# Does Newton's Law of Universal Gravitation work when particles are very close?

By Newton's Law of Universal Gravitation, the gravitational force between two particles is $Gm_1m_2/r^2$. Let's assume that the numerator is constant and happens to equal $1$.

Imagine that two particles that meet the above assumption are near each other in a vacuum. As they are colliding, the distance between them will at some point equal a Planck length. The force between them will then be: $1 /( 1.6 \times 10^{-35})^2\ \text{N}$. This is about $4\times 10^{69}\ \text{N}$.

Why doesn't this really happen?

• Why do you think this cannot happen? May 2, 2018 at 18:49
• Well, it seems like the formula implies that as the distance approaches zero, the force approaches infinity. This seems to happen regardless of mass. Rocks collide all the time and infinite force doesn't appear to be produced. May 2, 2018 at 18:58
• Both of these particles you've described would be black holes of quite a bit larger than one Planck length in radius.
– Chris
May 2, 2018 at 19:17
• Tests of Newtonian gravity get very tricky at length scales of tens of micrometers; here's a literature review by Adelberger et al., 2009. However, long before the Planck length, you have to start worrying about the strong and weak nuclear forces.
– rob
May 2, 2018 at 20:07

I think the only issue with your scenario is using Newton's law of gravity to calculate the value of the attractive force between the particles. There is no reason to think that this situation 'cannot' happen.

In the situation you've described, each particle has a mass of $122,406 \text{ }\mathrm{kg}$, which yields a Schwarzschild radius of $1.818\times10^{-22} \text{ }\mathrm{m}$. But the Plank length is considerably smaller, at $1.616\times10^{-35} \text{ }\mathrm{m}$. To be within one Plank length of each other, they would both need to be black holes, with each singularity inside the other's event horizon (in addition to its own, of course). In other words, this scenario involves two black holes merging together. We seem to be well beyond the range of validity for Newtonian gravity.

Can two singularities get this close to one another? I guess so. Interpreted literally, a 'singularity' in GR is a true point particle.

I am addressing your comment in this answer. The mass of a rock is not centred at a point. Even though the distance between the surfaces of the rock approaches zero, the force between them does not go to infinity. In case of point masses (which do not exist in reality) it can happen.

For example, the force between two uniform spheres is same as that of two point masses kept at their centre. When you make two spheres touch with each other, the force between them will be same as that of two point masses kept at distance equal to the sum of the radius of the two spheres, which is a finite force.

The answer is : no, it does not work at Planck scale. A reason is that particles are typically $10^{20}$ times larger than Planck length. A proton has a radius of the order of a femtometer.

This is like asking what happens if two stars are at a distance of 0.1 angstrom of each other !

It doesn't hold at such small separations. You need to then take into account electro-magnetism then and then at even smaller distances the strong and weak force.

Had Newton noticed this he might have been inspired to suggest a repulsive force at such small distances otherwise the distance between the boundaries of two particles would reduce to zero.

It may be the case that he had: He had a large corpus of notes and I'm not sure, even after 300 years have gone by, plus his fame as a physicist, that his corpus has been gone through in any thorough manner. This might change as his work is digitalised and put online for anyone to look at.