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The original question was evidently far too long to fit into the title, so I'll post it here in the body:

"If two non-ionic tungsten atoms spontaneously popped into existence at rest a hundred light years apart in the approximate center of a hypothetical non-expanding universe that is a billion light years across but utterly devoid of all other ordinary matter and energy, including "dark matter," would the gravity of the widely separated atoms eventually bring them together into a close encounter or even an outright collision? If so, how closely will each atom approach the speed of light just before that encounter?"

I honestly have wondered about this on occasion for the longest time (in truth, decades) and hope that my obvious ignorance of physics hasn't rendered the question as phrased trivial or otherwise uninteresting. I've tried to phrase it for maximum conceptual simplicity. Nor is it quite clear how to properly formulate this question for a straightforward Google query — my attempts thus far have attracted only a flood of irrelevant results that dance around the core question. The major search engines don't seem able to cope with so many common physics keywords. -_-

P.S. No, this is not a homework question! I'm just an uneducated dude with odd questions buzzing around in his head and some experience in writing clearly for commercial clients. o_o

Qmechanic
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owlsupport
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  • Given enough time they will eventually collide – R. Emery Jan 05 '21 at 19:53
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    @R.Emery - and, given the very small attractive force (and 'binding' energy), bounce off each other and head back to where they started from, doomed to repeat that over and over again... – Jon Custer Jan 05 '21 at 19:59
  • Related: https://physics.stackexchange.com/q/24309/2451 and links therein. – Qmechanic Jan 05 '21 at 20:03
  • Ah, thank you, R. Emery. I'd suspected as much, but one of the earliest lessons for any human of reasonable intelligence is to never assume anything about our weird and wacky universe, riddled as it is with extremely peculiar phenomena such as gravity and magnetism. Honestly, I never cease to feel wonder at the slightly unnerving spectacle of a strong rare-earth magnet hanging from the top of my hand, held by the magnetic field between it and another rare-earth magnet resting on my upturned palm. It seems as if few people indeed realize how truly strange is magnetism with which to begin! – owlsupport Jan 05 '21 at 20:03
  • Really, Jon Custer? I hadn't thought of that — seems to imply anyway that the collision velocity would be a rather low percentage of the speed of light. Tiny, even. Hmmm ... in any case, wouldn't infinitesimal internal differences cause them upon collision to bounce off in different directions from the original directions? Does that question make sense? – owlsupport Jan 05 '21 at 20:09
  • Thank you for the reference, Qmechanic. I'm reading that page carefully. I might even emerge from the experience with a few more permanent nuggets of knowledge embedded within my thick skull. ^_^ – owlsupport Jan 05 '21 at 20:13
  • Their free-fall time is about a trillion trillion trillion years. – G. Smith Jan 05 '21 at 20:40
  • Question from a math guy: how does this question make sense? Isn't this just a math problem? Rephrasing the question: Will two atoms in an otherwise empty universe where Newton's law of gravity applies be drawn together? Answer: yes, by construction. How is this exactly a Physics question? – Fernando Jan 05 '21 at 21:14
  • Holy crap, G. Smith! One wonders if the hypothetical phenomenon of “proton decay” would render the original question irrelevant or perhaps substantially more complicated before the passage of 10^36 years. ^^; – owlsupport Jan 05 '21 at 21:20
  • I don't understand why you are asking. If there are only two atoms in the universe there is literally nothing that could sow any doubt about it, is there? – Peter - Reinstate Monica May 13 '21 at 21:43

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The force of gravity has no limits, so both atoms will feel a tug from the other one, even if it's 100 light years away. The magnitude of the force will be very minor, around $10^{-96}$ Newtons. The total gravitational potential energy around $10^{-81}$ J, which will result in a collision speed of about $10^{-28}$ m/s. Even if the atoms closed the distance at their top speed the entire time, it would take $10^{36}$ years for them to collide, which exceeds the age of the universe by almost 30 orders of magnitude!

As another way to think about it (in reverse), particles that are very far apart and get drawn in by gravitation alone will collide at roughly escape velocity. Since a tungsten atom is very tiny, its escape velocity is miniscule - you need very little velocity to launch yourself as far away as you'd like, never to return.

NOTE: I need to double check some of these figures, I think I might have messed something up with the fact that there are two atoms moving toward a central point. Regardless, the forces and speeds involved will be very, very tiny.

Nuclear Hoagie
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  • Thank you! The bit about the "escape velocity" of each atom is especially illuminating. I now feel less uneducated and a bit schmarter! ^_^ – owlsupport Jan 05 '21 at 20:26
  • @owlsupport Indeed, I often like to think of gravity/orbit problems in the reverse to check intuition. 100 light years away isn't quite the same as the "infinite" distance achieved by an object traveling at escape velocity, but since the force of gravity is so incredibly weak at that point, it should be darn close. – Nuclear Hoagie Jan 05 '21 at 20:43
  • I think your collision speed is off by about 15 orders of magnitude. Please check your calculation. – G. Smith Jan 05 '21 at 20:49
  • Since we have no quantum gravity we cannot be sure if the answer is correct. Or is there an argument to exclude quantum effects? – my2cts Jan 05 '21 at 20:50
  • @my2cts Are you concerned that quantum gravity might be relevant when they are far apart or when they are close together? – G. Smith Jan 05 '21 at 20:59
  • @G.Smith On second thought I expect bound quantum orbitals to be very small, much smaller than the proton size, so irrelevant. – my2cts Jan 05 '21 at 21:04