physics, thought experiment

Spin to Win | Black Holes, Part 2

In the previous part of this series, I tried to analyze what it would be like to fly an Apollo Command Module into black holes of various sizes. This time, though, I’m going to restrict myself to a single 1-million-solar-mass black hole. The difference is that, this time, I’m going to let the black hole spin (at 98% of the maximum possible spin, which is pretty average for a fast-spinning hole). But I’m getting ahead of myself. Before I go on, here’s my vehicle:

Apollo 11.jpg

(From the website of the Smithsonian Air and Space Museum)

That’s the actual command module Michael Collins, Buzz Aldrin, and Neil Armstrong took to the Moon (minus the Plexiglas shroud, of course). It would fit in even a medium-sized living room. This time, the crew will consist of me, Jürgen Prochnow (Das Boot Jürgen, naturally. Captain’s hat and all), and Charlize Theron. I was gonna take David Bowie and Abe Lincoln along again, but frankly, I’ve put Bowie through enough, and Lincoln was just so damn grim all the time.

Anyway, back to the subject at hand: the scary monster that is a rapidly-spinning black hole. All the black holes I discussed in the last part were Schwarzschild black holes, meaning they had no spin or electric charge. This black hole, though, is a Kerr black hole: it spins. The spin means this trip is a whole new ballgame. We’re still going to die horribly, of course, but hey, at least it’ll be interesting.

The first difference is that you can get closer to the event horizon of a spinning black hole. For a non-spinning black hole, there are no stable circular orbits closer than one and a half times the radius of the event horizon (the Schwarzschild radius), because in order to be in a circular orbit any closer, you’d have to travel faster than light. For spinning ones, there’s a lozenge-shaped region outside the event horizon called the ergosphere (My first-born daughter will be Ergosphere Sullivan). Objects near a rotating hole (or any rotating mass, to a lesser extent) are dragged along with the hole’s rotation. But inside the ergosphere, though, they’re being dragged along so fast that, no matter what, they can’t stand still. Inside the ergosphere, you have to rotate with the hole, because traveling anti-spinward would require going faster than the speed of light.

Here’s roughly what a free-fall trajectory into our Kerr black hole would look like (looking down at the hole’s north pole):

098-kerr-black-hole-infall

The ergosphere is the gray part. The event horizon is the black part.

Jürgen and Charlize are suspicious of me, but I gave them my word that we’re just orbiting the black hole. To make some observations. For science, and all that. When they’re not looking, I’m gonna hit the retro-rocket and plunge us to our deaths. I feel like there’s a flaw in my reasoning, but I don’t have time for such things.

Even orbits don’t work the way they normally do, near a spinning hole. Orbits around ordinary objects are very close to simple ellipses or circles. But, sitting in our command module, here’s what our orbit looks like (starting from parameters that should have given us a nice elliptical orbit):

Kerr BH Stable Orbit.png

This is because, when we orbit closer to the hole, we get a kick from the spin that twists our orbit around.

From nearby, a non-rotating black hole looks like its name: a black circle of nothingness, surrounded by a distorted background of stars and galaxies. From our orbit around the spinning million-solar-mass hole, though, the picture is much different:

Orbiting a Kerr Black Hole.jpg

(Picture and simulation by Alain Riazuelo.)

In that picture, the hole’s equator rotates from left to right. The reason the horizon is D-shaped is that photons coming from that direction were able to get a lot closer to the horizon, since they were moving in the direction of the rotation. On the opposite side, the horizon is bigger because those photons were going upstream, so to speak, and many of them were pulled to a halt by the spin and then either pulled into the hole, flung away, or pulled into a spinward orbit. Black holes are bullies. Spinning ones say “If you get too close, I’m going to eat you. And if you’re standing within a few arm’s lengths, you have to spin around me, or else I’ll eat you.”

(Incidentally, if you read about the movie’s background, the black hole in Interstellar was spinning at something like 99.999999% of the maximum rate. Its horizon would have been D-shaped like the picture above, from up close. From a distance, it would have looked…well, it would have looked like it did in the movie. They got it right, because they hired Kip motherfucking Thorne, Mr. Black Hole himself, to help write their ray-tracing code.

Speaking of Interstellar, the fact that you can get so much closer to a spinning black hole than a non-spinning one (providing you’re orbiting spinward) means you can get much deeper into its time-dilating gravity well. That means, as long as the tides aren’t strong enough to kill you, you can experience much bigger timewarps. The only way to get the same timewarp from a non-rotating black hole is to apply horrendously large forces to hover just outside the horizon. It’s much more practical to do in the vicinity of a spinning hole. Well, I mean, it’s no less practical than putting a Command Module in orbit around a black hole.

According to the equations from this Physics Stack Exchange discussion, as Jürgen, Charlize, and I zip around the hole close to the innermost stable orbit, time is flowing upwards of four times slower than it is for observers far away. I’m gonna keep us in orbit for a week, to lull my crewmates into complacency, so I’ll have the element of surprise when I try to kill us all. Well, we think it’s a week. Everyone outside thinks we’re orbit for a month and change

Then, without warning, I flip us around, turn on the engines, and take us into the hole. Jürgen fixes me with those steely blue eyes and that pants-shittingly intimidating face he was doing all through Das Boot. Charlize spends fifteen seconds trying to reason with me, then realizes I’m beyond all help and starts beating the shit out of me. Did you see Fury Road? She can punch. Neither of them can do anything to stop me, though: we’re already seconds from death.

But because this is a big black hole, the tides are gentle, at least outside the horizon. They’re stronger than the tides the Moon exerts on the Earth (which are measured in hundreds of nanometers per second per second), but they’re not what’s going to tear us to pieces.

What’s going to tear us to pieces is frame-dragging. Let’s go back to the metaphor of the whirlpool. The water moves much faster close to the center than it does far away. Because your boat is a physical object with a non-zero size, when you get really close, the water on one side of your boat is moving significantly faster than the water on the other side, because the near side is significantly closer than the far side. This blog hasn’t had any horrible pictures recently. Here’s one to explain the frame-dragging we experience:

Horrible Frame Dragging.png

In this picture, the capsule orbits bottom-to-top, and the hole rotates clockwise (this is the opposite of the view in the orbit plots; in this picture, we’re looking at the hole from the bottom, looking at its south pole; the reason has nothing to do with the fact that I screwed up and drew my horrible picture backwards).

Space closer to the hole is moving faster than space farther from the hole. The gradient transfers some of the hole’s angular momentum to the capsule, which is bad news, because that means the capsule starts spinning. It spins in the opposite direction of the hole (counter-clockwise, in the Horrible Picture (TM)).

I say the spin is bad news because, from the research for “Death by Centrifuge“, I know that things get really messy and horrible if you’re in a vehicle that rotates too fast.

Here’s a fun fact: Neil Armstrong came perilously close to death on his first spaceflight. During Gemini 8, while Armstrong and crewmate Dave Scott were practicing station-keeping and docking maneuvers with an uncrewed Agena target vehicle, the linked spacecraft started spinning. Unbeknownst to them, one of the Gemini capsule’s thrusters was stuck wide-open. Thinking it was the Agena causing the problem, they undocked. That’s when the shit really hit the fan, though I think Armstrong probably described it more gracefully. A video is worth a thousand words: here’s what it looked like when they undocked. Before long, the capsule was tumbling at 60 revolutions per minute (1 per second), wobbling around all three axes.

Did you ever spin in a circle when you were a kid? I did. Did you ever try it again as an adult, just to see what it was like? I did. I spent the next fifteen minutes lying in the grass (because I couldn’t tell which way was down) wondering if I should just go ahead and puke. Human beings don’t handle rotation well. According to this literature survey (thanks to Nyrath of Project Rho for helping direct me to it; it was hell to try and find a proper paper otherwise), average people do okay spinning at 1.7 RPM. At 2.2 RPM, susceptible people will probably start puking everywhere. At 5.44 RPM, ordinary tasks become stressful, because, thanks to the Coriolis effect (that troublemaking bastard), things like limbs, bodies, and inner-ear fluid don’t move normally, which plays hell with coordination. Also, it makes you puke. At 10 RPM, even the tough subjects in the study were seriously distressed.

Armstrong and Scott were spinning six times faster. When you spin, your brain loses the ability to compensate for movements of the eyes: you lose the ability to stabilize the image on your retinas, and the world wobbles and jumps. That’s bad news, especially if, for instance, you’re stuck inside a metal can which is spinning way too fast, and the only way you can stop it spinning way too fast (so that you don’t die) is by focusing your eyes on buttons and moving your Coriolis-afflicted hands to press them. Armstrong was an especially tough, calm dude, and he managed it, even though both men were starting to have serious vision problems. He did what any good troubleshooter would do: he switched the thrusters off and then on again (more or less). That saved the mission.

