Matt Strassler [April 27, 2012]
In my article on energy and mass and related issues, I focused attention on particles — which are ripples in fields — and the equation that Einstein used to relate their energy, momentum and mass. But energy arises in other places, not just through particles. To really understand the universe and how it works, you have to understand that energy can arise in the interaction among different fields, or even in the interaction of a field with itself. All the structure of our world — protons, atoms, molecules, bodies, mountains, planets, stars, galaxies — arises out of this type of energy. In fact, many types of energy that we talk about as though they are really different — chemical energy, nuclear energy, electromagnetic energy — either are a form of or involve in some way this more general concept of interaction energy.
In beginning physics classes this type of energy includes what is called “potential energy”. But both because “potential” has a different meaning in English than it does in physics, and because the way the concept is explained in freshman physics classes is so different from the modern viewpoint, I prefer to use a different name here, to pull the notion away from any pre-conceptions or mis-conceptions that you might already hold.
Also, in a previous version of my mass and energy article I called “interaction energy” by a different name, “relationship energy”. You’ll see why below; but I’ve decided this is a bad idea and have switched over to the new name.
Preamble: Review of Concepts
In the current viewpoint favored by physicists and validated (i.e. shown to be not false, but not necessarily unique) in many experiments, the world is made from fields.
The most intuitive example of a field is the wind:
- you can measure it everywhere,
- it can be zero or non-zero, and
- it can have waves (which we call sound.)
Most fields can have waves in them, and those waves have the property, because of quantum mechanics, that they cannot be of arbitrarily small height.
- The wave of smallest possible height — of smallest amplitude, and of smallest intensity — is what we call a “quantum”, or more commonly, but in a way that invites confusion, a “particle.”
A photon is a quantum, or particle, of light (and here the term `light’ includes both visible light and other forms); it is the dimmest flash of light, the least intense wave in the electric and magnetic fields that you can create without having no flash at all. You can have two photons, or three, or sixty-two; you cannot have a third of a photon, or two and a half. Your eye is structured to account for this; it absorbs light one photon at a time.
The same is true of electrons, muons, quarks, W particles, the Higgs particle and all the others. They are all quanta of their respective fields.
A quantum, though a ripple in a field, is like a particle in that
- it retains its integrity as it moves through empty space
- it has a definite (though observer-dependent) energy and momentum
- it has a definite (and observer-independent) mass
- it can only be emitted or absorbed as a unit.
[Recall how I define mass according to the convention used by particle physicists; E = mc2 only for a particle at rest, while a particle that is moving has E > mc2, with mass-energy mc2 and motion-energy which is always positive. My particle physics colleagues and I do not subscribe to the point of view that it is useful to view mass as increasing with velocity; we view this definition of mass as archaic. We define mass as velocity-independent --- what people used to call ``rest mass'', we just call ``mass''. I'll explain why elsewhere, but it is very important to keep this convention in mind while reading the present article.]
The Energy of Interacting Fields
Now, with that preamble, I want to turn to the most subtle form of energy. A particle has energy through its mass and through its motion. And remember that a particle is a ripple in a field — a well-defined wave.
Fields can do many other things, not just ripple. For example, a ripple in one field can cause a non-ripple disturbance in another field with which it interacts. I have sketched this in Figure 1, where in blue you see a particle (i.e. a quantum) in one field, and in green you see the response of a second field.
Suppose now there are two particles — for clarity only, let’s make them ripples in two different fields, so I’ve shown one in blue and one in orange in Figure 2 — and both of those fields interact with the field shown in green. Then the disturbance in the green field can be somewhat more complicated. Again, this is a sketch, not a precise rendition of what is a bit too complicated to show clearly in a picture, but it gives the right idea.
Ok, so what is the energy of this system of two particles — two ripples, one in each of two different fields — and a third field with which both interact?
The ripples are quanta, or particles; they each have mass and motion energy, both of which are positive.
The green field’s disturbance has some energy too; it’s also positive, though often quite small compared to the energy of the particles in a case like this. That’s often called field energy.
But there is additional energy in the relationship between the various fields; where the blue and green fields are both large, there is energy, and where the green and orange fields are both large, there is also energy. And here’s the strange part. If you compare Figures 1 and 2, both of them have energy in the region where the blue and green fields are large. But the presence of the ripple in the orange field in the vicinity alters the green field, and therefore changes the energy in the region where the blue field’s ripple is sitting, as indicated in Figure 3.
Depending upon the details of how the orange and green fields interact with each other, and how the blue and green fields interact with each other, the change in the energy may be either positive or negative. This change is what I’m going to call interaction energy.
