I have tried finding the answer to this question for a while, still don't know. You have one atom a billion miles away from another, in a hypothetical universe where nothing else exists. These two atoms are traveling a quarter the speed of light away from each other, and in this universe, there is infinite space in both directions (no looping back around). Given an eternity, will these two atoms ever pull each other back to touching, or, after so much distance, the gravity well of each bit of mass just does not affect the other, and they never touch again? Even if the amount of gravity is so incredibly minute, does distance ever completely destroy the effect it has?
As we understand gravity, eventually the gravitational force will slow the two atoms, and they will attract each other. For extra credit, you can actually work out how long it takes the two atoms to meet each other. If you're interested, we can do it in this thread.
Well, assume you have two bodies of equal mass, m. The force between the two bodies is given by \(F = G \frac{m^2}{r^2}\) where G is newton's constant and r is the distance between the two particles. The acceleration of one of the masses (in the direction of the other mass) is given by \(a = \frac{G m}{r^2}\) You know the initial velocities, and the separations, and the accelerations, so you know everything.
BenTheMan may have read too fast, but this part of your question makes his answer wrong. For typical atomic masses \(m_1 = m_2 = 3 \, \text{ amu } \, = 5 \times 10^{-27} \, \text{ kg}\), they have a certain binding energy equal to \(U = - \frac{G m_1 m_2}{r}\) and a certain amount of kinetic energy, \(T = \frac{1}{2} \frac{(m_1 + m_2)m_1 m_2}{(m_1 + m_2)^2} v^2 = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} v^2 = \frac{1}{2} \frac{m_1}{1 + m_1/m_2} v^2 \). If \( T + U \ge 0\), then the atoms will go on forever, slightly slowing with every second but always getting farther away (after perhaps an initial close approach). But if \( T + U < 0\) then the two atoms will orbit each other in ellipses. So the minimum speed for separating atoms is \( T_{min} + U = 0 = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} v_{min}^2 - \frac{G m_1 m_2}{r}\) or \(|v_{min}| = \sqrt{\frac{2 G (m_1 + m_2)}{r}}\). But as you have it, G and the masses are small in human units, and r is large. \(|v_{min}| = \sqrt{\frac{2 \times 6.67 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \times 10^{-26} \, \text{kg}}{1.61 \times 10^{12} \, \text{m}}} = \sqrt{8.29 \times 10^{-49} \text{m}^2 \text{s}^{-2} } = 9.1 \times 10^{-25} \text{m}/\text{s}\). That's a little more than a millionth of an inch in a billion years. Any more than that and the atoms escape from each other. Since a quarter of the speed of light is more than that, the atoms obviously escape from each other. Lets say v = 0. Then the time it take for the two atoms to fall into each other is \(\sqrt{\frac{\pi^2 r^3}{8G(m_1+m_2)}} = \sqrt{\frac{4.11 \times 10^{37} \, \text{m}^3}{8\times 6.67 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \times 10^{-26} \, \text{kg}}} = \sqrt{7.71\times 10^{71} \, \text{s}^2} = 8.8 \times 10^{35} \, \text{s}\) or about 2 billion billion times the current age of the universe. But according to theory, that is likely to be correct upto a cosmological correction, which involves the fate of the universe. Since you have defined your universe to be flat and empty, I will ignore such things.
If Newton's gravitational laws hold at large distances, and the two atoms are moving away from each other at \(c/4\), they will attract each other forever and gradually slow down, but they'll also keep moving apart forever. Their speeds will get slower and slower, but always stay above a certain minimum velocity that they'll approach as time approaches infinity. I'm assuming here that we're talking about atoms with masses in the usual atomic ranges as we know them. If they're two super-heavy atoms (i.e. much, much, much heavier than stars or even galaxies), the force between them could be strong enough to eventually bring them to a halt and then back together. If the atoms are ultra-light atoms like normal, with virtually negligible gravities, the force between them will be pathetically weak and they will always be travelling at nearly \(c/4\) no matter how far you go into the future.
Interesting resonses to his question. However, I wonder why you posted this hypothetical. It is pretty well established that Newton's gravitational laws fail at galactic distances and is the basis for the creation of the hypothetical Dark Matter.
To clarify the above point, if gravity is Newtonian at large distances then there must be dark matter. If Newtonian gravity is modified at large distances then there is no need for dark matter. I think there's more evidence for the former but I will wait to be corrected on that.
Actually, it is pretty well established that any modifications to Newton's Laws also requires dark matter, as shown by the discovery of the Bullet Cluster. See http://en.wikipedia.org/wiki/Bullet_Cluster MacM---if you wish to discuss modification of Newton's laws, then you should do so in another thread.