Now, I don’t know how fast the black hole will spin us, because the math is very complicated. But considering it’s a black hole we’re dealing with, probably pretty fast. Like I said, nothing about black holes is subtle. At 10 RPM, I throw up. My vomit describes a curved Coriolis-arc through the cabin and splatters on the wall. Jürgen doesn’t throw up until 20 RPM (after all, he’s a seafaring submarine captain). Charlize doesn’t throw up until 25,because she’s a badass.

At 60 RPM, I’m already screaming my head off, hyperventilating, and desperately regretting my decision to plunge us into a black hole. I try to hit buttons (pretty much at random), but I can’t get my fingers to go where I want them, and I press all the wrong ones. Jürgen is trying to calm me down and telling me he wants proper damage reports, but in my panic, I’ve forgotten all my German. Charlize has written both of us off and is trying to re-orient us and thrust away from the horizon, but it’s already too late.

At 60 RPM, the centrifugal acceleration on the periphery of the CM is already over 7 gees. There’s probably a bit of metal creaking, but nothing too serious. Because the crew couches are only a foot or so from the center of mass, we only experience an acceleration of 1.2 gees. For the moment, our main problem is that we’re punching and/or throwing up on each other.

At 120 RPM, the command module is starting to complain. Its extremities experience 28 gees. Panels slam shut. A cable pops loose and causes a short that trips the circuit breakers and kills our power. Even in our couches, we’re feeling almost 5 gees. I’m making a face like this:

gloc-face-735x413.jpg

(Source.)

At 200 RPM, the heat shield, experiencing 77 gees of centrifugal acceleration, cracks and flies off. It’s possible that the kick it gets from leaving our sorry asses behind, combined with the kick from being in the hole’s ergosphere (sounds dirty) is enough to slingshot the fragments to safety. Kind of a moot point, though, since I only care about the human parts, and all of those have blacked out at 13 gees.

Somewhere between 200 RPM and 500 RPM, the hull finally tears open. The bottom dome is flung off, letting important things like our air, our barf bags, and possibly our crew couches, fly out. Not that it matters: at 83 gees, we’ve all got ruptured aortas, brain hemorrhages, and we’re all in cardiac arrest.

At 1000 RPM (16.7 revolutions per second), the command module is flung decisively apart. Thanks to conservation of angular momentum, all the pieces are spinning pretty fast, too. Jürgen, Charlize, and I, are very dead, and in the goriest of possible ways: pulped by centrifugal force, and then shredded as we were spun apart.

The fragments closer to the horizon appear to accelerate ahead of us. The parts farther away fall behind. Most of the fragments fall into the hole. Spaghettification takes a while: once again, a more massive hole has weaker tides near the horizon.

As for what happens as we fall into the really nasty part of the hole (because it was sunshine and jellybeans before…), physics isn’t sure. The simplest models predict a ring-shaped singularity, rather than a point-shaped one. Some models predict that the ring singularity might act as an actual usable wormhole to another universe. But it’s also possible that effects I don’t pretend to understand (which have to do with weird inner horizons only rotating holes have) blue-shift the infalling light, creating a radiation bath that burns our atoms into subatomic ash. Either way, we’re not going to be visiting any worlds untold.

Once again, I’ve killed myself and two much cooler people. At least I only did it once this time. In the next and final part, I’m going to spend a little more time playing around with far less realistic black holes. (WARNING! Don’t actually play with black holes. If you have to ask why, then you skipped to the end of both articles.)

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physics, Space, thought experiment

Houston, We Have Several Problems: Black Holes, Part 1

There aren’t many movies that I loved as a kid and can still stand to watch as a grownup. I remember loving Beauty and the Beast, but I don’t know if I could watch it now, because I’ve developed an irrational hatred of musical numbers. I certainly couldn’t watch that amazingly cheesy monster truck movie where the guy who drove Snakebite drugged the guy who drove Bigfoot. Apollo 13, though, aged well. It’s high on the list of my favorite space movies. Thanks to that movie (and the much weirder, but still pretty damn good Marooned), this vehicle is comfortably familiar to me:

Apollo_CSM_lunar_orbit.jpg

(From Wikipedia.)

That there is the Apollo 15 command-service module. The command module (astronaut container) is the shiny cone at the front. Because I had cool parents, I got to go to the Johnson Space Center as a kid and see an actual command module, in person. As far as spacecraft go, it’s pretty small. If you stole one, you could hide it in your average garage. Now there‘s a good scene for a movie: some mad scientist with a stolen command module in his garage, tinkering with it while 80’s electronic montage music plays. Turning it into a time machine or something.

Once again, I imagine many of you are wondering what the hell I’m talking about. Don’t fret, there’s always (well, almost always) a method to my madness. Earlier today, I got thinking about the classic thought experiment: What would it be like to fall into a black hole. That thought experiment normally involves just dropping some hapless human (usually without even a spacesuit) right into one. You guys know me, though: if I’m going to do a weird thought experiment, I’m going to go into far more detail than necessary. That’s the fun part!

The point is, I’m going to start off my series on black holes by imagining what would happen if you dropped a pristine command module into black holes of various masses. The command module needs a three-person crew, and my crew will consist of myself, a young David Bowie, and Abraham Lincoln. Despite what you may think, no, I’m not on drugs. This is all natural, which, if you think about it, is much scarier.

A Mini Black Hole

As XKCD pointed out, a black hole with the mass of the Moon would not be a terribly dramatic-looking object. Even accounting for gravitational lensing, it would be next to impossible to see unless you were close to it, or you were looking really hard, or you had an X-ray/gamma-ray telescope. Nobody knows for sure whether black holes with masses this small actually exist. They might possibly have formed from the compression of the high-density plasma that filled the early universe, but so far, nobody’s spotted their telltale radiation.

But me, Bowie, and Lincoln are about to see one, and far too close.

At 1,700 kilometers, we’ve already gone from flying to falling. There’s nothing exciting happening, though, because as physicists always say, from a reasonable distance, a black hole’s gravitational field is no different from the gravitational field of a regular old object of the same mass. The tidal acceleration (which is the real killer when it comes to black holes) amounts to less than a millionth of a gee. Detectable, but only just.

At 100 kilometers, the tides are noticeable. Lincoln’s top-hat (which he foolishly left untethered. There wasn’t time to explain space flight and free-fall to a 19th-century politician) is slowly migrating to the end of the cabin. The tidal acceleration (the difference in pull between the center of the CM and either of its ends) is similar to the gravity of Pluto. The command module is stretching, but no more than, say, a 747 flexes in flight. Only detectable with fancy things like strain gauges.

At 10 km, we’re really motoring. Traveling at 31 kilometers per second, we’ve broken the record set by Apollo 10 for the fastest-moving humans. Bowie’s crazy Ziggy Stardust hair is getting misshapen by differential acceleration, which by now amounts to almost a full gee. Anything untethered (Lincoln’s hat, Bowie’s guitar, my cup of coffee, et cetera) is stuck at either end of the cabin.

At 5 km, we’re all shrieking in pain. The stray objects at the ends of the cabin are starting to get smashed. We’re all being stretched front-to-back (since that’s how you sit in a command module). The tides are pulling the command module like a rubber band, but a command module’s a compact and sturdy thing, so apart from some alarming creaking of metal, and maybe some cracking in the heat shield, it’s still in one piece.

All three of us die very quickly not long after the 2.5 km mark. Since the university still won’t let me use their finite-element physics package (well, more accurately, they won’t let me in the physics department…), I can only conjecture what will kill us, but it’ll either be the fact that the blood in our backs is being pulled toward the black hole much harder than the blood at our fronts (probably turning us very nasty colors and causing lots of horrible hemorrhages), or the fact that the command module has been stressed beyond its limits and sprung a leak. I’d wager the viewports would shatter before anything else happened, as their frames start to bend out of shape.

Time dilation hasn’t really kicked in, even by 500 meters, so an observer at a safe distance would see the CM crumple in real-time. The cone collapses like an umbrella being closed. Fragments of broken glass, shards of metal, control panels, shattered heat shield, and pieces of a legendary rocker, a melancholic President, and an idiot are spilling out of the wreckage.

By 10 meters, the command module is no longer falling straight down. It’s started falling inward, compressing side-to-side even as it stretches top-to-bottom. The individual components and fragments cast off from the wreck stretch and pull apart. Metal panels tear in two. Glass shards crack. Bits of flesh tear messily in half. The fragments divide and divide and divide. As they approach the event horizon, they explode into purple-white incandescent plasma: because the atoms are falling inward towards a point, they slam sideways into each other at high speeds. All 13,000 pounds of command module and crew either help bulk up the black hole, or form a radiant accretion disk denser than lead and smaller than a grape.

