could you point us to some sort of an official statement on the 'nothing-at-stake problem'? No one seems to really be all that clear on what this problem actually is. From what I was told the person who first proposed it took it down?
You've been told some pretty weird stuff then. Nothing has been taken down, and it's not that complicated a point.
In POW to contribute to a consensus you must burn a resource, which means you must make an exclusive choice among all the possible consensus you could contribute to, to the exclusion of all others... and for your effort to not be wasted you should be spending it on the chain you think most likely to survive. In pure POS schemes, there is no such exclusivity created. This leads to fun outcomes like old stake holders can exit the system (sell their coins) and then sell their old keys to people go fork off the chain at a point in the past, at no cost to themselves. Someone who is later handed two histories— the real one and the simulated one— cannot distinguish them, they can tell— perhaps— that someone was naughty, but that doesn't help them decide which chain is the good one. There are a number of other related implications. A number of different modifications have been proposed, but so far all of them seem to be obfuscation and not actually fix the underlying issue, which seems a bit fundamental.
You can read more about this in Section 5 of
https://download.wpsoftware.net/bitcoin/asic-faq.pdfPPC was attacked utilizing this fact the moment POS mining became possible on it— a savvy miner tried all possible forks finding a sequence of forks which selected their stake as the winning stake as much as possible. PPC prevented this with block signing and discouraged it by hard forking the protocol so that POW blocks were required.
Don't the GPS satellites transmit an accurate time signal that could be used for this purpose?
GPS is unauthenticated. Any local-to-you jammer can spoof it with nothing more complex than a USRP and some software. It's also run by US space command, and the US has been quite up front that they are willing to manipulate or disrupt the signal to achieve military objectives, they're able to perform geo-targeted alterations of the signal too. It's pretty useful on average, but it's not a secure solution by itself.