius A*, has the mass of around 4.1 million Suns, which is *pretty* massive. But M87* weighs in at around 6.5 billion Suns, more than a thousand times bigger. M87* is massive even by supermassive standards.

Sagittarius A* is by far the closest SMB (a mere 26,000 light-years away), but of course there’s a galaxy in the way. To see it we have to look long-ways through the disk of the Milky Way, which is full of stuff. M87* on the other hand is a freaking monster, but it’s 54 *million* light-years away. That’s fairly nearby in galactic terms (you can see it on a dark night), but that’s still incomprehensibly distant. The light collected to create the picture of M87* was released when our branch on the tree of life was a bunch of tiny, arguably adorable critters, barely recognizable as primates, and expanding across a planet with a recent and notable lack of dinosaurs. In balance, it was easier to image M87*, a black hole a thousand times bigger and two thousand times farther away, than Sagittarius A*.

As a quick aside, that stunningly boring name, “M87”, comes from Messier (being French, his name is pronounced “mess-E-ay”). At the time astronomers were getting famous putting their names on comets (worked for Halley) and Messier wanted a part of the action. It turns out that to an 18th century telescope, there are lots of vague, blurry things in the sky that might be comets, but aren’t. So Messier wrote a list of things he didn’t want to look at twice, and that rejection list is what he’s remembered for now (incidentally, he also discovered some comets). The galaxy M87 is the 87th thing on Messier’s “ignore this” list.

The blurry doughnut is a cloud of in-falling gases that heat up and ionize as they fall in. M87*, despite being substantially larger than the orbit of Neptune, is much smaller than the cloud of stuff falling into it, so there’s a bottle neck as gas falls in and that means friction and compression and that means heat. That glow is a fire storm in space far bigger than our solar system. The dark in the middle isn’t the black hole itself, so much as a lack of gases. Newton’s gravity is proportional to the inverse square of the distance to whatever’s making the gravity; half the distance = four times the gravity, a third the distance = nine times the gravity. In Newton’s gravity, there’s no limit to how close you can be in orbit around something. But in Einstein’s (correct) formulation of gravity, the inverse square law is only an approximation that works in low gravity situations (basically, Newton is good enough for practically every application *except* black holes). It turns out that black holes have an innermost stable orbit, that’s 3 times the Schwarzchild radius (3 times the black hole’s radius). Anything that tries to orbit closer will just spiral in. The edge of that doughnut is the innermost stable orbit; the last chance gases have to glow before they get fast-tracked to the black hole itself.

The gas is about the same temperature all the way around M87*, but because it’s spiraling in it’s moving toward us on one side and away on the other. The side that’s moving toward us appears brighter because of an (usually difficult to notice) effect called “relativistic beaming“; *very* high speed material emits more light in the direction of motion (although, from its own perspective, it’s still emitting light evenly in every direction).

We were almost guaranteed to see that hot cloud of gas all the way around M87* because, in addition to all the other craziness going on, black holes bend light around them. This isn’t unique to black holes, we see “gravitational lensing” every time light passes close to something massive. In fact, one of the earliest confirmations of General Relativity was during the 1919 solar eclipse when stars were seen *around* the Sun that were actually *behind* it. So black holes aren’t the only things that can bend light beams, but they do elevate the art; instead of light being slightly deflected, it can completely change direction or even literally go into orbit. So when you see a stream of hot gas beside a black hole, it might actually be behind it. Regardless of how the disk of infalling hot gases are arranged, there will always appear to be gas around the black hole.

Gravitational lensing doesn’t just fun-house-mirror and confuse the image, it actually helps us here. The dark spot is three times the size of the black hole in reality (the innermost stable orbit), but the lensing effect magnifies it an makes it appear larger.