A 5-Solar-Mass Black Hole

In black hole thought experiments, the starting-point is usually a 1-solar-mass black hole. That makes sense: as far as astronomical objects go, the Sun is nicely familiar. But like I said before, scientists aren’t sure if there are any 1-solar-mass black holes anywhere in the Universe. As far as we know, all black holes in this mass range form from stars, and supernova leftovers smaller than something like 3 solar masses can still be supported by things like radiation pressure, degeneracy pressure, and the fact that atoms don’t like other atoms too close to them. In practice, there aren’t any known black hole candidates smaller than about 3 solar masses. There are a couple around 5 solar masses, and 5 is a nice round number, so that’s what I’m going with. We’re resetting this weird-ass experiment and dropping me, Bowie, and Lincoln into a stellar black hole.

At 380,000 kilometers (the distance from the Earth to the Moon), the tidal acceleration is detectable, but not noticeable. Good thing we’re free-falling (I should’ve made Tom Petty the third crewmember…), because if we were held stationary (say, by a magic platform hovering at a fixed altitude above the hole), we’d be flattened by a lethal 469-gee acceleration. Good thing, too, that we’re in an enclosed spacecraft: if the black hole has an accretion disk orbiting around it, we’re probably close enough that an unshielded human would be scalded to death by its heat, light, and X-rays. For our purposes, though, we’ll assume this black hole’s been floating through interstellar space for a long time, and has already cannibalized its own accretion disk, rendering it almost dark.

As we reach 10,000 km from the black hole, the same thing happens that happened with the mini black hole. David Bowie, who has gotten out of his seat to have a second tube of strawberry-banana pudding, finds it difficult to climb back to his couch against the gravity gradient. He’s being gently pelted by loose objects. Lincoln is just sitting in his couch looking very grave. I’m screaming my head off, so I miss all of this.

At 5,000 kilometers, the command module starts to creak. David Bowie is now stuck upside-down at the top of the cabin. I, having lost my shit and tried to open the hatch to end it quickly, have fallen to the bottom and broken my coccyx. Lincoln is still sitting in his couch looking very grave. We’re experiencing a total acceleration of over a million gees. If we tried to maintain a constant altitude, that million gees would turn us and the command module into a sheet of very thin and very gory foil. We’re moving almost as fast as the electrons shot from the electron gun of an old CRT TV.

Between 5,000 and 1,000 kilometers, the command module starts popping apart. It’s not as fast as the last time. First, the phenolic plastic of the heat shield cracks and pulls free of the insulation beneath. Then, the circular perimeter of the cone starts to crumple and wrinkle. (True story: the command module was built with crumple zones, just like a car, so that it didn’t pulp the astronauts too much when it hit the ocean at splashdown.) Not long after, the pressure hull finally ruptures, spewing white jets of gas and condensation in all directions like a leaky balloon. Then, the bottom of the pressure hull bursts. Think of a sledgehammer hitting a sheet of aluminum foil. All the guts of the command module spill out: wires, seats, guitars, apples (I was hungry), tophats, shoes, hoses, spare spacesuits, screaming idiots (I fell out). We’re moving 10% of the speed of light.

By 100 kilometers, the command module has spaghettified into a long stream of debris. The individual metal parts, although badly warped by being torn from their mountings, are mostly holding together, though they’re really starting to stretch. Anything softer is shattering/pulping/shredding. The black hole’s event horizon is the largest object in the sky: a fist-sized black disk of nothingness surrounded by a very pretty mandala of distorted stars and galaxies. It looks something like this:

scr00004

(Screenshot from the unbelievably awesome (and free) program Space Engine.)

We can’t see it, though: we’re all dead.

By 25 kilometers, we’re just a stream of fine dust hurtling towards the event horizon at close to the speed of light. For an observer at a great distance, our disintegration proceeds in slow motion, both from the massive speed at which we’re traveling, and from the time-dilating effects of extreme gravity.

As we scream through 14.8 kilometers, we’ve almost reached the event horizon. The individual atoms the command module used to be made of are accelerating apart, spraying the whole CM into a narrow stream of plasma. Outside observers, though, just see the incandescent dust slow to a halt, change color from electric-arc purple to brilliant blue-white to the color of the sun to the color of hot steel to red-hot to black. What happens when we hit the singularity not long after is anybody’s guess. By definition, at a singularity, the equations you’re working with just quit making sense.

Sagittarius A*

There’s a very massive and very dense thing at the center of the Milky Way. It has about 3.6 million times the mass of the Sun, and because there’s a star (poetically called S0-102) that orbits pretty close to it (relatively speaking, anyway: its closest-approach distance is still larger than the distance from the Sun to Pluto), we know it has to be quite small, and therefore quite dense. According to our current understanding of physics, any mass like that would inevitably collapse into a black hole no matter what. The short version: it’s probably a black hole. (Note 1: as of this writing in November 2016, radio astronomers have finally committed to using a gigantic virtual telescope to take a picture of the actual event horizon in 2017, which is awesome) (Note 2: Though the actual mechanism for their formation isn’t known, some astrophysicists have done simulations suggesting that they formed from super-massive stars in the early universe. These days, the largest stars are a few hundred solar masses, with the largest stars for which we have firm evidence weighing around 120 solar masses. That’s massive, but not super-massive. These super-massive primordial stars contained thousands of solar masses. The one in that article massed 55,500. Some may have exceeded a million.)

Because a black hole (or, at least, its event horizon) is as compact as you can make anything, black holes tend to be really small compared to normal objects of similar mass. A Moon-mass black hole would look like a black dust-grain. An Earth-mass one would look like a pea. A Sun-mass one (and remember, there’s a lot of stuff in the Sun) would be the size of a small town. The 5-solar-mass hole we considered a second ago would be the size of a city. Sagittarius A*, though, containing so much mass, is actually a proper astronomical-sized object: 15 times larger than the Sun. If some deity with a really sick sense of humor replaced the Sun with Sgr A*, it would hang in the sky, a little smaller than a fist at arm’s length. We would also all be dead. For many reasons: no sunlight, radiation from the accretion disk, and the fact that we’d be orbiting so fast that grains of dust would heat the upper atmosphere lethally hot.

At 1 AU (the average distance from the Earth to the Sun), Sgr A*’s event horizon is much larger in the sky than the Moon. We’re accelerating at 1800 gees, but we don’t feel it, because we’re in free-fall. The tidal acceleration is minuscule: less than a micron per second per second. Once again, we’re pretending the black hole has no accretion disk, because if it did, its radiation would probably have incinerated us by now.

By the time we pass through Mercury’s orbit (0.387 AU) (assuming we actually are in this nightmarish black-hole solar system), we’re going one-third the speed of light, accelerating at over 15,000 gees. The tides are still detectable only by specialized instruments.

By 0.1 AU, we’re moving three-quarters the speed of light. David Bowie is singing “Space Oddity”, because I’ve smuggled a durian fruit onboard and threatened to cut it open if he doesn’t. Lincoln is starting to get sick of this shit, but this just gives him that same grave expression he has in all his photographs. The tides are detectable by instruments, but probably not by our human senses. Hitting a stationary dust particle the size of a bacterium unleashes a burst of light as bright as a studio flash.

At 0.071 AU, we pass through the event horizon without even realizing it. Falling into a black hole intact is a little like having a gigantic black bag closed around you: the event horizon already covers more than half of the sky, thanks to the fact that the black hole bends light toward the horizon. The sky shrinks into an ever-diminishing circle in a black void. The circle grows brighter and bluer with every passing second.

By 0.05 AU, stray objects are drifting to the ends of the cabin again: the tides are finally picking up. David Bowie is holding me down and punching me repeatedly, because he’s sick of me resurrecting him and killing him over and over. Lincoln is letting him do it, because frankly, I’ve cracked his statesman’s patience with my bullshit. We’re only 30 seconds from the singularity.

Even at 0.01 AU, the tides aren’t stretching the capsule. The effects might be palpable, but they’re nothing compared to what we’ve already been through in the previous experiments. We’re riding down a shaft of blue-shifted light, concentrated into a point straight overhead: everything that’s fallen into the hole recently can’t help but curve inward, until it’s falling almost ruler-straight towards the singularity.

At one Sun radius from the singularity, the differential acceleration is approaching one gee. Things are starting to get uncomfortable. David Bowie has stopped punching me, because he’s fallen to the top of the capsule again. Lincoln, though, still has the strength to pick up a pen and stab me in the sternum. He’s cursing at me, and as I start to bleed to death, I observe that Lincoln is much more creative with his swears than I would have given him credit for.

