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.

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.

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, those pieces feel tiny forces. Second, I assumed that the object can be dismembered, and the results of the process leave us with a set of separated internal components of the original object. But these assumptions will generally fail, in one way or another, when there is a trapping force. The result is that the stored energy in the system can never be made zero. And so, energy conservation no longer allows us to conclude, even though we have put positive energy in to the system, that the initial stored energy was negative.

This does not prove that the initial stored energy is positive; it could be either positive or negative. 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 article (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.

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