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

I can understand that on small scales (within an atom/molecule), the other forces are much stronger, but on larger scales, it seems that gravity is a far stronger force; e.g. planets are held to the sun by gravity. So what does it mean to say that "gravity is the weakest of the forces" when in some cases, it seems far stronger?

When we ask "how strong is this force?" what we mean in this context is "How much stuff do I need to get a significant amount of force?" Richard Feynman summarized this the best in comparing the strength of gravity - which is generated by the entire mass of the Earth - versus a relatively tiny amount of electric charge:

And all matter is a mixture of positive protons and negative electrons which are attracting and repelling with this great force. So perfect is the balance however, that when you stand near someone else you don't feel any force at all. If there were even a little bit of unbalance you would know it. If you were standing at arm's length from someone and each of you had one percent more electrons than protons, the repelling force would be incredible. How great? Enough to lift the Empire State building? No! To lift Mount Everest? No! The repulsion would be enough to lift a "weight" equal to that of the entire earth!

Another way to think about it is this: a proton has both charge and mass. If I hold another proton a centimeter away, how strong is the gravitational attraction? It's about $10^{-57}$ newtons. How strong is the electric repulsion? It's about $10^{-24}$ newtons. How much stronger is the electric force than the gravitational? We find that it's $10^{33}$ times stronger, as in 1,000,000,000,000,000,000,000,000,000,000,000 times more powerful!

• This answer might be useful in response to a different question. Here it does not seem to jive with what the OP is asking. Then again, after seeing some of the OP's comments it is not clear to me what he was asking. – user346 Jan 31 '11 at 19:48
• This answer just shows how electromagnetic force is much stronger than gravitational force in an example. Do you mean that the meaning of the statement, "Gravity is the weakest force" is that it is the weakest in "many examples" ? – ashpool Jun 27 '14 at 11:47
• It may or may not also have been Feynman who said words to the effect that the question "why is gravity so weak?" can be rephrased as "why is the mass of the proton so small?". – Nathaniel Jul 28 '14 at 6:22
• Charge and mass are two different quantities. What amount of charge do you equate to what amount of mass to state that one force is stronger? – Mahathi Vempati Dec 26 '15 at 11:42
• @Tinkidinki: Another way to think of this is that mass is a measure of what warps space to generate gravitational force, while electric charges create electrical fields that can move objects with mass against or through a gravitational field. – Ernie Jan 13 '17 at 18:39

When we say that gravity is much weaker then the other forces we mean that its coupling constant is much smaller than the coupling constants of other forces.

Think about a coupling constant as a parameter that says how much energy there will be in per "unit of interacting stuff". This is a very rough definition but it will serve our purpose.

If you determine the coupling constants of all different forces, you discover that, in decreasing order, strong, eletromagnetic and weak forces are much, much stronger than gravity.

You need around $10^{32}$ (that is 100,000,000,000,000,000,000,000,000,000,000) times more "stuff interacting" to get around the same energy scale with gravity if you compare it with the weak force. Moreover, the difference between strong, weak and electromagnetic forces among themselves isn't nearly as extreme as the difference between gravity and the other forces.

• +1, this is basically the answer I would have posted if I hadn't already seen it here. (I think it would not be inappropriate to include a small bit of mathematical detail, though.) – David Z Jan 31 '11 at 2:33
• Yes, it answers what it means, but doesn't offer an explanation. I guess, though, he didn't ask for one... – Gordon Jan 31 '11 at 2:55
• Actually the Hierarchy problem is about the radiative corrections in the Higgs propagator. The large discrepancy of coupling constants is not a real problem. – Leandro Seixas Jan 31 '11 at 3:41
• @Leandro: that is just one particular hierarchy problem. In general @space_cadet is right, the problem in general is with understanding the magnitude of coupling constants which usually requires some fine-tuning or new physics. The actual mechanism which explains it (in your case either fine-tuned cancellation of bare mass by quadratic radiative corrections or the usual SUSY solution) is only secondary. By the way, another instance of a famous hierarchy problem is the one of the smallness of cosmological constant. – Marek Feb 3 '11 at 21:23
• This doesn't answer the question. To answer the question, you would have to explain what system of units you have in mind such that all four coupling constants have the same units and therefore can be ranked in size. – user4552 Apr 5 '13 at 4:10

I don't think that any of the existing answers fully answer this rather subtle question (correctly). If we just consider the interactions themselves, and not particular particles that they couple together, then there is no meaningful sense in which gravity is any weaker than any of the other forces. This simply follows from the fact that the gravitational coupling constant $$G$$ has different units than the coupling constants of all of the other Standard Model interactions, so the fundamental "strengths" of the interactions are incomparable.