The curse of always being right, I’ve been told, is that you’re never surprised and rarely invited to parties. The new black hole picture isn’t really a discovery, but it is a stunning accomplishment. Black holes, as we understand them today, were first theorized when Schwarzschild got ahold of the recently published equations of Einstein’s general theory of relativity, applied them to a dense sphere of matter, and followed where they led. Modern physics has an established track record of starting with basic assumptions, building verifiable theories, and then applying those theories to reach conclusions that everyone agrees are so completely nuts they can’t possibly be true. Black holes, which are real, are a great example of “so weird they can’t possibly be real”.

Today, you’d have a hard time finding an astronomer or astrophysicist who didn’t already firmly believe in the existence of black holes and their basic, overall properties. They quibble about the details, sure, but that’s scientists. There’s a long catalog of observations that already fell in line perfectly with the predicted nature of black holes. LIGO’s ongoing detections of gravitational waves are beautifully and precisely described by black holes merging. Stars in the galactic core get whipped back and forth at around 1% the speed of light (which is absurdly fast for anything that isn’t a particle) while orbiting a dark, compact “radio source”. Quasars, some of the brightest, highest energy things in the universe, are exquisitely described by gases falling into SMBs, but are totally mysterious otherwise.

So what did astronomers expect to see when they finally got a direct image of an actual black hole? Basically what we’ve seen. The simulations used to predict what a real picture would look like were disappointingly accurate.

It would have been great if the picture had been completely different. Like a smiley face or something. Science advances when we’re wrong or surprised or both. What this new picture does is add another verification win to the theory of General Relativity, which is good. It’s not enough to have a clever theory that always works, you also have to give it every possible new opportunity to fail.

So as far as theory goes, we haven’t gained too much. This was kind of the “eat your vegetables” of empirical physics. What we gained, we gained through the effort it took to get this picture. The reason this hasn’t be done before is that it’s really, *really* hard to do. M87* is about 54 million light-years away and about 38 billion km across, which makes it 42 microarcseconds across. There are 360 degrees in a circle, 60 arcminutes in a degree, 60 arcseconds in an arcminute, and a million microarcseconds in an arcsecond. 42 microarcseconds is the angle across a single hair from 500 km away.

But here’s the problem: that’s too small. There’s a fundamental limit to the resolution of a telescope called the “diffraction limit” that’s built into the wave nature of light. The more perfectly your telescope is constructed, the closer you can get to this limit, but you can never do better. If D is the diameter of your telescope’s main lens or mirror and W is the wavelength of the light you’re using, then the smallest angle (in radians) you can resolve is about , so shorter wavelengths and bigger telescopes produce sharper pictures. The now-famous black hole image was taken using microwave light with a wavelength of 1.3 mm, which means that to get a fuzzy circle with a dark center (“Hey, a black hole!”) instead of a fuzzy smudge (“Hey, hot gas with maybe a black hole!”) you need a telescope that’s a few thousand km across. Point of fact: that’s too big.

The Earth happens to be a few thousand km across, so we just need a way to use the whole thing. Enter the Event Horizon Telescope (EHT), an international collaboration of telescopes that points at M87* with an “effective aperture” that’s about the size of the Earth. A regular telescope creates images by collecting light and focusing it on a detector. A light wave from a distant source rolls in like a wall, bounces off of the mirror, and arrives at the focal point all at once. This is done automatically by the geometry if the mirror is parabolic, since every parallel path takes the same amount of time to get to the focus (that’s just one of those handy properties that parabolas have).

If parts of the mirror happens to be missing, that’s fine. The wave still arrives at the focus all at once, but some of it just passes on through. So the fact that the EHT doesn’t need to cover an area the size of Earth, it just needs to have pieces spread out across the Earth. But there’s still a huge problem; the light waves rolling in from M87* are never collected at a focal point. Instead, the data received by each telescope and array is recorded, collected in a single location (often by loading hard drives onto a plane and *physically* bringing the data together), and processed using a technique called very-long baseline interferometry (VLBI). “Very-long baseline” refers to the fact that the distance between the individual telescopes is “very long”. “Interferometry” refers to the fact that the collected light waves need to interfere.