Where's my post about the possibility that the gravitational force could be limited in space, at galactic distances, by a "reachability" factor in the formula??? Seems to has been deleted without any justification. By the way, in relation to the article at wikipedia, I think the "Bullet Cluster" image could be explained in other way...
You seem to have a quick mod trigger. My post made no suggestion about modifiction of Newton's laws. I merely commented on his introduction saying "If Newton's laws apply......" When it is known they do not. Further you assume to much when you conclude Dark Matter is required. That implies it is the only solution. It is not. You seem to just be promoting the current arbitrary patch work in physics. The truth of the matter is current science has the wrong idea about the orgins of gravity. That "IS" another subject however. So chill your mod pen.
Dark matter has multiple kinds of evidence; the rotation rates of galaxies, the Bullet Cluster and perturbations in the CMB. The existence of uncharged matter has long been considered in particle physics so dark matter is one thing which explains many in a natural way. All other attempts can, at best, explain one of the three phenomena just mentioned and almost all of them fail to manage to explain even one of them. Dark matter isn't a 'patch', it's a perfectly logical extension to our view of the universe. Coming up with a modification to gravity so as to fit precisely rotation curves is both ad-hoc and extremely inelegant. That is 'patching' a theory.
I don't think so. I must agree: under current situation of Physics Science those kinds of thoughts are catalogued as "Pseudoscience". No, thanks. I have more important things to worry about. I'm not surprised. Just for a moment I forgot the politics/philosophy of the Forum...
Well you've heard of William of Ockam's "admonition against unnecessary hypotheses." http://en.wikipedia.org/wiki/Occam's_razor When dark matter and MOND are attempts to explain why there seems to be more gravity than accounted for in stars and other matter which we can sense, you propose that we need your idea which explains why there is less gravity than Newton or Einstein theories would indicate. This is clearly an "unnecessary hypotheses" as it attempts to explain a situation contradicted by all attempts to gather evidence. "[T]he possibility that the gravitational force could be limited in space" is similar to the possibility that there is a conspiracy of coral and fungus which actually draws lines of longitude and latitude on the Earth, as well as all the country names as depicted on your classroom globe. It is similar to the possibility that immiscible gasses in the high atmosphere are responsible for the gold and purple tilework of the sky. These are hypotheses that explain only counterfactual situations. If I have belabored the point, please feel free to delete my post. Thank you.
Ha, ha, ha... So you don't have problem to invent an unexisting kind of unknown thing in the 90% of the Universe nor to say Newton's dynamics equation is wrong. For you the sacred untouchable things are the field's formula and Einstein, isn't it? Surelly I must not waste my time on this. Bye.
The essence of politics is the twin activities of seeking and exercising power. The power on this forum is wielded by the administrators. The philosophy (please do us all a favor and turn on your spell checker) which they want us to follow is to rebuild this website as the place of science that it was ten years ago when I joined. When in doubt, we are to be guided by the scientific method. The Rule of Laplace (or Sagan's law for TV junkies) tells us that an extraordinary assertion must be supported by extraordinary evidence before we are obliged to treat it with respect. One can debate the precise definition of "extraordinary," but certainly an assertion that contradicts any of the fundamental theories that shape our understanding of the laws of nature falls into that category. This is not to say that those theories are not occasionally overturned, or more commonly expanded (such as from Newtonian to Einsteinian physics), but those efforts are always backed up by extraordinary evidence (such as astronomical measurements made by more sensitive instruments than Newton had). The Moderator of this board has determined that your assertions, which challenge the theories of physics, are not backed up by extraordinary evidence. For this reason they have been sent to the Pseudoscience board. That is a place for the discussion of such ideas, and you are free there to try to convince us that you have found new evidence to back up the hypotheses. It is not equivalent to being treated with the disrespect that Laplace/Sagan would permit, because that would have meant consignment to the Cesspool or simply deletion. I'm not much of a physicist but I am a little better versed in biology and I have followed the so-called "creation science" movement. I personally found that their evidence did not come close to qualifying as extraordinary. At best it's based upon poorly reviewed papers from third-rate universities. On the average it's based upon wishful thinking. And at worst it's downright fraudulent, based upon a subset of fossils carefully selected to distort the record. Most of the evidence for extraordinary assertions in any science--from physics to psychology--falls into one of these three categories. Quite often it takes more than a layman's expertise to sort it out. Unfortunately a free press often makes these assertions available directly to laymen.
