One of the problems for humans having settlements off-Earth is the huge cost of getting the supporting materials there, and the great bulk of that cost is actually in getting the stuff out of Earth’s gravitational field. If you look at a chemical rocket, you start with a huge monster, and cart up only trivial amounts, the reason being that the great bulk of the initial mass is the mass of the fuel necessary to get the rockets going, and the mass of the metal needed to contain the fuel.
One proposal to get around this is the space elevator. The idea of this is simple in concept. At about 35,800 km above any point on the equator the orbital angular velocity is the same as that of the Earth, and hence you have a point in space where a satellite is always above the same point. At the risk of annoying some physicists, if we reduce the problem to one dimension, the centrifugal force arising from the orbital motion is exactly the same as the force heading towards the Earth and the angular velocity is the same. Now, suppose we put a cable between Earth and this geostationary point, and make the cable strong enough that we can run an elevator up and down it. Now the work done is the same as using an elevator, and an electric motor can power it. But as it stands, this won’t work because where before the centripetal force was that of earth’s gravity, now it has the force from the weight of the cable added to it. This can be corrected by adding corresponding centrifugal force, achieved by extending the cable further and attaching a massive body to it. As long as it stays put, its centrifugal force will cancel the weight of the cable, so if all this is done carefully, you have an elevator cable that you can run things up and down and transfer everything to a geostationary satellite.
From an economic point of view, the space elevator should lift material up there at least 8 times cheaper than the most favourable prediction from rockets, and its capital cost is estimated to be about $20 billion. So, why do I think this is a non-starter?
First, there is the issue of materials. The cable appears to need to be at least 40,000 km long. Something like titanium is far too weak for the task, but it has been speculated that carbon nanotubes might be satisfactory. However, you are not going to make a 40,000 km long nanotube, so some sort of composite will be required. The strength is that of the weakest part. The composite has to be at least as strong as the nanotube, and adhere as strongly, and also retain that strength indefinitely despite space weathering. I do not believe such a material is possible.
The next problem is, where do you put this cable while you are making it? You have a single length that is 40,000 km or so long. Where do you store it while you are making it? Assuming you make it in sections, how do you know the joins will be strong enough? How do you know there are no weaknesses deep within the cable? Then specifically where do you assemble it? How? A coil? How flexible are such nanotubes, and why is the composite sufficiently elastic? What is the proposed radius of the coil? Since it has to be on the equator, how many notice the equator is basically wet?
Now, suppose this huge cable is coiled up somewhere, how do you get it into position? You cannot really take it up with a rocket because the exhaust will ignite your carbon. Oops! On top of that, the rocket has to go up and counter the Earth’s rotation. But just suppose you get this massive weight up there, how do you hook it to the counterweight? The asteroid, recall, is NOT in geostationary orbit but will have quite a relative velocity. Your propulsion unit has to arrive at exactly the right time, with thrusters supporting the whole weight of the cable, and somehow this has to be joined to the asteroid while still supported by the rocket until the junction is firm. And even if you think you can manage this, how can you be sure that nothing will go wrong? One slip, one miscalculation, and 40,000 km of cable comes hurtling back to Earth. That is enough to wrap itself around the planet, causing serious damage to anything in its path. Then, supposing all this can be done, how do you get it down again safely at the end of its working life? In part because I have designed and overseen the construction of a chemical plant, and have seen what can happen with engineering, and I know one should always start off small, to iron out the bugs. That is not possible here and I just do not have sufficient faith in such a one-off engineering feat.
ianmillerblog 
I’ve heard of this idea, and it did seem improbable to me, even though I didn’t think it out as well as you have here. Reading about the logistics gave me a queasy feeling. Thanks for an interesting post!
One of the things I have often seen, Audrey, is that people come up with some quite interesting ideas, until you start to examine how you would actually do it. I have been consulting to a number of ventures that a end up doing engineering, and of course on a much smaller scale than this, and it is surprising how many mistakes get made. With this, one overlooked fact could mean disaster.
This is the official line of those who don’t want to dream progress:
http://io9.com/5984371/why-well-probably-never-build-a-space-elevator
However, it all boils down on being unable to build carbon nanotubes which are long enough. Carbon nanotubes are strong enough, 300 times more than the strongest steel, but too short. In the article above, they are viewed as little different from soot. However, in the same year half a meter long tubes were made.
It is one thing to make a tube half a meter long, completely different to make them nearly 40,000 km long. And of course you have to make a bundle of them.
More interesting,from the dynamical point of view, why is this not metastable? Recall, the tethering asteroid is not at a geostationary point, but is held where out is by tension in the cable. Should the cable wobble, why should not resonance waves be enough to knock it off course?
I am not against progress – I just think there are better options.
Yes, well tricks, new tricks can be used. By bombarding with electrons, sticky strands can be made, 20 times stronger (that was in 2013). Also one can start with a SKY HOOK, not a full space elevator. One will have, to see how to master the wobbling.