That interference is important; it’s not enough to just report what each telescope sees. Getting the wave front to arrive at the detector all at once is a really important part of what makes a telescope work. After all, waves from what you’re *not* looking at also arrive at your detector, just not all at once. Parts of the wave arriving at different times interfere destructively and cancel each other out. So there are two big hurdles: combining the data while also knowing *exactly* when a given light wave sweeping across Earth arrives at each different telescope’s detector. You need to be able to tell the difference between one wave front and the next, and if the next wave front is 1.3 mm behind and traveling at the speed of light, then you need to reliably distinguishing between events 4 picoseconds apart (4 *trillionths* of a second). So, every telescope needs a shiny new atomic clock and a *really* fast camera. You begin to get a sense of why the data consolidation looks more like a cargo shipment than an email attachment; trillions of snapshots every second of not just the waves you’re looking for, but also the waves that will cancel out once all the data is processed. Add to that the challenge of figuring out (generally after the fact) where every detector is to within a fraction of a millimeter even as their orientation changes and the Earth rotates, and you’ve got a problem.

Or *had* a problem anyway. The picture came out just fine. The two black holes that appear the largest from Earth are Sagittarius A* (because it’s close) and M87* (because it’s a monster), and they’re both right on the edge of what can be effectively imaged using Earth based arrays of telescopes like the EHT. But if the overarching history of technology is any indication, the first time is almost impossible and the thousandth is embarrassingly easy. We’ll get better pictures of M87* and passable pictures of Sagittarius A* soon. Once we set up an array of space telescopes throughout cislunar space (the volume inside the Moon’s orbit) we’ll get pictures of the SMBs in the cores of every nearby galaxy and that’s when the science really gets started. The cost low Earth orbit has dropped by 99% since 1980 and the price of spaceflight in general is on schedule to drop substantially more in the decades to come, so a fleet of space telescopes is not such a big ask.

That blurry picture of M87* is the first stumbling steps into a profound new kind of astronomical research that will be totally routine sooner than you think.

The Milky Way picture is from here.

The simulated black hole image is from here.

]]>Once time travel has/will have been invented, you’d think that said inventor could just go back in time and show off their invention, or give it to some ancestor (or even become some ancestor) so they could be born into old money.

But despite being both common and world-changing, time travel is intrinsically very low-key. In the 20th century the world population increased from 1.6 to 6 billion people, and even though time travelers account for about 3 of those 4.4 billion new folk, evidence for their presence is almost impossible to find. It turns out that time machines are just like every other machine; they don’t exist if they’re not invented. So whatever else anyone does with a time machine, it didn’t/won’t affect the invention of time machines themselves.

In 1992 Steven Hawking derived the “chronology protection conjecture“, which posits that “closed time-like curves” are impossible. Moving along a time-like path is what you (and every other chunk of matter in the universe) are doing right now; moving slower than light and experiencing time in the usual way. Moving along a closed time-like path is like going for a walk in the woods and following a trail that returns you home yesterday. Hawking showed that closed time-like curves produce “feed back” that destroys everything involved. In other words: Timecop rules.

Ever empirical, on June 28th, 2009 Doc Hawk threw a party for time travelers and (to ensure *only* time travelers showed up) he kept it secret until June 29th, when he sent out invitations. Save the date!

To his bemused shock, Hawking’s soiree was very well attended. He claims to have met “people” from as far afield as 70189324233 AD, the year in which the invitation, as well as the spaciotemporal coordinates of Earth, were unceremoniously overwritten and forgotten during “The Great System Update”.

At his party, the Hawk discovered three things. First, time travel is not just possible, but easy. Second, closed time-like curves are impossible, but that’s not how time travel works. And third, time travelers don’t leave much evidence behind, because they couldn’t if they tried (and don’t when they do).

