# Relation between binding energy and inertial mass

1. Jul 7, 2012

### Gulli

This question has been bugging me for a while now. I roughly understand how the Higgs mechanism gives elementary particles their rest mass and I also understand that gravity couples to all forms of energy, including binding energy in a nucleus or atom. I also know most of the mass of a system (such as an atom) comes from binding energy between its components, not from the rest masses of those components.

What I don't understand is how binding energy affects the inertial mass of the system. Why is the inertial mass not different from the gravitational mass, with the inertial mass being equal to the rest masses of the components and the gravitational mass equal to the total energy of the system? Do the disturbances in the electroweak and/or strong fields associated with binding energy couple to the Higgs field somehow so that the total inertial mass of the system is affected?

2. Jul 7, 2012

### mathman

Not true. If you add up the masses of the nucleus of an atom's components (protons plus neutrons), it will be greater than the mass of the nucleus. The difference is the binding energy, which holds the nucleus together.

Inertial mass is equal to gravitational mass - no one has measured any difference.

3. Jul 7, 2012

### Gulli

Protons and neutrons are not elementary particles. The difference is large when you compare the mass of three quarks to that of one proton. In any case the magnitude of the difference doesn't really matter, only the fact that there is a difference.

Yes, but that wasn't my question, my question was why is there no difference?

4. Jul 8, 2012

### netheril96

Gravitational mass is actually a classical concept because it is the "charge" appearing in Newton's gravitational law. There is no analogue in general relativity.

5. Jul 8, 2012

### Gulli

I know, Einstein postulated that inertial mass and gravitational mass are one and the same. Experiments confirm this, but why? How does binding energy cause inertial mass? That question seems to be avoided in every source I come across, even the ones that meticulously explain how an elemental particle's rest mass is caused by the Higgs mechanism. I know that when in physics many sources avoid a question or state that it's "trivial", that's often a sign the author doesn't know either because and/or it's very complicated, I'm hoping that's not the case this time.

6. Jul 8, 2012

### netheril96

I don't understand your question. You are not familiar with mass-energy equivalence?

7. Jul 8, 2012

### Gulli

I am, but how would that answer my question?

8. Jul 8, 2012

### netheril96

If you are, then it should be no surprise to you that mass contains contribution from energy.

9. Jul 8, 2012

### Gulli

I know it does, but what's the mechanism (apart from the Higgs mechanism for the rest mass)? How does the interaction through massless photons between an electron and a proton make a hydrogen atom more difficult to move around than separate protons and electrons?

10. Jul 8, 2012

### kmarinas86

So-called binding energy isn't actually energy in the binding state. It's the energy required to break up the binding state. It's identical to the energy released in the formation of the bond.

High nuclear binding energy per nucleon represents energy lost to the environment. Look at the nuclear binding energy curve:

http://www.a-levelphysicstutor.com/nucphys-binding-energy.php

http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/nucbin.html

http://demonstrations.wolfram.com/EffectiveAndInertialMassesOfAPhotonNearABlackHoleForAFamilyO
So, the "inertial mass" is maintained. It is simply transferred to the photon.

11. Jul 8, 2012

### kmarinas86

You don't "add masses of a nucleus". You add masses to form a nucleus. This is known as fusion, and the byproducts of fusion carry away inertial mass.

The idea that protons and neutrons have the same mass before and after they fuse is completely wrong.

Finally, there is no literal addition of energy that represents the binding energy. So-called binding energy is in fact a negative change in potential energy, which is not a coincidence, since energy is lost, not gained, in the formation of an attractive bond (and bonds are attractive, by definition).

12. Jul 8, 2012

### D H

Staff Emeritus
Because that's what mass is. Mass is bound energy.

Because physicists say so.

Most inquisitive kids learn that if they keep pestering their parents or teachers with a chain of what/when/why/how questions, they will inevitably get that "because I said so" answer.

Physicists say this too, only the words they use are "it's axiomatic." In this case, the equivalence of inertial mass, active gravitational mass, and passive gravitational mass is embodied in the Einstein's equivalence principle. Experimental physicists test strong statements such as this. The equivalence of inertial mass and gravitational mass has been tested to the nth degree. Theoretical physicists try to find deeper explanations of these axiomatic statements. For example, the conservation laws are deeper than Newton's 3rd law, Noether's theorems are deeper than the conservation laws. Explaining the equivalence of inertial mass and gravitational mass are equivalent is part of the holy grail of physics. This explanation would be a part of the so far non-existant theory that ties quantum mechanics and general relativity. Until this theory is developed and verified, we're stuck with "because we said so."

That's a different question. It's not the Higgs field. The Higgs gives elementary particles their masses. The strong interaction gives protons and neutrons their masses. There is a whole lot of energy bound up in those protons and neutrons. That's why protons and neutrons are considerably more massive than than the sum of the masses of the quarks that form them. Mass is bound energy.

13. Jul 8, 2012

### Gulli

Photons (and gluons) with non-zero inertial mass would solve my question, but since they don't couple to the Higgs field, what is the origin of their inertial mass?