It's true that for some applications, it's simplest to work in Planck units where $$\hbar = c = G = 1$$ - but that statement simply reflects the fact that no dimensionless ratios can be formed out of those constants, so this simultaneous assignment is possible. It's therefore incorrect to say (as claimed in another answer) that the strength of gravity is "equal" to the strength of any other interaction in any nontrivial way. It's simply incomparable, neither weaker nor stronger. You could just as easily choose a system of units (like SI!) in which either coupling constant's numerical value is arbitrarily larger than the other's.

To meaningfully compare the strength of gravity to that of the other interactions, you need to consider the specifics of the Standard Model (SM) matter fields. The dimensionally meaningful statement is that $$G M_H^2/(\hbar c) = (M_H/M_P)^2 \approx 10^{-34} \ll 1$$, where $$M_H$$ is the Higgs mass and $$M_P$$ the Planck mass. (You get similarly small numbers if you plug in the mass of any other SM particle.) Different types of physicists will find different natural ways to interpret this inequality.

To a (phenomenological) particle physicist, the natural mass scale is the Higgs/SM scale $$M_H$$ (and the natural velocity and action scales are $$c$$ and $$\hbar$$). From this perspective, $$G = M_P^{-2} \approx 10^{-34}$$, and the natural question is

"Why is the gravitational interaction between SM particles so much weaker than the other SM interactions? Or equivalently, why is the Planck mass so huge relative to the SM scale?"

To a quantum gravity theorist, who is less concerned with the details of the Standard Model, the natural mass scale is the Planck mass $$M_P$$. From this perspective, $$M_H \approx 10^{-17}$$, and the natural question is

"Why is the Higgs mass so tiny relative to the Planck scale?"

This is what Wilczek meant when he said "The question is not 'why is gravity so weak?' The question is 'why is the electron mass so small?'." I'll leave it to the philosophers to debate whether this is actually a "better" formulation of the question. The "hierarchy problem" really encompasses both formulations of this question, but is usually formulated from the latter perspective, and phrased in terms of "unnaturally fine-tuned radiative corrections to the Higgs propagator" and a bunch of other jargon involving the renormalization group, whose details are orthogonal to the OP's question.

Another answer states another common misconception, which is that the weakness of gravity simply stems from the irrelevancy/nonrenormalizability of the gravitational interaction. To see why this explanation is seriously incomplete, it's useful to consider yet another type of physicist's perspective: that of a condensed-matter theorist. The key point is that irrelevant operators (in the technical sense of the word) are only irrelevant (in the colloquial sense of the word) at energies far below the microscopic "UV cutoff" energy scale.

In this case, the cutoff scale is the Planck scale, so it is indeed straightforward to show from general arguments that gravity is very weak at energies far below the Planck scale. But this doesn't really answer the question; it just pushes it back to the question "Why do SM particle interactions occur at energies so far below the 'natural' Planck scale?" In a generic system, we would expect the low-lying excitations (the elementary particles) to have mass gaps on the order of the microscopic energy scale (the Planck scale). Only very close to a phase transition does the mass gap almost close and field theory become applicable. So to a condensed-matter physicist, the natural formulation of the hierarchy problem is

"Why are the Planck-scale interactions so finely tuned to lie near a phase transition, thereby allowing us to use field theory to accurately describe the low-lying excitations (the elementary particles)?"

Gravity seems stronger because it's always attractive. Of the other 3 interactions:

• Electromagnetism has positive and negative charges, so it only manifests macroscopically when there is a charge imbalance.
• The weak and strong interactions are intrinsically short-ranged.
• Though color (i.e. strong force related) charges come in 3 (plus 3 anti) flavors they share with electric charges the ability to form "neutral" bodies (indeed confinement requires this), the phenomena called "color transparency" (known in meson systems and theorized in baryons) takes advantage of this even inside the range of the strong force. In principle bodies could be assembled that are on aggregate "weak neutral", though not using a small integer number of bits. – dmckee --- ex-moderator kitten Jan 31 '11 at 16:13

This is indeed something one has to be careful about, because, after all, gravity scales with the mass of the particles in question, whereas the other forces scale with the electric charge or the magnetic moment. It appears that one compares apples with pears.

