# Can two planets in an empty universe meet/be pulled together?

For this question assume that the entire universe is completely empty. The universe is not expanding or contracting, it is completely motionless.

Only two identical earths without moon are left in the entire universe and they are 5 billion light years apart. From their starting position the earths do not spin or move.

Would these planets ever meet/pulled together? Or are the their respective gravities too small to have any meaningful effect?

• For a good 5 billions ly they ignore each other. – Alchimista Feb 9 at 12:34
• I want to say that probably you consider the entire universe as to be the entire visible and/or observable universe. The accepted answer is indeed satisfactory only under several assumption. It is more to answer about two generic masses in a static enormous room. Else cosmology sets in – Alchimista Feb 9 at 12:54
• I am familiar with the differences, but for this question to work i needed to be sure there where no outside factors affecting the two planets. – Tom Feb 9 at 13:24
• Is the universe expanding? – userLTK Feb 9 at 15:05
• It is not I'm sorry this was not mentioned. – Tom Feb 9 at 17:41

Yes, they would experience gravitational attraction. It would take a long time for them to collide... the formula is

$$\sqrt{\frac{d^3}{2G(m_1+m_2)}}$$ where $$d$$ is the distance, m1 and m2 are the masses of the planets, and G is the Gravitational constant. This gives a time of about $$10^{23}$$ years, much much longer than the universe has existed. This assumes Newtonian mechanics. Relativity would not change the conclusion much.

There is no known upper limit to gravity, and plenty of indirect evidence that it has no upper limit.

• The collision speed of a test particle falling to Earth from infinity is equal to the escape velocity, about 11 km/s. For 2 Earth-mass bodies, the speed will be similar, I think you can just multiply it by $\sqrt 2$, but I'm too sleepy to do the algebra right now. ;) – PM 2Ring Feb 9 at 20:16
• Can you add a source for that equation? If I set $d$ to either the total distance between the two, or half that, I don't get the same answer as from numerical integration. There might need to be a slice or two of pie in there somewhere; physics.stackexchange.com/a/14702/83380 and also physics.stackexchange.com/a/90722/83380 – uhoh Feb 10 at 3:39
• What (indirect) evidence is available that gravity still acts 5 billion light years away? – Martijn Weterings Feb 12 at 21:39
• Hi @JamesK, I believe your equation is incorrect. I can edit it and do the maths for you if you don't mind, but from the links in my comment above it should be pretty easy for to adapt here. – uhoh Feb 13 at 2:48
• I just did the algebra. The relative collision speed of 2 Earths is identical to the terminal speed of a test particle, so 11.186 km/s. – PM 2Ring Feb 13 at 21:05

Yes: Given that the universe isn't expanding at a faster rate than the speed of the gravity effect of those objects, then the gravity effect of them would be able to reach one another.

Even then it would take a very long time for the gravity effects of the objects to reach one another, so the objects would just stay there motionless for ages(in this case 5 billion light years) before starting to move towards eachother.

http://www.nowykurier.com/toys/gravity/gravity.html

• The last part isn't ok.At least not in general. See for instance researchgate.net/publication/…. Anyway the Q is edited and a static universe is now specified. So the problem is reduced to test masset at infinity or whatever distance so the answer should be easier – Alchimista Feb 10 at 9:01

YES!

Gravity will eventually pull them together. The thing is that gravity exerts itself over very vast distances, although the force decreases exponentially over distance. BUT, two motion less massive objects will attract each other, without any other influence whatsoever.

• "Exponentially" in a casual day to day sense but not in a mathematical sense. Gravitational attraction decays more quickly than inverse linear, doubling the distance more than halves the attraction, but not as fast as exponential. In is inverse quadratic which, in the world of maths, is slow compared to exponential. – badjohn Feb 15 at 9:08