# 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 since time immemorial and has never moved before.

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

Would these planets ever meet or at least be 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
• Surely this is more of a physics question? – Fattie Feb 12 at 14:50

## 3 Answers

Yes, they would experience gravitational attraction. It would take a long time for them to collide... the formula (derived here and shown here) is:

$$t = \frac{\pi}{2} \sqrt{\frac{d^3}{2G(m_1+m_2)}}$$

where $$d$$ is the initial distance between the two planets of mass $$m_1$$ and $$m_2$$ 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
• What (indirect) evidence is available that gravity still acts 5 billion light years away? – Martijn Weterings Feb 12 at 21:39
• 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
• "long time for them to collide" they'll only collide certainly if they are both "stationary" to begin with, or, have certain relative vectors. if they are moving to begin with, they'll be attracted but in many cases will end up in a (huge, mammoth) orbit around each other. – Fattie Feb 15 at 11:38
• There is still the problem that there is indeed an effective upper limit to gravity. If two objects are too far apart, due to expansion their separation will increase faster than the speed of light. There are parts of the universe that will never experience gravity or light from other parts of the universe. See this answer and answers to Am I attracting Pluto? – uhoh Feb 28 at 8:33

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