However, I believe the declaration that gravity is the "weakest" of the forces stems indeed solely from its irrelevancy on the scale of particle physics.

• other forces don't scale ? sure ? – iamgopal Jan 31 '11 at 12:14

The Randall-Sundrum model explains it. The other forces are confined to the brane which we consider to be our universe. The brane is embedded in higher dimensional space where some of the dimensions may be compactified, but others could be larger or even infinite (a 5 dimensional anti-de Sitter space in which a (3+1)dim brane is embedded.All particles except the graviton are bound to the brane.) Higher dimensional space is called the bulk. If gravity is not confined to our brane and can penetrate into the bulk, that would explain its weakness. The problem with the extreme difference in strengths of the forces is termed the hierarchy problem (weak force=$10^{32}$grav force). There are other explanations involving supersymmetry.

• To explain the down-vote: the question isn't about why gravity appears weaker. It's the fact that, in my mind, gravity doesn't appear weaker. – Smashery Jan 31 '11 at 4:04
• Dear @Smashery please quit down-voting answers based on perfectly good physical arguments. Otherwise it appears you not looking for answers but only an affirmation of your pre-existing beliefs. – user346 Jan 31 '11 at 19:41
• @Smashery: Your headline question seems to ask why gravity is considered the weakest force. In your elaboration you seem to be asking why you think it should be stronger. Arrghh, I am not a telepath. An obvious reason would be that we evolved brains on one massive mother of a planet and gravity is accretive. Any number of other explanations as to why you think gravity should be stronger come to mind, but none of them are flattering. – Gordon Feb 1 '11 at 6:10
• @space_cadet: Thanks for the support. I thought he wanted to learn something, not that I am the Amazing Kreskin. – Gordon Feb 1 '11 at 6:12
• My apologies if you took the downvote personally; like I said elsewhere, you are obviously knowledgeable about the topic; but the gap in my knowledge was more that "to my intuition, gravity doesn't seem weak at all." Sorry if that wasn't clear in the question. I've since upvoted another of your excellent answers. – Smashery Feb 1 '11 at 6:20

Gravity is weak because the masses of elementary particles are so small. Gravity has a natural mass unit, $m_p~=~\sqrt{\hbar c/G}$, the Planck mass, which is about $10^{-5}$ g. The proton is $22$ orders of magnitude less massive. So the stuff which makes up the world is elementary particle “styrofoam stuff” which gravity couples to.

This can be seen as well with IIA strings and their S-dual heterotic strings. Those heterotic strings just do not like to stay on our brane, which they have no end points to form Chan-Paton factors or Dirichlet boundary conditions on the brane with. They slip through our brane as if nothing is there. Their S-dual strings are open strings on the brane, but with puny masses --- far less than the Planck mass or the mass corresponding to the string tension.

• To explain the down-vote: the question isn't about why gravity appears weaker. It's the fact that, in my mind, gravity doesn't appear weaker. – Smashery Jan 31 '11 at 4:03
• Then look for answers on a psychology site, not a physics site. We're supposed to be answering physics questions, not explaining how your mind works. – pho Jan 31 '11 at 14:33
• @Smashery this is the best answer to your question. To repeat myself, please do not down-vote unless you actually have a good reason to do so. And if you do do so, don't proclaim it. It hurts your credibility. – user346 Jan 31 '11 at 19:42
• @space_cadet Not sure why this is such a crazy request by @Smashery. He wants an explanation of how to reconcile everyday phenomena with commonly cited theory: "Gravity seems to be the most important force in my life, yet people say it's the weakest. Explain." Do you see how talking about Chan-Paton factors might not be a satisfactory response? – spencer nelson Jan 31 '11 at 20:25
• Hi @Spencer. If you were to ask a physicist today the question "why is gravity the weakest force" then of the many ways of stating the problem, one way is to note (as @Lawrence did) the tremendous difference between the natural mass scale of gravity and that of the standard model. This is what is referred to as the "hierarchy problem". Ask a postdoc or a professor about it and I'm sure they would be delighted to explain. The tidbit about S-duality is only one of many ways to find a resolution to this question, another one being the Randall-Sundrum model as @Gordon mentions in his answer. – user346 Jan 31 '11 at 21:02

