# Is most mass from KE or PE?

I know most invariant mass is from the interaction of quarks with the gluon field, but is that kinetic energy or potential energy, or does this question not make sense? Am a college freshman but I want to understand this. Thank you!

## Is the question sensible?

Let's start by examining the question

"Does it even make sense to ask what fraction of the mass comes from potential and kinetic energies?"

in general before we try to work out the answer for the strong interaction in particular

### Long-range forces

For gravity, electrostatics, and the nuclear force (AKA "strong nuclear force" or "residual strong force") the force and potential vanishes at large distance. The result is that systems held together by these forces (i.e. bound) have negative potential energy.

But of course the kinetic energy is positive.

So the mass defect of these systems is caused by the potential energy but partially relieved by kinetic energy.

The question has—at best—an unsatisfying answer for long-range forces.

### The strong force

The real strong force (the one that holds baryons together) is different. It exhibits both confinement (which requires that the potential grow rapidly at large distances) and asymptotic freedom (which means the potential is constant at very short distances). Moreover it is experimentally determined to have a potential roughly of the form $$U_\text{strong}(r) \approx kr \;. \tag{1}$$ As a result in this system both the kinetic and potential energy are positive.

The question makes sense for the strong interaction.

## A tool to find the answer

The Virial theorem is a relationship in classical mechanics between the mean kinetic energy of a system and a particular average over the inner product of the forces on particles and the position of the particles (which is called the "virial of system") as long as the system meets certain requirements (many bound systems qualify).

If the force laws at work between the particles are central and proportional to a power of distance between the particles (that is of the form $$U = kr^n$$), then the theorem can be extended to give a relationship between the average kinetic energy of the system and the average potential energy of the system.

Using the force law in (1) to get the relationship for the strong force we find1 $$\langle T \rangle = \frac{1}{2} \langle U \rangle \;,$$

and if we write the dynamic energy $$E_d$$ of the system2 we get \begin{align} E_d &= \langle T \rangle + \langle U \rangle \\ &= \frac{3}{2} \langle U \rangle \;. \end{align}

So, long story short one-third of the dynamic energy is kinetic and two-thirds is potential.

1 Writing $$T$$ for kinetic energy and $$U$$ for potential energy.

2 Which is to say the part of the mass that comes from the behavior of the partons rather than the mass of a valence quarks.

• 1) If potential energy can be negative, can something have negative mass? Is this possibly how virtual particles have negative energy? – Alexander Wu Jun 2 '18 at 2:06
• 2) In classical mechanics you can set an arbitrary zero value for potential energy. But that would mean mass is arbitrary. Does that no longer work in relativity? – Alexander Wu Jun 2 '18 at 2:08
• The mass of a system made from multiple parts can be less than the total mass of all the parts. In the context of nuclear physics the difference is called the "mass defect", but this mechanism neither allows negative absolute masses nor explains the virtual particles of quantum field theory (which should be treated as "real" only with care and within well defined contexts physics.stackexchange.com/q/162845/520 physics.stackexchange.com/q/185110/520 ). – dmckee Jun 2 '18 at 2:45
• And in relativity you can still take potential energies to have arbitrary zeros for many purposes, but there is a correct choice for computing masses. – dmckee Jun 2 '18 at 2:46