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Imagine that, for unknown reasons, the Earth has left the solar system. It has been a while now since the event. It's so cold outside that the polar caps met at the equator and the atmosphere froze solid.

Some people survived in bases built underground or on the ocean floors. They use geothermal and fusion power to stay alive.

People from distant bases need to communicate wih one another. For example, some folks in the middle of the Atlantic seafloor might wish to communicate with some people that are in an underground base a few kilometers/miles beneath what used to be Alaska.

I initially thought of submarine fiber optics cables, but those require maintenance that might not be available in the post-apocalypse. What other methods could these deep bases (underwater and underground) use to communicate over vast distances?

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    $\begingroup$ Radio is out, with no atmosphere and no sun to ionize it. $\endgroup$ – Jon Custer Dec 7 '20 at 2:27
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    $\begingroup$ Radio would be out? Maybe some, but radio doesn't need an atmosphere unless you're thinking short-wave. Electromagnetic transmission is just photons at different wavelengths. The communication satellites are likely still in orbit around the planet. In fact, without an atmosphere, they might actually work better. Though not relevant to the question, cell phones might have problems only because the towers would have trouble in the intense cold or might not be capable of radiating waste heat without the atmosphere - but that has nothing to do with EM transmission intrinsically. $\endgroup$ – JBH Dec 7 '20 at 4:44
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    $\begingroup$ I think Jon Custer's point is that, without an atmosphere for over-the-horizon propagation, radio becomes line-of-sight only (like a laser) and only reaches as far as the horizon visible to the transmitter (about 36km at a height of 100m above ground level). Also, depending on how long it took for the atmosphere to condense out, communications satellites in low earth orbit may have deorbited already from remaining atmospheric friction; they aren't long lived, IIRC. $\endgroup$ – GrumpyYoungMan Dec 7 '20 at 14:28
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    $\begingroup$ @JBH comsats are solar powered. By the time the atmosphere freezes out, they'll be long dead. $\endgroup$ – Zeiss Ikon Dec 8 '20 at 12:14
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    $\begingroup$ @JBH Those weren't comm relays, though; they were weapons or telescopes (designed to point at the ground) -- and even so, they probably have a working lifetime of a couple decades, 40-50 years at most. How long would it take to freeze the atmosphere? Comparable time, I'd guess. $\endgroup$ – Zeiss Ikon Dec 8 '20 at 14:59
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If there's still anything left in orbit (e.g. dead satellite), an optical or radio signal could be passively bounced off it, e.g. https://ham.stackexchange.com/questions/8897/could-uhf-earth-satellite-earth-bouncing-of-signals-off-of-this-large-metallized ([EDIT] For this, you would want your bounce satellite to be in a https://en.wikipedia.org/wiki/Graveyard_orbit so that it's around for a good, long time.) Similarly, it's also possible to bounce signals off the moon, if that's still present, but the increased distance makes it more challenging. This is probably the highest bandwidth option.

Between ocean bases, something similar to the ELF radio system used for submarine communications may be usable , see https://en.wikipedia.org/wiki/Communication_with_submarines#Extremely_low_frequency. However, only a few letters per minute can be sent (not a typo) so it is very limited.

Depending how sci-fi you're willing to go, communication using neutrinos has been demonstrated to be possible: https://physicsworld.com/a/neutrino-based-communication-is-a-first/. Because neutrinos pass through matter easily, they can go through the entire planet. However, that also makes them extremely difficult to detect, making the both transmitting and receiving equipment extremely bulky and complex. The data rate would be rather low (maybe a handful of kilobits per second?) but you could probably fudge a slightly higher data rate without the reader noticing.

[EDIT] Without a terrestrial fiber or tower network or an active satellite network, I can't think of any way that anything like today's Internet speeds would be practical in the proposed scenario. In the cases listed above, I'd expect communications something like the early 1980s Internet, where messages are completely text.

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Number one option here is ELF radio.

No, not talking about willowy folks with long ears: ELF is Extremely Low Frequency, and it propogates mainly along the ground-air (or ground-vacuum) interface, as well as penetrating the ground and sea to a considerable extent. The down sides are that it's only useful for Morse code or very slow data rate digital communications, and the transmitting antenna generally needs to be a physically large grid of wires.

ELF has been used by the US Navy to communicate with submarines in a manner that doesn't require them to surface to receive radio (as would be the case at more conventional frequencies) -- power requirements are modest, and the grid is fairly tolerant of damage; in addition, the receiving antenna can be nothing more than a long wire.

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