"Weakness" of gravitational force

Question

I often hear that gravitational force is much "weaker" than electroweak and strong forces. But how can you compare the strength of interactions without the parameters like mass, charge on which it depends? Surely the gravitational force can be made as (practically) strong as one wants by increasing the mass?

Answer 1

For forces that can be expressed in terms of a quantum field theory, you can compare the size of the coupling constants (which are dimensionless). In short this means that perturbative expansions of weaker forces are well represented by a small number of leading terms because the series converges quickly, while those of stronger forces require more terms (or in the case of the strong interaction don't converge at all in the naive application).

This spares us the philosophical quandary of comparing dimensional coefficients in force laws.

So far, however, there is no quantum field theory of gravity. But it is possible to work out the scale of the gravitational coupling constant by observing scattering problems for which there is a close analogy in E&M. I suppose that this in effect this means that we are comparing the force-law coefficients written in natural units ($c = \hbar = G = 1$).

Answer 2

You compare the size of the constants in the force law. For example $$F_g = \frac{GMm}{r^2},$$ and $$F_E = \frac{kQq}{r^2}.$$ In SI units $k= 9 x 10^9$, while $G = 6.67 x 10^{-11}$. That means to have an equivalent force, you have to assemble a mass twenty orders of magnitude (in SI units) larger than a collection of charges.

Basically, this, along with the fact the electric force has both negative and positive charges is responsible for most of the interactions we see in everyday life, and in astronomy. You wonder why gravity is the dominant force on large scales? It's because, even though it's smaller than the E/M force, you cannot create a large collection of like charges because they all repel each other while you can assemble large massive bodies. However, on small scales, like the atomic scale, a few charges are enough to create molecules without being pulled together by their mutual gravity.

Answer 3

The proposition ''gravitational force is much weaker than electroweak and strong forces'' is wrong , at least if you mean is ALWAYS weaker. For example, for a charged objects like an electron you can simply make the ratio between the force of gravity $ F_g=\frac{GmM}{r^2} $ and the electric force $ F_e=\frac{kqQ}{r^2} $. You find a difference of $ \simeq 40 $ orders of magnitude. But the result is inverted for a very massive object or for a very energetic one.

The point is that you don't have some adimensional coupling constant to compare, like with the other forces. So the comparing is energy dependent (or equivalently mass dependent).