So the way I see it now is this: an elementary particle that is standing still (relative to the observer and all that) has zero kinetic energy and would therefore be expected to have zero mass (since mass depends on energy exclusively), except when it interacts with the Higgs field which imparts the particle with some sort of rest energy, even in the absence of kinetic energy. Elementary particles that don't couple to the Higgs field (or alternatively: whose progenitors have undergone the symmetry breaking properties of the Higgs field and happened to end up in the "no rest mass" bin), like photons and gluons, would still be expected to have no energy and no mass at all when standing still, however since particles that don't couple to the Higgs field always travel at the speed of light we don't have to think about such a weird situation. Composite particles gain additional mass because of the kinetic energy of the (time average number of) elementary particles mediating the binding force(s) at work, plus any rest mass these mediating elementary particles may have. The Higgs field is then merely a convenient (and apparently correct) way to explain why some particles have a rest mass and others don't. Photons and gluons have inertial masses because of their kinetic energy but are somehow "born" at the speed of light, before their inertial mass could hamper their acceleration towards c (though that interpretation would make it possible to deccelerate a photon, which I'm pretty sure should be impossible).

Is this somewhat correct?

14. Jul 8, 2012

### D H

Staff Emeritus
Photons and gluons have zero inertial mass.

No. It is completely incorrect.

Massless particles always go at c. The concept of being at rest with respect to a photon doesn't make sense. Massive particles always go at less than c. The concept of being at rest with a massive particle does make sense. It is a frame in which the kinetic energy of the particle is zero. Just because kinetic energy is zero does not mean mass is zero. It just means that velocity is zero.

15. Jul 8, 2012

### Voltz

Lot of people talking about binding energies of nucleons but I believe the OP is talking about the binding energy in elementary particles - that is the bound energy in the strong force holding quarks together to form protons and neutrons.
Mass and Energy are two sides of the same coin, anything with energy has mass, that's why even photons react to gravity - binding energy is a form or 'mass' and because inertial and gravitational mass have been empirically observed to be the same then binding energy manifests as both 'forms' of mass

16. Jul 8, 2012

### Gulli

Yes, that's the kind of energy I was talking about.

Don't they have effective inertial mass (E/c^2), seeing as they are a source of gravity and gravitational and inertial mass are supposed to be equal? also, something's gotta give: if gluons don't have inertial mass themselves how can they contribute to the inertial mass of a system of quarks held together by gluons and how can they interact with gravity?

Yes, that's what I said: we don't have to worry about what a photon standing still would be like because it always travels at c.

But wouldn't it mean exactly that, if the Higgs mechanism didn't exist? An elementary particle's energy is the sum of its rest mass and its kinetic energy, when standing still there would be no kinetic energy and without the Higgs mechanism there would be no rest mass.

Last edited: Jul 8, 2012
17. Jul 8, 2012

### Staff: Mentor

In the usual convention nowadays in which "mass" = "invariant mass" a.k.a. "rest mass", the mass of a system of particles generally does not equal the sum of the masses of its parts. The mass of a system is the total energy of the system divided by c2, in the reference frame in which its total momentum is zero (i.e. the system as a whole is at rest even though its individual parts may be moving around). The total energy includes the energy equivalents of the masses of the component particles, the potential energy of the system, and the kinetic energies of the component particles (again, in the reference frame in which the total momentum is zero).

We don't have to use the rest frame of the system in order to calculate the mass. The following equation works in any inertial reference frame:

$$m_{system} c^2 = \sqrt {E_{system}^2 - (p_{system} c)^2}$$

which reduces to the description above if $p_{system} = 0$.

Also note that in the fundamental equations of general relativity, the Einstein field equations, mass does not appear explicitly. As you're probably aware, what we call "gravity" is a manifestiation of spacetime curvature. The curvature depends on the stress-energy tensor which has components corresponding to energy and momentum. Mass enters into the picture by contributing to the energy of a system; but it's not the only contribution to the energy.

Last edited: Jul 8, 2012
18. Jul 8, 2012

### cosmik debris

Can you explain why the binding energy of a nucleus is negative but positive in a nucleon? What I mean is, the sum of the mass of the components of a nucleus is more than that of a bound nucleus, but the sum of the quark masses is less than that of a bound nucleon. I think that is right. It seems the binding energy is opposite in sign.

19. Jul 9, 2012

### Gulli

How about this: the Higgs mechanism gives rest mass to all particles except photons and gluons (that I know of). Particles that don't get rest mass from the Higgs mechanism always travel at the speed of light and don't have conventional inertial mass: they can't be made to move faster or slower, however they can change direction (which is also a form of acceleration), something they will resist as if they had inertial mass. A composite particle is being held together by a cloud of gluons or photons (composed of streams that are exchanged between those elementary particles with a rest mass) with zero net momentum in the rest frame of the composite particle. The composite particle as a whole has more inertial mass than the rest masses of all its elementary particles combined because the gluon or photon cloud has inertial mass because the speed of light is finite and hence gluons or photons that are "underway" between two elementary particles with rest mass, (while those particles with rest mass are being accelerated), will still favor an earlier equilibrium position, pushing the elementary particles with rest mass back a little. I think this is similar to having a closed box with on the inside perfect mirrors bouncing a lot of photons back and forth: moving the box away from you will cost more energy than you would expect based on the rest mass of the box alone, because you'll have more photons bouncing against the side nearest to you and less against the opposite side while the box is accelerating. So a single photon or gluon may not have any conventional inertial mass, but a cloud of photons or gluons does: accelerating the "barycenter" of the cloud requires an amount of energy that scales with the energy content of the cloud.