# Gravity's limits

Suppose there are only two hydrogen atoms in the entire universe. Suppose further that they are both situated at the very limits of the cosmos, equally distant. Would they exert an attractive force on one another?

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Yes. It would be miniscule. But yes. – Michael Brown Feb 20 '13 at 23:08
@MichaelBrown I tried various more detailed interpretations of the question in my answer, but I'm not sure if the latter two are correct. – Shivam Sarodia Feb 20 '13 at 23:24
"Equally distant" when there are exactly two things in the universe makes no sense at all. – Olin Lathrop Feb 21 '13 at 0:06

Yes. By Newton's Law of Universal Gravitation:

$Force = \frac{G{m_1}{m_2}}{r^2}$

where $G$ is the gravitational constant (a relatively small number), $m_1$ is the mass of one body (in this case one of the hydrogen atoms), $m_2$ is the mass of the second body (the other hydrogen atom), and $r$ is the distance between them.

This force will obviously be small:

1. $G$ is a small constant, symbolizing the relatively weak strength of gravity relative to other forces.

2. $m_1$ and $m_2$ are really small: they are just hydrogen atoms, which have little mass.

3. $r$ is really large, since the atoms are separated by a large distance. However, since this incredibly large number is in the denominator of the expression, it will make the force even smaller.

Because of this, the gravitation force will be minuscule - but it will still exist.

I'm not an expert (in any way) in relativity, but general relativity dictates that the speed of gravitational waves are limited by the speed of light just as other propagations containing information are. Because of this, the disturbance of the first hydrogen atom would take a long time to reach the second atom, but would do so anyway at some point far in the future.

Further integrating cosmology into the picture, I believe it could be argued that the atoms would not have a gravitational attraction. The universe is expanding faster than light can travel, and this has several interesting effects. First, we from Earth have a "cosmic horizon," which is the maximum distance we can see. Since the universe expands faster than light, as light speed out away from the Earth, the universe expands even faster than the light wave, meaning that we can only get access to so much of the universe before the expansion is too fast for our light wave to catch up. Similarly, two hydrogen atoms on opposite ends of the cosmos would be expanding apart faster than light. This means that even light, the fastest form of information transfer possible, would be unable to get from one atom to the other, meaning that the two would not exert a force on each other (as gravity too would be unable to traverse this distance). However, if you account for this, then you must also consider that it would be impossible to place the two atoms, for after placing one, the other side of the universe would expand too fast for you to catch up to it, even at the speed of light.

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You are correct about the relativistic effects, which I neglected above. You can actually include the cosmological constant (somewhat inaccurately, be warned) as a repulsive modification of the gravitational potential: $\Phi=-\frac{GM}{c^{2}r}-\frac{\Lambda}{6}r^{2}$. (Einstein's equations are nonlinear so it would be a mistake to try and add these expressions for multiple bodies.) So at some cosmological distance the attraction becomes a repulsion. – Michael Brown Feb 20 '13 at 23:36
Should also mention that there is a weak electrostatic attraction between the hydrogen atoms. It starts out stronger than gravity but decreases faster with distance, so at some distance there will be a change over and gravity becomes more important - but I don't know what that distance is. :) – Michael Brown Feb 20 '13 at 23:42