At 150,000 km, we start to get woozy on account of the blood in our bodies pooling in all the wrong places. We narrowly avoid a collision with Matthew McConaughey in a spacesuit, who has gone from muttering about quantum data to describing the peculiar aging patterns of high-school girls.

At 50,000 km, moving very, very nearly the speed of light, the command module finally starts to disintegrate. Seconds later, we spaghettify, just as before, and strike the singularity. And, once again, we run afoul of the fact that physicists have very little idea what happens that deep in a gravity well. For reference, at 100 meters from the singularity (and ignoring relativistic effects and pretending we can use the Newtonian equation for tides down here), the differential acceleration is measured with twenty-digit numbers. If the capsule were infinitely rigid and didn’t spaghettify, by the time its bottom touched the singularity, the tides would be measured in 25-digit numbers. If, somehow, we’d survived our trip to the singularity, we’d be accelerating so fast that, thanks to the Unruh effectempty space would be so hot we’d instantly vaporize.

And here’s where physics breaks down. If I’m reading this paper right, the distance between any point and the singularity is infinity, because space-time is so strongly curved near it.

Imagine that space is two-dimensional. It contains two-dimensional stars with two-dimensional mass. That curves two-dimensional space into a three-dimensional manifold. The gravity well (technically, the metric) around a very dense (but non-black-hole) looks roughly like this:

GravityPotential.jpg

(From Wikipedia.)

If you measure the circumference of the object, you can calculate its diameter: divide by two times pi. But when you measure its actual diameter, you’ll find it’s larger than that, because of the way strong gravitation stretches spacetime. In the case of a black hole, spacetime looks more like this:

Fig1.png

(From the paper cited above.)

The cylindrical part of the trumpet is (if I’m understanding this correctly) infinitely long. The “straight-line” distance through the black hole, on a line that just barely misses the singularity, is much larger than you’d expect. But the distance through the black hole, measured on a line that hits the singularity is infinite. All lines that hit the singularity just stop there.

But, to be honest, I really don’t know what it’d be like down there. Nobody does. The first person to figure out what gravity and particle physics do under conditions like that will probably be getting a shiny medal from some Swedes.

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All kinds of explosions.

As you’ve probably worked out by now, I’m a big fan of explosions. Since this ridiculous blog began, I’ve blown up rabbits (don’t worry–hypothetical rabbits), stellar-mass balls of gold, grains of neutronium, and I’ve nearly blown up the earth with an ultrarelativistic BB gun. As you’ve probably also figured out, I’m a big fan of bizarre thought experiments, thought experiments based on ideas boiled and distilled down to their absolute essentials. With that in mind, let’s blow some more shit up!

But before we can get started, we have to decide what an explosion is. For the purpose of this thought experiment, we’re going to use a one-meter-diameter sphere of space anchored to the surface of the Earth, just touching the ground. The explosions will consist of energy (in the form of photons of an appropriate wavelength) magically teleported into this sphere. (I still haven’t decided what symbol to use for the magic teleporter in my Feynman diagrams.) With that cleared up, let’s get blastin’.

The smallest possible explosion.

It’s pretty difficult to decide on the lower limit for the size of a one-meter-diameter explosion. First of all, that one-meter sphere is already full of all kinds of energy: solar photons, the kinetic energy of air molecules moving at high speeds, the rest-mass energy of those air molecules, et cetera. But even if the sphere were completely evacuated, quantum mechanics tells us that the lowest amount of energy that a volume of space can possess is greater than zero: its zero-point energy or vacuum energy.

To simplify things (and to keep me from having to learn the entirety of gauge field theory while writing a blog post), we’re going to say that the lowest-energy explosion we can create has the energy of the longest-wavelength (and therefore lowest-energy) photon we can fit in the sphere: 1 meter. That’s towards the high-frequency (short-wavelength) end of the radio spectrum. It’s not quite a microwave (those have wavelengths on the order of a centimeter), but it’s shorter than the photons used to transmit FM radio signals. Needless to say, it would impart a pretty much un-measurable quantity of energy to our sphere of air. A 1-meter photon carries 1.986×10e-25 joules of energy, the same energy as an oxygen molecule puttering along at a grandmotherly 6 miles per hour (10 kph).

The most efficient possible explosion.

But while we’re using our magic vacuum-fluctuation laser (lots of magic in this article…), we might as well see how big an explosion we can make with a single photon. We know the minimum is 1.986×10e-25 joules. But what’s the highest-energy photon we can stick in there? (Sounds like a plot to a horrible science porno…) I’m not a physicist, but I would guess that it’s a photon with a wavelength of 1 Planck length. Here’s my logic: the Planck length is the smallest length that makes sense according to our current laws of physics. To carry energy, an electromagnetic wave must change over time (or space, which work out to be part of the same thing). In order to take two different values, the wave’s crest and trough must be at least 1 Planck length apart. Of course, weird things happen on the Planck scale, so who knows if an electromagnetic field varying by a large quantity over 1 planck length would even make sense, or even behave like a photon, but you can say this with some certainty: a photon with a wavelength shorter than that doesn’t make a lot of sense.

A photon with a wavelength this short would be off-scale, as far as the electromagnetic spectrum goes. By definition, it would be a hard gamma ray, but that’s only because we humans don’t have access to the high-energy regions of the spectrum, so we say “Anything with a wavelength between A and B is an X-ray. Anything with a wavelength between B and zero is a gamma ray. Stupid gamma rays. Who needs ’em?”

This is a gross oversimplification, but photons tend to prefer to interact with objects that are roughly their same size. Visible photons interact with the electron clouds of  large molecules (like chlorophyll, which is good if you like oxygen). Infrared photons interact with large molecules directly, making them rotate or move. Radio-frequency photons interact only with big crowds of mobile electrons, like you find in plasma or radio antennas. In the other direction, ultraviolet photons interact with atoms and their bonds, either knocking outer electrons loose or breaking the bonds. X-ray photons interact with the tightly-bound electrons in lower energy states (I’m afraid to say “closer to the nucleus,” because that’s not quite right, and people smarter than me will make fun of me; but we’ll pretend that’s how it works to speed things along). Gamma-ray photons interact with the nucleus directly. They can push the nucleus into strange high-energy states or, if they’re the right wavelength, pelt protons and neutrons loose.

The physics of extremely short-wavelength gamma rays is much more complicated. That’s partly because, once a gamma ray passes an energy greater than 1022 keV (the kinds of energies you get when you heat something to several billion Kelvin) have enough mass-energy that they can spontaneously turn into matter: a 1023 keV gamma ray can briefly become an electron and a positron, which rapidly annihilate to re-form the gamma ray. The higher your photon’s energy gets, the more important these bizarre transformations become. Basically, once you get above 1022 keV, your photons start behaving less and less like light and more and more like matter.

Our Planck-length photon would (probably) be small enough to pass through all ordinary matter. It might pop a quark loose of a proton or a neutron, but I don’t really know. This photon would carry an energy far greater than 1022 keV. Indeed, its energy wouldn’t be measured in the peculiar particle-physics unit of the electronvolt, but rather in freakin’ megawatt-hours. This photon would add as much energy to our sphere as the explosion of three tons of TNT, which would form a cube large enough to contain our sphere with room to spare. From one…single…photon. Physics is scary.

Explosions, both conventional and nuclear (ain’t I posh?).

If you Google the phrase “largest conventional explosion,” you’ll probably dig up an article on Operation Sailor Hat, in which the U.S. Navy built a 500-ton Minecraft-style sphere of TNT:

It made a real mess of the decommissioned test ships anchored just offshore, and left a crater in Hawai’i that still exists to this day. It remains one of the largest intentional non-nuclear explosions humans have ever created. This isn’t relevant to our thought experiment, but I’m easily distracted by 500-ton piles of TNT.

In everyday life, a “conventional explosion” is one created by a chemical reaction of some kind. This ranges from the rapid burning of a lot of wheat dust in a grain silo to the bizarre high-tech cube-shaped molecule octanitrocubane. An explosion that gets most of its power from the fission of radioactive elements (or the fusion of light elements) is a nuclear explosion. Simple.

But you can think of it another way: the difference between conventional chemical explosions and nuclear explosions is a matter of temperature. Explosives like TNT produce gases with temperatures of thousands of degrees Kelvin. Nuclear explosions produce temperatures in the million to hundred million Kelvin range.

Since we’ve got a magic sphere of air we can pump energy into (how convenient!), let’s make one of each kind of explosion!