Hawking later wrote, “*Dear Diary, [I] wasn’t sure about actually buying champagne for the affair, since I knew (or thought I knew) that this was all a [silly stunt]. I’m glad I did! Time travelers are a cagey lot and the evening didn’t really get into full swing until the 7th or 8th crate was opened. A man (perhaps?) who introduced himself as the Designate Demithrall of the North Antarctic Seasteader Federation in 4372, mentioned that the key to time travel is my own work on imaginary time and that it’s ‘obvious really, if you think about it’. This is remarkable! But in the sober light of da [sic.] I can’t help wondering if the Designate Demithrall wasn’t drunk or sarcastic or both. Forty-forth century humour is really hard to read.*”

When a time traveler intends to give instructions to someone in the past to help them be the first person to build a time machine, they inevitably and accidentally don’t. The retro-self-cohesion principle of the time-line prevents grandfather paradoxes, so neither time travelers nor machines can change the logic of their own history. In other words: *not* Back to the Future rules. For example, if you go back in time to kill your own grandfather, then you won’t exist to go back in time and do said killing. You have to come from somewhen. Inescapably, you’ll either get the wrong guy or fail to get the right guy. In other words: Bill and Ted rules. Time travel is possible, and even common, but you can’t change things so much as confirm them. In the archetypal example, Rufus goes back in time to ensure the Wild Stallions succeed in bringing about peace and enlightenment throughout the universe, and he knows they do because he was/will be there to help.

In the same way that you don’t (presently) worry about your murderous unborn grandchildren, the inventor of time travel is immune to hints. No matter how many time travelers they may incidentally meet, none of them will ever get past general pleasantries; the topic of time travel is logically verboten. The same holds for common knowledge. Presumably, the reason that you can’t go online and find the schematics for a (functional) time machine is that the future inventor of time travel doesn’t live in a cave. The first thing they’re likely to do before getting down to work is a quick internet search to see if they’re reinventing the wheel (or flux capacitor), so all the universe must conspire to make that internet search fail. Like time travelers themselves, the idea has to come from somewhen. Being aware of this tautological time travel truth, and possibly having read Hawking’s published diary, the Designate Demithrall was most likely safeguarding the logical consistency (and existence) of the very conversation he was in by filling it with sarcasm and misdirection.

So if you ever meet anyone who claims to be a time traveler and makes no attempt to support their claim, then they’re probably telling you the truth. Time machines are more common then cellphones, but they’re literally impossible to talk about. And if you yourself are a time traveler, remember that we ran/will run out of prosecco about halfway through Hawking’s thing, so BYOB.

]]>You’d be hard pressed to find someone who disagrees with the first rule, and if you press hard you’ll find that almost everyone has been confused (at least once) by the third. In a nutshell, if you try to define the multiplication rules any other way, arithmetic stops working in a big hurry. Or at least, you have to scrap a lot of other math that’s incredibly useful.

When you multiply X by a positive integer Y you’re adding it to itself Y times. So it makes sense that a positive times a positive is positive. For example, . By looking at 3 (or any other number you like) multiplied by smaller and smaller integers you see a pattern:

Every time you multiply by a one-smaller-number, you take away another 3. Following the pattern, we take 3 away from zero and make a decent guess at how that pattern should continue:

Do this for a couple of different numbers and you can construct a multiplication table. Like this one!

If the “negative times a negative” quadrant on the lower left were all negative instead of positive (e.g., ““), then the rows and columns that go through it will suddenly have to switch patterns (e.g., “increasing by 3’s” to “decreasing by 3’s”) when they pass zero. In some sense, the rules for signs are set up so that multiplication tables like this follow a nice, simple pattern.

So, “, , and ” is a clean, reasonable way to define multiplication. But does it work with the rules of arithmetic?

In particular, the distributive property, which says that , is one of the backbone rules upon which all of arithmetic is built. In fact, this property is literally the thing that defines the relationship between addition and multiplication! For example,

because “3” is *defined* as “3=1+1+1”. Losing the distributive property basically means you need to go home and start designing a new (and worse) kind of math from scratch.

For positive numbers there’s no issue, because (practically) everyone is fine with the “ rule”. For example, and .

