Of Particular Significance

# Chapter 9, Endnote 6

• Quote: When it comes to rest mass, the whole can be greater than the sum of its parts. (It can also be less than the sum of its parts, as is the case in many familiar contexts.)

• Endnote: In nuclei, atoms, chemical bonds, and planets around stars, the stored energy that holds the system together is negative.

### Why Binding Energy is Typically Negative

In a typical object that is held together by forces — say, for example, the Earth itself — the stored energy of the object is negative. We can see this by imagining separating the object into its component parts, as follows.

Imagine we dismember the object, moving its pieces very far from one another, so that the forces between them become tiny. The separation of the components will require we expend some energy, working against the forces trying to hold the object together. Once the components of the object are so far apart that they no longer pull on each other, no energy is stored between them anymore.

But energy is conserved. Since the stored energy at the end of the dismemberment is zero, and we have put positive energy in to carry our the dismemberment, it must be that the initial stored energy of the object was negative.

Such an argument holds for humans, rocks, tables, stars, planets, molecules, atoms, and most composite objects that we encounter. And yet, there are crucial exceptions to it.

### Why Sometimes It Isn’t

While this argument holds true for almost all familiar and unfamiliar objects held together by forces, there are hidden assumptions that can fail when there is a trapping force, as for a proton’s or neutron’s quarks and gluons.

• First, I assumed that once the object is dismembered into small pieces very far apart, any forces between those pieces have become tiny.
• Second, I assumed that the object can in fact be dismembered in a simple way, and the results of the process leave us with a set of separated parts that were the internal components of the original object.

But these assumptions will generally fail, in one way or another, when there is a trapping force. Without these assumptions we cannot conclude, as we did above, that the stored energy in the dismembered system is zero. If there are still strong forces between the parts, then there may still be a lot of energy associated with whatever generates those forces. If the parts obtained in the dismemberment are not the parts that were inside the system to start with, then there are additional contributions to the energy that may have affected the dismemberment. And if we can’t dismember the system at all, then the whole argument falls apart. In any of these cases, we can’t so simply use energy conservation to conclude that the initial stored energy was negative, even though we have put positive energy into the system in our effort to dismember it.

A failure of the argument, however, does not prove that the initial stored energy is positive. It could be either positive or negative. It depends on the system.

For protons and neutrons, it turns out to be positive, for reasons that are not simple to explain. I have given some insight into why this is so in this post (itself based on this post and this post.) But if up and down quarks had much larger rest masses than they actually do, then the stored energy would have been negative. And thus the positive stored energy of protons and neutrons is not a matter of principle, but a matter of details.

What happens when we try to dismember a proton or neutron? It simply can’t be broken up into its component parts. Because the quark rest masses are so small compared to a proton’s or neutron’s rest mass, what happens is that any effort to extract a quark from a proton splits the proton into two or more hadrons (typically, in the language of this endnote 5 of this chapter, one baryon and one or more mesons). Below I’ve reproduced Fig. 1 from that article:

### An Aside About Quarks With Lots of Rest Mass

Incidentally, if instead all of the quark rest masses in nature were very large compared to the scale at which the strong nuclear force becomes strong, the proton really would be just made from three quarks. But we would instead find it impossible to dismember the proton by separating the quarks, because the proton would not break as it does in the picture above. Instead, it would stretch out into one or more long “strings” — tubes of chromoelectric flux, as described for instance in this post. Stretching it in this way would require an enormous amount of energy. At no point could the proton be dismembered into its three quarks with no force between them, and so the argument given above would still not work — even though in this case the binding energy of this heavy-quark “proton” would indeed be negative.

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