For the conventional explosion, we’ll heat the air to a temperature of 5,000 Kelvin (about as hot as the surface of the sun). Our sphere has a volume of 0.52 cubic meters, and air has a volumetric heat capacity of about 0.001 joules per cubic centimeter per Kelvin. Therefore, heating the air to 5,000 kelvin will require 2.6 million joules. We’re talking about a decent-sized bang here, equivalent to just over half a kilogram of TNT. Not exactly spectacular, but certainly not the kind of explosion you want to hold in your hand. (Actually, thinking about it, why would you want to hold any explosion in your hand? That was silly of me.)

For the nuclear explosion, we’ll heat our sphere to 10 million Kelvin, which is a compromise, since nuclear explosions are complicated beasties and their temperature varies very rapidly in the span of microseconds. I should point out that I’m still using a volumetric heat capacity of 0.001 joules per cubic centimeter per Kelvin, which is absurd. At these temperatures, we’re not just making molecules move faster–we’re breaking them apart. And blasting electrons off the individual atoms. Both of these things take energy, so heating the plasma to 10 million Kelvin will actually require more energy input than my idealized equation suggests. But I digress. We’re looking at an explosive energy of about 5.2 billion Joules, which is about one and a quarter tons of TNT. I’m disappointed. Even the smallest nuclear weapon ever made had a yield larger than that. (Incidentally, the smallest nuclear weapon ever made (by the U.S., at least) was the W54 warhead. This is what they used in the “Davy Crockett,” which was a nuclear warhead launched from a fucking bazooka. Think about that.)

But the real temperature of a young nuclear fireball (good name for a band) is probably higher than this. After the fission of the nuclear material is complete, there’s a brief period where the nuclear explosion is mostly made up of high-energy nuclear radiation (hard x-rays, gamma rays, and particles). Then, it gets absorbed by the evaporating bomb casing and the surrounding air and (mostly) turns into heat. It’s not unreasonable to assume that the temperature in a baby nuclear fireball (bad idea for a child’s doll) is closer to 100 million Kelvin, which works out to almost twelve and a half tons of TNT. Still not nuclear in the traditional sense, but certainly more than enough to bust up a whole city block.

Turning it up to 10.

I’m tired of messing around. I want a real boom. We’re sticking a full megaton into that sphere, and damn the consequences!

Well, with energy densities this high, you can’t really even pretend that the volumetric heat capacity equation applies. Even at 100 million Kelvin, we were getting into the region where atoms are stripped of most or all of their electrons, which is where matter stops behaving even remotely the way it does at ambient temperatures. Much hotter, and we’re going to start splitting the atoms themselves.

Instead, to describe the conditions in our magic sphere when 1 megaton is pumped into it, I’m going to make two assumptions: the energy is deposited over the course of 1 nanosecond, and the superheated sphere radiates like a thermodynamic blackbody, which is to say the same way red-hot steel does and stars (mostly) do.

An energy release of 1 megaton over 1 nanosecond gives us 4.184 trillion trillion watts of radiated power. The sphere will very briefly shine as bright as a small star, and will have a surface temperature of 69 million Kelvin. Then, ba-boom, say goodbye to your city. Next time you build a city, don’t go letting people like me build magic energy-spheres in the middle. I hope you’ve learned your lesson.

This thing goes to 11.

Sorry. Couldn’t help myself.

Let’s stick in 50 megatons now. That’s about the energy released by the Tsar Bomba, a Soviet hydrogen bomb that produced a mushroom cloud that rose above the top of the stratosphere. For comparison, really impressive explosions and big thunderstorms and volcanic eruptions and such are lucky if they can make it into the stratosphere at all, let alone breaking all the way through the damn thing.

I can do all the math here just like before: 210 trillion trillion watts, a temperature of 185 million Kelvin, yadda yadda yadda. The thing is, as I wrote in “A Piece of a Neutron Star“, up to a point, an explosion is an explosion. The energy of an explosion gets spread over a large volume fairly quickly, and ordinary fluid dynamics and thermodynamics take over. Ever notice how a conventional explosive produces a mushroom cloud that looks an awful lot like the mushroom from a nuclear explosion? That’s because both are powered by the same thing: a rising plume of hot air. While it’s true that in the nuclear explosion there’s a lot more air and it’s a hell of a lot hotter, it’s easy to see that the two are related. They’re part of the same spectrum. A nuclear explosion is like a conventional explosion, only it’s larger, and it produces a lot more radiant heat.

This thing goes to 111.

This scaling law (which real physicists have investigated in great detail) seems to hold for larger and larger energies. Small asteroid impacts create explosions that are very much like nuclear explosions (with some extra effects from high-velocity ejecta and from the entry trail and so on). Even impacts like Chicxulub (which may or may not have helped kill off the dinosaurs, but certainly ruined everybody’s day for several thousand years) produce a fairly ordinary shockwave, and then a fireball that reaches an enormous, but reasonable, size before cooling to ordinary temperatures. The excellent Impact Effects Calculator tells me that a Chicxulub-like impact would produce a visible fireball with a radius of about 66 kilometers. This would reach up through the stratosphere, and about halfway to the edge of space, so it would probably be flattened, with a blurry, undefined edge at the top, but it would still be very much like what it is: a 100 million megaton explosion.

But like all scaling laws, the explosion spectrum eventually gives out. Once you start imagining blasts with the same kinetic energy as, say, a 100-kilometer-wide stone asteroid, you’ve passed over a threshold. Beyond this threshold, the assumptions that let us imagine our earlier explosions break down. Assumptions like “Shockwaves and heat plumes travel through the atmosphere, but the atmosphere doesn’t go flying off or anything” and “The crust is firmly attached to the rest of the Earth.” (It’s nice that we live in a time when we can make assumptions like that. Not that we could live in a time when the crust wasn’t attached, but it’s nice to know we (probably) have that kind of stability). We move out of the realms of meteorology and geology and into the realms of astrophysics. When you talk about 100-kilometer asteroid impacts, you’re no longer talking about “an asteroid hitting the Earth.” You’re talking about “two celestial bodies colliding.” Things like organic life and atmospheres crumble (and burn and evaporate) before energies like this.

Which is a (very) roundabout way of saying that a 90-billion-megaton explosion isn’t really an explosion anymore. It’d blow the atmosphere off a pretty big portion of the planet, peel back the crust, and pave some or all of the Earth in magma. Larger explosions could put sizeable dents in the Earth, or blow off enough material to push it into a different orbit. Not that we care: we’ll all be dead. Again. Sorry. Planet-killing is a bad addiction to have.

Hey! Where’d my explosion go?

This transition from fluid dynamics to astrodynamics has an ultimate limit. You can only squeeze so much energy into a 1-meter-diameter sphere and have it come back out. For this, you have Albert Einstein and Karl Schwarzschild to thank.

Energy and mass are equivalent. Not only can one be converted into the other, but they also produce and react to gravity in the same way. Because of this, if you try to cram more than 3.029e43 Joules (7239 trillion trillion megatons) of energy into a sphere 1 meter across, you won’t get an explosion at all. Your energy, no matter what form it’s in, will vanish behind an event horizon. This is bad news for the Earth. Partly because this energy will weigh 56 times as much as the Earth (over a fifth the mass of Jupiter), and will therefore screw up its orbit and either freeze us or boil us. Of course, the slow and painful destruction of the entire human race, all of our infrastructure and achievements, everything you and I and everybody else cares about, and also all life on earth, is the least of our problems. Because now we’ve got a 1-meter black hole sitting in the middle of a public park somewhere (I moved it away from downtown, to be nice. Sorry.) The Earth will swirl into the hole like a sinkful of water down a drain. That’s not just me failing to be poetic, that’s about how it’ll be. The whole Earth wont’ fall into the hole in an instant, nor will it be instantly spaghettified. Remember that, until you get close to it, a black hole behaves like any other object of the same mass. So the bulk of the Earth will fall towards the hole (thereby putting the hole at the middle of the planet) and everything else will collapse around it. We will, briefly, have a planet cracked and hollowed like a broken Kinder egg. Then we will have a ball of incandescent gas around a tiny black hole. Then we will have an accretion disk whose molecules do not resemble the farmers, sailors, priests, road signs, eggs benedict, and fire ants they were once part of. This is the point at which the explosion stops being an explosion. This is the limit.

Hey, this thing goes to 10.999.

That’s no fun. Well, I tell a lie–black holes are great fun to play with. (From a large distance.) But we want explosions, not black holes! We can talk about black holes some other time (hint hint). What if we put a little less than 3.029e43 Joules into our magic sphere? If that happens, boy oh boy do we get some fireworks!

