Draupnir: the jewelry with periodic backups
I love killing time by reading Wikipedia, especially its articles about myth. I just read about Draupnir, Odin's golden arm ring that created eight copies of itself every nine days. Now I can't stop drawing geeky analogies to periodic data backups or cron jobs that log statistics periodically.
Assuming that only the original Draupnir can replicate, assuming US$20 per gram of gold, and taking an educated guess 400 gram arm bracelet, I calculate that Draupnir makes 2.5 million dollars per year. This is about 130 kg per year, for a volume of about 6.5 liters per year.
Before I did those calculations, I thought such a ring would make a good set-up for a Twilight Zone episode or other fable involving the devil, because the ring would eventually bury its owner, the surrounding countryside, the Earth, and who knows what else in gold rings, but not before it drives the price of gold down to nothing. Clearly, though, nothing even close to that will happen in the case I just described. The world supply of gold is in the millions of kilograms per year. $2.5 million a year is quite posh for one person, but the trillion dollar U.S. economy wouldn't even feel that. You need millions of dollars per minute to come close to affecting huge systems like that.
Furthermore, I'm actually embarrassed that I thought the mass of gold from only one Draupnir could have an effect on the astronomical scale. By five billion years, when the Sun is going through that red giant business, Draupnir will have made 640 billion kg of gold, close to the mass of world oil production for one year. This is 33 million cubic meters, a cube of gold 300 meters on a side: larger than NASA's Vehicle Assembly Building in every dimension, but not far off, relatively speaking. This is an absolutely pathetic number next to engulfing the Earth. To make a mass of gold the size of the Moon or Earth would take a single Draupnir 1020 to 1022 years. Nobody knows for sure what the Universe will be like then, but it will be sufficiently boring that the collection of gold rings won't matter much.
If each new ring can create new rings, however, we have a profoundly different situation. Remind me to ask Odin whether Draupnir works in the linear way or the exponential way. Anyway, suppose Odin, or Old Scratch, gives you Draupnir on January first. In a month, you'll have made as much as you would in a year with only the single replicating ring: a cool million dollars. But by mid March you'll have a million times that, even. Now you're in the neighborhood of the U.S. annual economy. You could have a 50 meter ball of solid gold if you melted down all the rings, and they weigh a billion kilograms: heavier than a supertanker, and more than the total amount of gold the world has ever mined. Congratulations on pushing around the economy, but it's time to push around something bigger.
By June, you'll have that building-sized hunk that it took billions of years to make the old-fashioned way, and by August you'll have the mass of the Moon in solid gold. Now you have very little choice in the shape of your pile. You certainly can't defend it from any thieves at this point, but why would they be interested in it? Rings are probably even getting liquefied under each other's weight, but I will assume that they can still create copies of themselves. By September, you'll have the mass of the Earth. Of course, the rings will totally envelop the Earth when they self-copy again nine days later, like clockwork. In October, as they continue to create mass, presumably magically out of nothing, their collective mass will pass that of the Sun. I'll make another assumption that Odin allows flagrant violation of physical laws such as the conservation of mass, but only in the case of Draupnir. You may also be wondering about the velocity of the gold ball's surface as it instantaneously creates new rings. Do not mock Odin with such questions.
I'm not an astrophysicist, so I don't know exactly what happens when we get stellar masses of gold together. I bet at least some part of the ball will see some thermonuclear action from fusionable material that it accretes from space or the Sun or whatever. Once again, I'll assume that plasma-phase former Draupnir copies and even nuclear-reacted no-longer-gold particles of rings can make copies of more rings. One of the other problems is that gold doesn't like to do fusion. It likes to fission, or something. Oh, well. Given my assumptions, in a couple weeks I think we'll have enough gold in one place to put out the Sun. Last but not least, after the new year, the Schwarzschild radius for this mass of gold will surpass its measured radius, well, assuming it has the uniform density of metallic gold, which it won't, but in any case, it will become a supermassive black hole. I hope there's a Neiman-Marcus inside this event horizon so I can spend my infinite "wealth."
Moral: The Universe is very big, but it is no match for exponentials and silliness.
