Landscape of Indefinitely Old Earth

Question

If we somehow fix the Sun's luminosity indefinitely at current levels, stabilize the Moon's orbit so that it no longer drifts away, and match the solar tides precisely with lunar tides so Earth's rotation no longer slows down, what would happen to Earth's landscape eventually? The tectonic activities will eventually cease. What happens billions of years after that? Will the continents eventually sink below the sea and leave only archipelagos?

EDIT: My bad. I should have stated more clearly my intention. I'm primarily interested in geological processes in geological time horizons after tectonic ceases completely, not cosmological or other theoretical physical scenarios. The emphasis is on sufficiently long after tectonics ceases, not what ifs of a future truly eons after that. So, here's my restatement of my problem.

If we hold every other variable constant, e.g. sun's luminosity, Earth and Moon's orbits, magnetic field, replenish the water lost from Callisto, etc, how would the landscape evolve?

Assuming we still have rainfalls and plant growth continuing indefinitely after tectonics ceasing, will rivers eventually become fjords and deserts eventually have all their sands blown off, flooded and eventually become shallow seas? Is it possible that, even with well-forested areas, billions of years of rainfalls eventually leach the whole continent away ion by ion, so that we have nothing but a global ocean in the end?

Answer 1

Will the continents eventually sink below sea

Wait long enough, and there will be no sea.

Water vapor in the earth's atmosphere is subject to photolysis by UV light emitted by the sun. This eventually produces atomic hydrogen, which will slowly escape into space via Jeans escape, because some proportion of the atomic hydrogen at in the exosphere will have a thermal velocity greater than Earth's escape velocity.

Combined with water loss through subduction into the mantle via plate tectonics, the surface of the Earth will eventually end up dessicated. This will happen as a matter of course to the regular Earth, speeded up somewhat thanks to the Sun's luminosity naturally increasing as it gets older, but over a long enough timescale it should happen even under regular solar illumination and will be accelerated if any other process causes global temperatures to rise.

Future ice ages will slow the process, but once that hydrogen's gone it is impractical to get it back in sufficient quantity.

Wait long enough, and I suspect you'll end up with a nitrogen and CO₂ atmosphere over a dead, rocky world.

Answer 2

Due to time limitations, this is all off the top of my head rather than citations or back of the envelope numerical calculations, apologies in advance.

So you're fixing solar activity, Earth's rotation and Earth's magnetic field, but allowing everything else to proceed. After nigh infinity years, the fate of Earth is unknown due to limitations to our current knowledge of physics... I'll take the liberty of adding a few additional assumptions: that the expansion of the universe remains "on its current course" and we don't have runaway acceleration into a "dark rip" scenario, and that protons do not eventually decay. One possible outcome could be that the entire planet eventually converts itself into a ball of iron due to quantum tunnelling, and then into neutron planet (not if sure there's enough mass, not sure about the weak nuclear equilibrium) and then a tiny black hole due to more of the same, which then evaporates into photons via Hawking radiation. These would take a cosmically absurd amount of time.

However, you seem to be more interested in earlier events that take mere billions and trillions of years, particularly the cessation of plate tectonics. Current speculations include the idea that Earth might lose all of its oceans in a billion years, at which point plate tectonics ceases due to reduced lubrication. However, these models account for increased atmospheric escape due to the sun becoming hotter. Since you're holding solar activity constant, I suspect this process will take considerably longer, and it's no longer clear to me that tectonics will continue until Earth loses its oceans and atmosphere and their erosive processes.

When tectonic activity does cease at some point, I don't think it's clear that continental formation will cease immediately. The planet could still have enough internal thermal energy to drive ongoing volcanic activity, only instead of plate edges, now it's more of a haphazard "hot spot" pattern like on the planet Venus. Volcanic activity of the "flood basalt" type could still form new continents. Once the enough energy has been lost for the volcanic activity to end, that may also be the end of the carbon cycle. If the Earth still has an atmosphere at this point, all of its carbon content may end up in rocks, leaving nothing for photosynthesis, leading to the end of all life as we know it. TLDR, long before it's an iron ball, I think the Earth will be a lifeless rock, devoid of ocean or atmosphere, but possibly not eroded to be completely smooth (furthermore, without erosion, new craters from astronomical impacts tend to stick around).