The first thing that’ll happen is that the spacetime around the sphere will go from the gentle curvature (gravity) produced by the Earth, Sun, and other celestial bodies to a violent light-bending time-warping curvature. This does not end well for us. (why would you think it would? You’re silly.) A single powerful pulse of gravity waves races out at the speed of light, turning the Earth to gravel in a few milliseconds. Right on its heels comes a wave of radiation with almost the power of a supernova (almost, as in, 0.3 times as energetic as a supernova. Serious shit.) The Earth does not have time to fall into the region of ridiculous energy density and turn it into a black hole. That’s because the Earth is busy turning into a pancake of purple-white plasma and racing outwards at high speed. If the pancake hits anything, the flash will destroy the solar system. (Which sounds like it should be a line from Spaceballs). If it doesn’t hit anything, the supernova will destroy the solar system.

The moral of today’s story is: Don’t trust a man who says he has a magic sphere. He might kill everything.

The other moral of today’s story is: If you study a little physics, you can take any thought experiment to its absolute logical limit. Which, as I’m discovering, is pretty damn fun.

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A great big pile of money.

When I was little, there was always that one kid on the playground who thought he was clever. We’d be drawing horrifying killer monsters (we were a weird bunch). I would say “My monster is a thousand feet high!” Then Chad would say “My monster is a mile high!” Then I would say “Nuh-uh, my monster is a thousand miles high!” Then Taylor would break in, filling us with dread, because we knew what he was going to say: “My monster is infinity miles high!” There would then follow the inevitable numeric arms race. “My monster is infinity plus one miles high!” “My monster is infinity plus infinity miles high!” “My monster is infinity times infinity miles high!” Our shortsighted teachers hadn’t taught us about Georg Cantor, or else we would have known that, once you hit infinity, pretty much all the math you do just gives you infinity right back.

But that’s not what I’m getting at here. As we got older and started (unfortunately) to care about money, the concept of “infinite money” inevitably started coming up. As I got older still and descended fully into madness, I realized that having an infinite amount of printed money was a really bad idea, since an infinite amount of mass would cause the entire universe to collapse into a singularity, which would limit the number of places I could spend all that money. Eventually, my thoughts of infinite wealth matured, and I realized that what you really want is a machine that can generate however much money you want in an instant. With nanomachines, you could conceivably assemble dollar bills (or coins) with relative ease. As long as you didn’t create so much money that you got caught or crashed the economy, you could live really well for the rest of your life.

But that’s not what I got hung up on. I got hung up on the part where I collapsed the universe into a Planck-scale singularity. And that got me thinking about one of my favorite subjects: weird objects in space. I’ve been mildly obsessed with creating larger and larger piles of objects ever since. Yes, I do know that I’m weird. Thanks for pointing that out.

Anyway, I thought it might be nice to combine these two things, and try to figure out the largest pile of money I could reasonably accumulate. My initial thought was to make the pile from American Gold Eagle coins, but I like to think of myself as a man of the world, and besides, those Gold Eagles are annoyingly alloyed with shit like copper and silver, and I like it when things are pure. So, instead, I’m going to invent my own currency: the Hobo Sullivan Dragon’s Egg Gold Piece. It’s a sphere of 24-karat gold with a diameter of 50 millimeters, a mass of 1,260 grams, and a value (as of June 29, 2014) of $53,280. The Dragon’s Egg bears no markings or portraits, because when your smallest unit of currency is worth $53,280, you can do whatever the fuck you want. And you know what? I’m going to act like a dragon and pile my gold up in a gigantic hoard. But I don’t want any arm-removing Anglo Saxon kings or tricksy hobbitses or anything coming and taking any of it, so instead of putting it in a cave under a famous mountain, I’m going to send it into space.

Now, a single Dragon’s Egg is already valuable enough for a family to live comfortably on for a year, or for a single person to live really comfortably. But I’m apparently some kind of ridiculous royalty now, so I want to live better than comfortably. As Dr. Evil once said, I want one billion dollars. That means assembling 18,769 Dragon’s Eggs in my outer-space hoard. Actually, now that I think about it, I’m less royalty and more some kind of psychotic space-dragon, which I think you’ll agree is infinitely cooler. 18,769 Dragon’s Eggs would weigh in at 23,649 kilograms. It would form a sphere with a diameter of about 1.46 meters, which is about the size of a person. Keen-eyed (or obsessive) readers will notice that this sphere’s density is significantly less than that of gold. That’s because, so far, the spheres are still spheres, and the closest possible packing, courtesy of Carl Friedrich Gauss, is only 74% sphere and 26% empty space.

You know what? Since I’m being a psychotic space-dragon anyway, I think I want a whole golden planet. Something I can walk around while I cackle. A nice place to take stolen damsels and awe them with shining gold landscapes.

Well, a billion dollars’ worth of Dragon’s Eggs isn’t going to cut it. The sphere’s surface gravity is a pathetic 2.96 microns per second squared. I’m fascinated by gravity, and so I often find myself working out the surface gravity of objects like asteroids of different compositions. Asteroids have low masses and densities and therefore have very weak gravity. The asteroid 433 Eros, one of only a few asteroids to be visited and mapped in detail by a space probe (NEAR-Shoemaker), has a surface gravity of about 6 millimeters per second squared (This varies wildly because Eros is far from symmetrical. Like so many asteroids, it’s stubbornly and inconveniently peanut-shaped. There are places where the surface is very close to the center of mass, and other places where it sticks way up away from it.) The usual analogies don’t really help you get a grasp of how feeble Erotian gravity is. The blue whale, the heaviest organism (living or extinct, as far as we know) masses around 100 metric tons. On Earth, it weighs 981,000 Newtons. You can also say that it weighs 100 metric tons, because there’s a direct and simple equivalence between mass and weight on Earth’s surface. Just be careful: physics dorks like me might try to make a fool out of you. Anyway, on Eros, a blue whale would weigh 600 Newtons, which, on Earth, would be equivalent to a mass of 61 kilograms, which is about the mass of a slender adult human.

But that’s really not all that intuitive. It’s been quite a while since I tried to lift 61 kilograms of anything. When I’m trying to get a feel for low gravities, I prefer to use the 10-second fall distance. That (not surprisingly) is the distance a dropped object would fall in 10 seconds under the object’s surface gravity. You can calculate this easily: (0.5) * (surface gravity) * (10 seconds)^2. I want you to participate in this thought experiment with me. Take a moment and either stare at a clock or count “One one thousand two one thousand three one thousand…” until you’ve counted off ten seconds. Do it. I’ll see you in the next paragraph.

In those ten seconds, a dropped object on Eros would fall 30 centimeters, or about a foot. For comparison, on Earth, that dropped object would have fallen 490 meters. If you neglect air resistance (let’s say you’re dropping an especially streamlined dart), it would have hit the ground after 10 seconds if you were standing at the top of the Eiffel Tower. You’d have to drop it from a very tall skyscraper (at least as tall as the Shanghai World Financial Center) for it to still be in the air after ten seconds.

But my shiny golden sphere pales in comparison even to Eros. Its 10-second fall distance is 148 microns. That’s the diameter of a human hair (not that I’d allow feeble humans on my golden dragon-planet). That’s ridiculous. Clearly, we need more gold.

Well, like Dr. Evil, we can increase our demand: 1 trillion US dollars. That comes out to 18,768,769 of my golden spheres. That’s 23,649,000 kilograms of gold. My hoard would have a diameter of 14.6 meters and a surface gravity of 29.6 microns per second squared (a 10-second fall distance of 1.345 millimeters, which would just barely be visible, if you were paying close attention.) I am not impressed. And you know what happens when a dragon is not impressed? He goes out and steals shit. So I’m going to go out and steal the entire world’s economy and convert it into gold. I’m pretty sure that will cause Superman and/or Captain Planet to declare me their nemesis, but what psychotic villain is complete without a nemesis?

It’s pretty much impossible to be certain how much money is in the world economy, but estimates seem to be on the order of US$50 trillion (in 2014 dollars). That works out to 938,438,439 gold balls (you don’t know how hard I had to fight to resist calling my currency the Hobo Sullivan Golden Testicle). That’s a total mass of 1.182e9 kilograms (1.182 billion kilograms) and a diameter of 54 meters (the balls still aren’t being crushed out of shape, so the packing efficiency is still stuck at 74%). 54 meters is pretty big in human terms. A 54-meter gold ball would make a pretty impressive decoration outside some sultan’s palace. If it hit the Earth as an asteroid, it would deposit more energy than the Chelyabinsk meteor, which, even though it exploded at an altitude of 30 kilometers, still managed to break windows and make scary sounds like this:

This golden asteroid would have a surface gravity of 0.108 millimeters per second squared, and a 10-second fall distance of 0.54 centimeters. Visible to the eye if you like sitting very still to watch small objects fall in weak gravitational fields (and they say I have weird hobbies), but still fairly close to the kind of micro-gravity you get on space stations. I can walk across the room, get my coffee cup, walk back, and sit down in 10 seconds (I timed it), and my falling object would still be almost exactly where I left it.

