By Green Horn

2011-04-11 11:20:24 8 Comments

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What does it mean to say “Gravity is the weakest of the forces”?

It is said nuclear force is the strongest force in nature.. But it is not true near a black hole where gravitational force exceeds nuclear force.. So which is the strongest force in nature?


@Lawrence B. Crowell 2011-04-11 12:47:10

There are two elements to this question. The first has to do with the strength of gravitation relative to other forces of nature. It is not hard to see that gravity is extremely weak. One can see this mathematically in the ratio of the magnitude of the gravitational and electromagnetic forces $$ F_g~=~\frac{GMm}{r^2},~F_e~=~\frac{q_1q_2}{4\pi\epsilon_0 r^2}, $$ for $M$ and $m$ the masses of a proton and electron and $q_1~=~-q_2~=~e$. The ratio is about $F_e/F_g~\simeq~10^{39}$. A more direct semi-empirical test is to drop an elastic ball from a height $h$. The ball accelerates by gravity to the floor, where upon hitting the floor it very quickly reverses the direction of motion and returns to some height $h’~<~h$. The electromagnetic forces which hold the molecules together in the ball and floor in a very small time period induce a change in momentum $\Delta p~=~ma\Delta t$, so the force $ma~>>~mg$ for $g~=~9.8m/s^2$ the acceleration of gravity near the Earth’s surface. It also takes the entire mass of the Earth to give rise to the acceleration of Earth’s gravity, while it take a few grams or maybe a kilogram of mass to induce the acceleration of the ball through molecular forces, which are ultimately electromagnetic. The nuclear force is about $100$ times the electromagnetic force, though it is short ranged and confined to the scale of hadron length $\sim~10^{-13}cm$.

Gravity becomes large if a huge amount of mass is compacted into a small volume. The mass equivalent of the sun compressed into a volume of about $1.5km$ radius becomes a black hole. The electromagnetic force does not tend to do this. A concentration of lots of charge is repulsive and requires another force to bind it together. In the case of the nucleus this is the emergent nuclear force of baryons and mesons, which is a low energy scale aspect of QCD.

The second aspect of this problem has to do with the variation of the relative strength of forces at different energy scale of interaction. The function $G$, such as a propagator function, for the interaction varies with interaction coupling parameter $g$ as $G(\partial G/\partial g)^{-1}$ and where the coupling strength depends upon the energy scale $e$, often denoted by $\mu$, as a beta function $$ \frac{\partial g}{\partial ln(\mu)}~=~\beta(g). $$ This equation emerges from the dependency of terms on the renormalization cut-off, which would require a long discussion to break out. The connection to the propagator function above is the Callin-Symanzik equation for an n-point propagator $$ \Big(\lambda\frac{\partial}{\partial\lambda}~+~\beta(g)\frac{\partial}{\partial g}~+~n\gamma\Big)G(x_1,x_2,\dots,x_n,g,m)~=~0, $$ for $\gamma$ a field scaling factor, and $\lambda$ the scaling bound or reference --- often the cut-off. This equation is similar to a Navier-Stokes equation, and it is generalized into renormalization flow equations. The $\beta$-function connects this most effectively with supersymmetry and strings. The running coupling constants $g$ which depend on the energy scale converge in a fairly natural way near the Planck scale of energy. This means the electromagnetic, or electroweak coupling, and gravitational coupling converges with the strong nuclear force at extremely high energy. In this way the forces of nature tend to “fuse” into a single interaction with a large symmetry group and with a single interaction strength.

@Luboš Motl 2011-04-11 12:12:30

The gravitational force exerted by a black hole is strong because the force is proportional to the mass and the black hole as a large mass. In the same way, the gravitational field of the Earth may beat a weak enough magnet - because the Earth is large and the magnet is small and we're comparing apples and oranges.

But fundamentally speaking, among particles with masses that correspond to elementary particles, gravity is the weakest force. For example, the gravitational force between two electrons is $10^{43}$ times weaker than the electrostatic one. The weak nuclear force is as strong as the electromagnetic one at very short distances - but exponentially drops at distances much longer than the W-boson wavelength. The strong nuclear force is the strongest among the four forces.

It seems that the weakness of gravity relatively to all other forces, when evaluated at the level of elementary particles, is a general principle that has to hold in any consistent quantum theory of gravity, see

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