To compare the strength of e.g. the electromagnetic force and the gravitational force one should not simply compare the difference of strength for a specific particle, because there are too many (electron, proton, myon, etc.) and all these comparisons would yield a different number. Instead let's compare gravity and EM force using a scale given by fundamental constants. For gravity this means, we choose the gravitational force between two particles of Planck mass $m_P=\sqrt{\hbar c/G}$ and for the EM force we compute the force between two particles of Planck charge $q_P=\sqrt{4\pi\epsilon_0 \hbar c}$. If we compute the forces for these two cases respectively we get $$F_g = G \frac{\hbar c/G}{r^2} = \frac{\hbar c}{r^2}\\ F_{EM} = \frac{1}{4\pi\epsilon_0} \frac{4\pi\epsilon_0 \hbar c }{r^2} = \frac{\hbar c}{r^2}$$ So we see the gravitational force and the electromagnetic force are actually of equal strength and range when they are compared at their natural scale.

In this sense gravity is not the weakest force. Instead the puzzle here can be rephrased to: Why is the mass of the known particles so low compared to the Planck mass, or why is the charge of the known particles so high compared to the Planck charge?

For example the ratio between planck mass and electron mass (the proton mass should not be used for comparison, because a proton is not a point particle and it's mass is made complicated by effects of QCD) is $$\frac{m_P}{m_e} = 2.389 \cdot 10^{22}$$ whereas the ratio between planck charge and the charge of the electron is $$\frac{q_P}{q_e} = 11.706$$ The ratio of these ratios is $2.040 \cdot 10^{21}$ and why this is such a high number is the real puzzle, and I don't know any theory which could explain this number.

It should be noted that the so called Mass-to-charge ratio for an electron is of order $10^{11}$, for a hypothetical particle with a Planck charge and a Planck mass it is accordingly of order $10^{-11}$ $$\frac{q_e}{m_e} = 1.759 \cdot 10^{11} \frac{C}{kg} ,\ \frac{q_P}{m_P} = 8.617 \cdot 10^{-11} \frac{C}{kg} = \sqrt{\frac{4\pi\epsilon_0}{G}}$$ which I think is a quite remarkable "symmetry".

UPDATE: The numbers shown here might not be familiar. But I want to note that the fine-structure constant is $$\alpha^{-1} = \left(\frac{q_P}{q_e}\right)^2 = 11.706^2 = 137.036$$ and the gravitational fine structure constant is $$\alpha_G^{-1} = \left(\frac{m_P}{m_e}\right)^2 = (2.389 \cdot 10^{22})^2 = 5.709 \cdot 10^{44}$$

• I don't understand. Your last line would imply that $q_E m_P/m_e q_P = 1.759/8.617$ ~ O($1$), right? But the lines above that would say that $q_E m_P/m_e q_P= 2.389*10^{22}/11.706$ ~ O($10^{21}$) . Right? – Rococo Jan 13 '17 at 16:46
• You are right. I missed a minus sign. $q_P/m_P = 8.617\cdot 10^{-11}$. I fixed the paragraph. – asmaier Jan 13 '17 at 18:24
• Your claim that "the gravitational force and the electromagnetic force are actually of equal strength and range when they are compared at their natural scale" is trivial and misleading, because their apparently equality just comes from the fact that you are working in units designed to make them both equal to 1. In fact they have different units and are incomparable, so you make choose units in which their numerical values are as near or far from equality as you want. – tparker May 8 '19 at 2:02

Not exactly an answer but a resource that bears relevance to this question: Richard Feynman had a few words to say about the notion in his messenger lectures series, lecture 1, around the 48:20 time mark

Gravity is the weakest force as its coupling constant is small in value. Gravity cannot be felt by us in daily life because of the huge universe surrounding us. Electromagnetic force is undoubtedly stronger as it deals with microscopic particles (electrons, protons). Gravity is always attractive in nature. It is a long range force among all other interactions in nature.

• ""gravity cannot felt by us in daily life "" Rofl I recommend You lay down under an apple tree for some hours! – Georg Aug 3 '11 at 9:17
• Or jump out from the window. – Abhimanyu Pallavi Sudhir Jun 27 '13 at 4:00
• Try to walk or sit in the air :) – Xtro Mar 4 '14 at 17:38
• This doesn't quite deserve the downvotes it received. There is a reason to downvote, which is that this answer is not an answer to the question. However, that you cannot feel gravity is exactly correct. You feel weightless when you jump out of a window. There's nothing to feel because you can't feel gravity. Nothing can. When you splat and hit the pavement moments later, that's not gravity. That's the normal force you are feeling. – David Hammen Oct 2 '14 at 11:35