Obviously, we need to go bigger. Most small asteorids do not even approach hydrostatic equilibrium: they don’t have enough mass for their gravity to crush their constituent materials into spheres. For the majority of asteroids, the strength of their materials is greater than gravitational forces. But the largest asteroids do start to approach hydrostatic equilibrium. Here’s a picture of 4 Vesta, one of the other asteroids that’s been visited by a spacecraft (the awesome ion-engine-powered Dawn, in this case.)

(Image courtesy of NASA via Wikipedia.)

You’re probably saying “Hobo, that’s not very fucking spherical.” Well first of all, that’s a pretty damn rude way to discuss asteroids. Second of all, you’re right. That’s partly because of its gravity (still weak), partly because its fast rotation (once every 5 hours) deforms it into an oblate spheroid, and partly because of the massive Rheasilvia crater on one of its poles (which also hosts the solar system’s tallest known mountain, rising 22 kilometers above the surrounding terrain). But it’s pretty damn spherical when you compare it to ordinary asteroids, like 951 Gaspra, which is the shape of a chicken’s beak. It’s also large enough that its interior is probably more similar to a planet’s interior than an asteroid’s. Small asteroids are pretty much homogenous rock. Large asteroids contain enough rock, and therefore enough radioactive minerals and enough leftover heat from accretion, to heat their interiors to the melting point, at least briefly. Their gravity is also strong enough to cause the denser elements like iron and nickel to sink to the center and form something approximating a core, with the aluminosilicate minerals (the stuff Earth rocks are mostly made of) forming a mantle. Therefore, we’ll say that once my golden asteroid reaches the same mass as 4 Vesta, the gold in the center will finally be crushed sufficiently to squeeze out the empty space.

It would be convenient for my calculations if the whole asteroid melted so that there were no empty spaces anywhere. Would that happen, though? That’s actually not so hard to calculate. What we need is the golden asteroid’s gravitational binding energy, which is the amount of energy you’d need to peel the asteroid apart layer by layer and carry the layers away to infinity. This is the same amount of energy you’d deposit in the asteroid by assembling it one piece at a time by dropping golden balls on it. A solid gold (I’m cheating there) asteroid with Vesta’s mass (2.59e20 kg) would have a radius of 147 kilometers and a gravitational binding energy of about 1.827e25 Joules, or about the energy of 37 dinosaur-killing Chicxulub impacts. That’s enough energy to heat the gold up to 546 Kelvin, which is less than halfway to gold’s melting point.

But, you know what? Since I don’t have access to a supercomputer to model the compressional deformation of a hundred million trillion kilograms of close-packed gold spheres, I’m going to streamline things by melting the whole asteroid with a giant draconic space-laser. I’ll dispense with the gold spheres, too, and just pour molten gold directly on the surface.

You know where this is going: I want a whole planet made of gold. But if I’m going to build a planet, it’s going to need a name. Let’s call it Dragon’s Hoard. Sounds like a name Robert Forward would give a planet in a sci-fi novel, so I’m pleased. Let’s pump Dragon’s Hoard up to the mass of the Earth.

Dragon’s Hoard is a weird planet. It has the same mass as Earth, but its radius is only 66% of Earth’s. Its surface gravity is 22.64 meters per second squared, or 2.3 earth gees. Let’s turn off the spigot of high-temperature liquid gold (of course I have one of those) for a while and see what we get.

According to me, we get something like this:

Gold Tectonics

The heat content of a uniform-temperature sphere of liquid gold depends on its volume, but since it’s floating in space, its rate of heat loss depends on surface area (by the Stefan-Boltzmann law). The heat can move around inside it, but ultimately, it can only leave by radiating off the surface. Therefore, not only will the sphere take a long time to cool, but its upper layers will cool much faster than its lower layers. Gold has a high coefficient of thermal expansion: it expands more than iron when you heat it up. Therefore, as the liquid gold at the surface cools, it will contract, lose density, and sink beneath the hotter gold on the surface. It will sink and heat up to its original temperature, and will eventually be displaced by the descent of cooler gold and will rise back to the surface. When the surface cools enough, it will solidify into a solid-gold crust, which is awesome. Apparently, my fantasies are written by Terry Pratchett, which is the best thing ever. I’ve got Counterweight Continents all over the place!

Gold is ductile: it’s a soft metal, easy to bend out of shape. Therefore, the crust would deform pretty easily, and there wouldn’t be too many earthquakes. There might, however, be volcanoes, where upwellings of liquid gold strike the middle of a plate and erupt as long chains of liquid-gold fountains. It would behave a bit like the lava lake at Kilauea volcano, in Hawai’i. See below:

What a landscape this would be! Imagine standing (in a spacesuit) on a rumpled plain of warm gold. To your right, a range of gold mountains glitter in the sun, broken here and there by gurgling volcanoes of shiny red-hot liquid. Flat, frozen puddles of gold fill the low spots, concave from the contraction they experienced as they cooled. To your right , the land undulates along until it reaches another mountain range. In a valley at the foot of this range is an incandescent river of molten gold, fed by the huge shield volcano just beyond the mountains. Then a psychotic space-dragon swoops down, flying through the vacuum (and also in the face of physics), picks you up in his talons, carries you over the landscape, and drops you into one of those volcanoes.

Yeah. It would be something like that.

As fun as my golden planet is, I think we could go bigger. Unfortunately, the bigger it gets, the more unpredictable its properties become. As we keep pouring molten gold on it, its convection currents will become more and more vigorous: it will have more trapped heat, a larger volume-to-surface area ratio, and stronger gravity, which will increase the buoyant force on the hot, low-density spots. Eventually, we’ll end up with convection cells, much like you see in a pot of boiling water. They might look like this:

Benard Cells

Those are Rayleigh-Bénard cells, which you often get in convective fluids. I used that same picture in my Endless Sky article. But there, I was talking about supercritical oxygen and nitrogen. Here, it’s all gold, baby.

Eventually, the convection’s going to get intense enough and the heat’s going to get high enough that the planet will have a thin atmosphere of gold vapor. If it rotates, the planet will also develop a powerful magnetic field: swirling conductive liquid is believed to be the thing that creates the magnetic fields of Earth, the Sun, and Jupiter (and all the other planets). This planet is going to have some weird electrical properties. Gold is one of the best conductors there is, second only to copper, silver, graphene, and superconductors. Therefore, expect some terrifying lightning on Dragon’s Hoard: charged particles and ultraviolet radiation from the Sun will ionize the surface, the gold atmosphere (if there is one) or both, and Dragon’s Hoard will become the spherical terminal in a gigantic Van de Graaff generator. As it becomes more and more charged, Dragon’s Hoard will start deflecting solar-wind electrons more easily than solar-wind protons (since the protons are more massive), and will soak up protons, acquiring a net positive charge. It’ll keep accumulating charge until the potential difference explosively equalizes. Imagine a massive jet or bolt of lightning blasting up into space, carrying off a cloud of gold vapor, glowing with pink hydrogen plasma. Yikes.

After Dragon’s Hoard surpasses Jupiter’s mass, weird things will begin happening. Gold atoms do not like to fuse. Even the largest stars can’t fuse them. Therefore, the only things keeping Dragon’s Hoard from collapsing altogether are the electrostatic repulsion between its atoms and the thermal pressure from all that heat. Sooner or later, neither of these things will be enough, and we’ll be in big trouble: the core, compressed to a higher density than the outer portions by all that gold, will become degenerate: its electrons will break loose of their nuclei, and the matter will contract until the electrons are squeezed so close together that quantum physics prevents them from getting any closer. This is electron degeneracy pressure, and it’s the reason white dwarf stars can squeeze the mass of a star into a sphere the size of a planet without either imploding or exploding.

The equations involved here are complicated, and were designed for bodies made of hydrogen, carbon, and oxygen instead of gold (Those short-sighted physicists never consider weird thought experiments when they’re unraveling the secrets of the universe. The selfish bastards.) The result is that I’m not entirely sure how large Dragon’s Hoard will be when this happens. It’s a good bet, though, that it’ll be somewhere around Jupiter’s mass. This collapse won’t be explosive: at first, only a fraction of the matter in the core will be degenerate.As we add mass, the degenerate core will grow larger and larger, and more and more of it will become degenerate. It will, however, start to get violent after a while. Electron-degenerate matter is an excellent conductor of heat, and its temperature will equalize pretty quickly. That means that we’ll have a hot ball of degenerate gold (Degenerate Gold. Add that to the list of possible band names.) surrounded by a thin layer of hot liquid gold. Because of the efficient heat transfer within the degenerate core, it will partly be able to overcome the surface-area-versus-volume problem and radiate heat at a tremendous rate.  The liquid gold on top, though, will have trouble carrying that heat away fast enough, and will get hotter and hotter, and thanks to its small volume, will eventually get hot enough to boil. Imagine a planet a little larger than Earth, its surface white-hot, crushed under a gravity of 500,000 gees, bubbles exploding and flinging evaporating droplets of gold a few kilometers as gaseous gold and gold plasma jet up from beneath. Yeah. Something like that.