But if you insist on using the rule ““, then you’ll find the distributive property doesn’t work. For example, and . The discerning eye will note that 5≠25, so . In other words, we need to use “” in order for arithmetic to work.

And if you begrudgingly allow the “” rule, but refuse to accept the “” rule, then consider this: and .

On a case-by-case basis, it’s not obvious that a negative times a negative should be positive. But when you look at lots of examples and the number system overall, you find that the “” rule is kinda hard to avoid. Using a different rule means asking a lot of hard questions, like: What is negativeness? Which rules of arithmetic are worth keeping? What is the sound of negative two hands unclapping?

Using the wrong rules is a good, practical, and genuinely useful training in what *not* to do. It’s well worth your time to shake off the shackles of mundanity and conformity, so that you can forge into a world of new discoveries. Specifically, you’ll discover why arithmetic’s shackles are usually left unshaken.

According to the Earth-based observer, the spaceship will arrive at the star in 5 years. However, because of time dilation, the spaceship’s clock will only read 3 years of elapsed time on arrival. To an astronaut on the spaceship, the distance to the star appears to be just 2.4 light years because it took him just 3 years to get there while traveling at 80% light speed.

This situation is sometimes explained as a consequence of length contraction. But what is it that’s contracting? Some authors put it down to space itself contracting, or just distance contracting (which it seems to me amounts to the same thing) and others say that’s nonsense because you could posit two spaceships heading in the same direction momentarily side by side and traveling at different speeds, so how can there be two different distances?

So what is the correct way to understand the situation from the astronaut’s perspective?

**Physicist**: Space and time don’t react to how you move around. They don’t contract or slow down just because you move fast relative to someone somewhere. What changes is how you perceive space and time.

When you measure the length of something in space (in other words, “normally”), the total length isn’t just the length in the x or y directions, it’s a particular combination of both that works out exactly the way you’d think it should. When you measure the length of something in *spacetime*, the total length isn’t just the length in the space or time directions, it’s a particular combination of both that works out in more or less the opposite of how you’d think it should.

We don’t talk about the three dimensions of space individually, because they’re not really distinct. The forward, right, and up directions are a good way to describe the three different dimensions of space, but of course they vary from perspective to perspective. Just call someone from the opposite side of the planet and ask them “What’s up?” and you’ll find yourself instantly embroiled in irreconcilable conflict. Everyone can agree that it’s easy to pick three mutually perpendicular directions in our three-dimensional universe (try it), but there’s no sense in trying to specify which *specific* three are the “true” directions.

A meter stick is a meter long (hence the name), so if you place it flat on a table and measure its horizontal length (with a… tape measure or something), you’ll find that its horizontal length is 100cm and its vertical length is zero. Given that, you could reasonably divine that it must be 100cm long. But if you tilt it up (or equivalently, tilt your head a bit), then the horizontal and vertical lengths change. There’s nothing profound happening. To handle a universe cruel enough to allow such differing perspectives we use the “Euclidean metric”, , to find the total length of things given their lengths in each of the various directions. The length in any given direction (x, y, z) can change, but the *total* length (d) stays the same.

Einstein’s big contribution (or one of them at least) was “combining” time and space under the umbrella of “spacetime”, so named because Germans love sticking words together in a traditional process called (roughly translated) stickingwordstogethertomakeonereallylongdifficulttoreadandoftunpronounceableword.

The different spatial dimensions are equivalent. To see for yourself, walk north and south, then walk east and west. Unless you’re carrying a compass, you shouldn’t notice any difference. But clearly time is different. To see for yourself, first walk north and south, then walk to tomorrow and back to yesterday. So when someone cleverly volunteers “we live in a 4 dimensional universe!”, they’re being a *little* imprecise. Physicists, who love precision slightly more than being understood, prefer to say “we live in a 3+1 dimensional universe!” to make clear that there are three space dimensions and one time dimension.

