Technology

Yes, we have the theory of what is required to build one but every material we have (including graphene and carbon nanotubes) is about 2-3 orders of magnitude below the tensile strength that is required for a space elevator on earth. Add in the fact that the longest graphene and carbon nanotube we can currently produce is in the mm range and we need it to be ~50,000 km and perfect at the atomic level we would be at best decades away from production if they could be used.

Ironically the best place for us to begin is in space.

Building space elevator on the moon is much, much easier (1/6 G and no atmosphere) and Mars is also a much easier proposition than on earth (1/3 G and 1/1000th the atmospheric pressure).

I fully expect that if humans ever build space elevators the first one will not be on earth.

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I'm in the middle of an MA in astrophysics...

You'd want tensile strength rather than compressive. The trick is to anchor a counterweight out beyond your target distance and let it pull the weight of the cable up rather than building a tower. Think of swinging a ball on a string rather than building a skyscraper. Assuming a sufficiently sized counterweight you can support the weight of the anchoring cable plus whatever else you want to hang off of it (space dock, elevator terminus, etc.)

If you build it tall enough, centrifugal force will start pulling on it. Building it that way though... But yeah. Doubt the right material exist atm.

stacked tall enough...? The space elevator concept is a geostationary node orbiting earth directly above a fixed point, with cables running between them. Not a gigantic skyscraper up into the sky. What am I missing?

Goats! For the love of God, stay away from the spiders!

Probably never will be.