But in a chemical sense, my huge pile of gold is still gold. The nuclei may be uncomfortably close together and stripped of all of their electrons, but the nuclei are still gold nuclei. For now. Because you know I’m going to keep pumping gold into this ball to see what happens (That’s also a line from a really weird porno movie.)

White dwarfs have a peculiar property: the more massive they are, the smaller they get. That’s because, the heavier they get, the more they have to contract before electron degeneracy pressure balances gravity. Sirius B, one of the nearest white dwarfs to Earth, has a mass of about 1 solar mass, but a radius similar to that of Earth. When Dragon’s Hoard reached 1.38 solar masses, it would be even smaller, having a radius of around 3000 kilometers. The stream of liquid gold would fall towards a blinding white sphere, striking the surface at 3% of the speed of light. The surface gravity would be in the neighborhood of 2 million gees. If the gravity were constant (which it most certainly would not be), the 10-second fall distance would be 2.7 times the distance between Earth and moon. Now we’re getting into some serious shit.

Notice that I specified three significant figures when I gave that mass: 1.38 solar masses. That is not by accident. As some of you may know, that’s dangerously close to 1.39 solar masses, the Chandrashekar limit, named after the brilliant and surprisingly handsome Indian physicist Subrahmanyan Chandrasekhar. (Side note: Chandrashekhar unfortunately died in 1995, but his wife lived until 2013. Last year. She was 102 years old. There’s something cool about that, but I don’t know what it is.) The Chandrasekhar limit is the maximum mass a star can have and still be supported by electron degeneracy pressure. When you go above that, you’ve got big trouble.

When ordinary white dwarfs (made mostly of carbon, oxygen, hydrogen, and helium) surpass the Chandrasekhar limit by vacuuming mass from a binary companion, they are unable to resist gravitational contraction. They contract until the carbon and oxygen nuclei in their cores get hot enough and close enough to fuse and make iron. This results in a Type Ia supernova, which shines as bright as 10 billion suns. It’s only recently that our supercomputers have been able to simulate this phenomenon. The simulations are surprisingly beautiful.

I could watch that over and over again and never get tired of it.

Unfortunately, even though it’s made of gold (which, as I said, doesn’t like to fuse), the same sort of thing will happen to Dragon’s Hoard. When it passes the Chandrasekhar limit, it will rapidly contract until the nuclei are touching. This will trigger a bizarre form of runaway fusion. The pressure will force electrons to combine with protons, releasing neutrinos and radiation. Dragon’s Hoard will be heated to ludicrous temperatures, and a supernova will blow off its outer layers. What remains will be a neutron star, which, as I talked about in The Weather in Hell, is mostly neutrons, with a thin crust of iron atoms and an even thinner atmosphere of iron, hydrogen, helium, or maybe carbon. Most or all of the gold nuclei will be destroyed. The only thing that will stop the sphere from turning into a black hole is that, like electrons, neutrons resist being squeezed too close together, at least up to a limit.

But you know what? That tells us exactly how much gold you can hoard in one place: about 1.38 solar masses. So fuck you, Taylor from kindergarten! You can’t have infinity dollars! You can only have 0.116 trillion trillion trillion dollars (US, and according to June 2014 gold prices) before your gold implodes and transmutes itself into other elements! So there!

But while I’m randomly adding mass to massive astronomical objects (that’s what space dragons do instead of breathing fire), let’s see how much farther we can go.

The answer is: Nobody’s exactly certain. The Chandrasekhar limit is based on pretty well-understood physics, but the physics of neutron-degenerate matter at neutron star pressures and temperatures (and in highly curved space-time) is not nearly so well understood. The Tolman-Oppenheimer-Volkoff limit (Yes, the same Oppenheimer you’re thinking of.) is essentially a neutron-degenerate version of the Chandrasekhar limit, but we only have the TOV limit narrowed down to somewhere between 1.5 solar masses and 3 solar masses. We’re even less certain about what happens above that limit. Quarks might start leaking out of neutrons, the way neutrons leak out of nuclei in a neutron star, and we might get an even smaller, denser kind of star (a quark star). At this point, the matter would stop being matter as we know it. It wouldn’t even be made of neutrons anymore. But to be honest, we simply don’t know yet.

Sooner or later, though, Karl Schwarzschild is going to come and kick our asses. He solved the Einstein field equations of general relativity (which are frightening but elegant, like a hyena in a cocktail dress) and discovered that, if an object is made smaller than its a certain radius (the Schwarzschild radius), it will become a black hole. The Schwarzschild radius depends only on the object’s mass, charge, and angular momentum. Dragon’s Hoard, or rather what’s left of it, doesn’t have a significant charge or angular momentum (because I said so), so its Schwarzschild radius depends only on mass. At 3 solar masses, the Schwarzschild radius is 8.859 kilometers, which is only just barely larger than a neutron star. Whether quark stars can actually form or not, you can bet your ass they’re going to be denser than neutron stars. Therefore, I’d expect Dragon’s Hoard to fall within its own Schwarzschild radius somewhere between 3 and 5 solar masses. Let’s say 5, just to be safe. There are suspected black holes with masses near 5.

That’s the end of Dragon’s Hoard. The physics in the center gets unspeakably weird, but the gold-spitting space dragon doesn’t get to see it. He’s outside the event horizon, which means the collapse of his hoard is hidden to him. He just sees a black sphere with a circumference of 92.77 kilometers, warping the images of the stars behind it. It doesn’t matter how much more gold we pour into it now: it’s all going to end up inside the event horizon, and the only noticeable effect will be that the event horizon’s circumference will grow larger and larger. But fuck that. If I wanted to throw money down a black hole, I’d just go to Vegas. (Heyo!) Dragon’s Hoard isn’t getting any more of my draconic space-gold.

But one last thing before I go. Notice how I suddenly went from saying Schwarzschild radius to talking about the event horizon’s circumference. That’s significant. Here’s a terrible picture illustrating why I did that:

CurvedSpacetime

Massive objects create curvature in space-time. Imagine standing at the dot on circle B, in the top picture. If you walk to the center-point along line A, you’ll measure a length a. If you then walk around circle B, you’ll get the circle’s circumference. You’ll find that that circumference is 2 * pi * a. The radius is therefore (circumference) / (2 * pi) But that only holds in flat space. When space is positively curved (like it is in the vicinity of massive objects), the radius of a circle will always be larger than (circumference) / (2 * pi). That is to say, radius C in the bottom picture is significantly longer than radius A in the top one, and longer than you would expect from the circumference of circle D.

In other words, the radius of a massive object like a star, a neutron star, or a black hole, differs from what you would expect based on its circumference. The existence of black holes and neutron stars has not actually been directly confirmed (because they’re so small and so far away). It is merely strongly suspected based on our understanding of physics. The existence of spacetime curvature, though, has been confirmed in many experiments.

Imagine you’re standing in a field that looks flat. There’s a weird sort of bluish haze in the center, but apart from that, it looks normal. You walk in a circle around the haze to get a better look at it. It only takes you fifteen minutes to walk all the way around and get back to your starting point. The haze makes you nervous, so you don’t walk straight into it. Instead, you walk on a line crossing the circle so that it passes halfway between the haze and the circle’s edge at its closest approach. Somehow, walking that distance takes you twenty minutes, which is not what you’d expect. When you walk past the haze at a quarter-radius, it takes you an hour. When you walk within one-eighth of a radius, it takes you so long you have to turn back and go get some water. Each time, you’re getting closer and closer to walking along the circle’s radius towards its center, but if you actually tried to walk directly into its center, where the haze is (the haze is because there’s so much air between you and the stuff beyond the haze, which is the same reason distant mountains look blue), you would find that the distance is infinite.

That’s how black holes are. They’re so strongly-curved that there’s way more space inside than there should be. The radius is effectively infinite, which is why it’s better to talk about circumference. As long as the black hole is spherically symmetric, circumferences are still well-behaved.

But the radius isn’t actually infinite. When you consider distance scales close to the Planck length, Einstein’s equations butt heads with quantum mechanics, and physicists don’t really know what the fuck’s going on. We still don’t know what happens near a black hole’s central singularity.

Incidentally, the Planck length compared to the diameter of an atom is about the same as the diameter of an atom compared to the diameter of a galaxy. The Universe is a weird place, isn’t it?

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