But while time and space are different, they’re not completely separate. In very much the same way that the forward direction varies between perspectives, the “future direction” also varies. And in the same way that rotating perspectives exchanges directions, moving at different velocities exchanges the time direction and direction of movement. The total “distance” between points in spacetime is called the “interval”, L. For folk familiar with the Euclidean metric, the “Minkowski metric” should look eerily familiar: . Some folk will flip the sign on this, , because it makes it a little easier to talk about the time experienced on a particular path (in fact, I’m gonna do that in a minute), but the important thing is not the sign of this equation, it’s that it’s *constant* between different perspectives. It should bother you that can be negative, but… don’t worry about it. It’s fine.

If you’re wondering, the spacetime interval is a direct consequence of rule #1 in relativity: the speed of light is the same to everyone. The short way to see this is to notice that if you find the interval between the start and end points of a light beam’s journey, the interval is always zero because . The long way to see why the interval is what it is, is a little long.

There are two things to notice about the spacetime interval. First, that “c” is the speed of light and it *basically* provides a unit conversion between meters and seconds (or furlongs and fortnights, or whatever units you prefer for distance and time). So 1 second has an interval of about 300,000 km (one “light-second“), which is most of the distance between here and the Moon. It turns out that the speed at which light travels comes from the “c” in this equation. So the speed of light is dictated by the nature of space and time (as described by the Minkowski metric), not the other way around. Which is good to know.

Second and more important is that negative. That *really* screws things up. It is arguably responsible for damn-near all of the weird, unintuitive stuff that falls out of special relativity: time dilations, length contractions, twin paradoxes, Einstein’s haircut and marriages, everything. In particular (and this is why the exchange between distance and duration is so unintuitive), if is constant, then when d increases, so does t.

This is in stark contrast to regular distance, where if is constant, an increase in x means a decrease in y. Picture *that* in your head and it makes sense. Picture relativity in your head and it doesn’t.

Now brace yourself, because here comes the point. The original question was about a journey that, from the perspective of Earth, was d = 4 light-years long, at a speed of v = 0.8c, and taking t = 5 years. The beauty of using “light units” (light-years, light-seconds, etc.) is that the spacetime interval is really easy to work with. The interval between the launch and landing of the spaceship is:

So the interval is L = 3 light-years.

Like regular distance, the power of the spacetime interval is that it is the same from all perspectives. From the perspective of the spaceship the launch and landing happen in the same place. It’s like a narcissist on a train: they get on and get off in the same place, while the world moves around them. So d = 0 and it’s just a question of how much time passes:

So, t = 3 years because 3 light-years divided by the speed of light is 3 years.

So just like changing your perspective by tilting your head changes the horizontal and vertical lengths of stuff (while leaving the total length the same), changing your perspective by moving at a different speed changes length-in-the-direction-of-motion and duration (while keeping the spacetime interval the same).

That’s time dilation (5 years for Earth, but 3 years for the spaceship). Length contraction is a little more subtle. Normally when you measure something you get out your meter stick (or yard stick, depending on where you live), put it next to the thing in question and boom: measured. But how do you measure the length of stuff when you’re moving past it? With a stopwatch.

So, like the original question pointed out, if it takes you 3 years to get to your destination, which is approaching you at 0.8c, then it must be 3×0.8 = 2.4 light-years away. Notice that in the diagram with the planets above, on the left they’re 5 light-years apart and on the right they’re 2.4ish light-years apart (measure horizontally in the space direction).

It feels like length contraction should be more complicated than this, but it’s really not. You can get yourself tied in knots thinking about this too hard. After all, when you talk about “the distance to that whatever-it-is” you’re talking about a straight line in spacetime between “here right now” and “over there right now”, but “now” is a little slippery when the “future direction” is relative. Luckily, “time multiplied by speed is distance” works fine.

There are a few ways to look at the situation, but they all boil down to the same big idea: perspectives moving relative to each other see all kinds of things different. Reality itself, space and time and the stuff in it doesn’t change, but how we view it and interact with it doesn’t *quite* follow the rules we imagine.