The possibility of negative shifts in the energy of the blue and green field’s interaction, due to the presence of the orange ripple (and vice versa) — the possibility that interaction energy can be negative — is the single most important fact that allows for all of the structure in the universe, from atomic nuclei to human bodies to galaxies. And that’s what comes next in this story.
The Earth and the Moon
The Earth is obviously not a particle; it is a huge collection of particles, ripples in many different fields. But what I’ve just said applies to multiple ripples, not just one, and they all interact with gravitational fields. So the argument, in the end, is identical.
Imagine the Earth on its own. Its presence creates a disturbance in the gravitational field (which in Einstein’s viewpoint is a distortion of the local space and time, but that detail isn’t crucial to what I’m telling you here.) Now put the Moon nearby. The gravitational field is also disturbed by the Moon. And the gravitational field near the Earth changes as a result of the presence of the Moon. The detailed way that gravity interacts with the particles and fields that make up the Earth assures that the effect of the Moon is to produce a negative interaction energy between the gravitational field and the Earth. The reverse is also true.
And this is why the Moon and Earth cannot fly apart, and instead remain trapped, bound together as surely as if they were attached with a giant cord. Because if the Moon were very, very far from the Earth, the interaction energy of the system — of the Earth, the Moon, and the gravitational field — would be zero, instead of negative. But energy is conserved. So to move the Moon far from the Earth compared to where it is right now, positive energy — a whole lot of it — would have to come from somewhere, to allow for the negative interaction energy to become zero. The Moon and Earth have positive motion-energy as they orbit each other, but not enough for them to escape each other.
Short of flinging another planet into the moon, there’s no viable way to get that kind of energy, accidentally or on purpose, from anywhere in the vicinity; all of humanity’s weapons together wouldn’t even come remotely close. So the Moon cannot spontaneously move away from the Earth; it is stuck here, in orbit, unless and until some spectacular calamity jars it loose.
You may know that the most popular theory of how the Earth and Moon formed is through the collision of two planet-sized objects, a larger pre-Earth and a Mars-sized object; this theory explains a lot of otherwise confusing puzzles about the Moon. Certainly there were very high-energy planet-scale collisions in the early solar system as the sun and planets formed over four billion years ago! But such collisions haven’t happened for a long, long, long time.
The same logic explains why artificial satellites remain bound to the Earth, why the Earth remains bound to the Sun, and why the Sun remains bound to the Milky Way Galaxy, the city of a trillion stars which we inhabit.
The Hydrogen Atom
And on a much smaller scale, and with more subtle consequences, the electron and proton that make up a hydrogen atom remain bound to each other, unless energy is put in from outside to change it. This time the field that does the main part of the job is the electric field. In the presence of the electron, the interaction energy between the electric field and the proton (and vice versa) is negative. The result is that once you form a hydrogen atom from an electron and a proton (and you wait for a tiny fraction of a second until they settle down to their preferred configuration, know as the “ground state”) the amount of energy that you would need to put in to separate them is about 14 electron-volts. (What’s an electron-volt? it’s a quantity of energy, very very small by human standards, but useful in atomic physics.) We call this the “binding energy” of hydrogen.
We can measure that the binding energy is -14 electron-volts by shining ultraviolet light (photons with energy a bit too large to be detected by your eyes) onto hydrogen atoms, and seeing how energetic the photons need to be in order to break hydrogen apart. We can also calculate it using the equations of quantum mechanics — and the success of this prediction is one of the easiest tests of the modern theory of quantum physics.
But now I want to bring you back to something I said in my mass and energy article, one of Einstein’s key insights that he obtained from working out the consequences of his equations. If you have a system of objects, the mass of the system is not the sum of the masses of the objects that it contains. It is not even proportional to the sum of the energies of the particles that it contains. It is the total energy of the system divided by c2, as viewed by an observer who is stationary relative to the system. (For an observer relative to whom the system is moving, the system will have additional motion-energy, which does not contribute to the system’s mass.) And that total energy involves
- the mass energies of the particles (ripples in the fields), plus
- the motion-energies of the particles, plus
- other sources of field-energy from non-ripple disturbances, plus
- the interaction energies among the fields.
What do we learn from the fact that the energy required to break apart hydrogen is 14 electron volts? Well, once you’ve broken the hydrogen atom apart you’re basically left with a proton and an electron that are far apart and not moving much. At that point, the energy of the system is
- the mass energies of the particles = electron mass-energy + proton mass-energy = 510, 999 electron-volts + 938,272,013 electron-volts
- the motion-energies of the particles = 0
- other sources of field-energy from non-ripple disturbances = 0
- the interaction energies among the fields = 0
Meanwhile, we know that before we broke it up, the system of a hydrogen atom had energy that was 14 electron volts less than this.
Now the mass-energy of an electron is always 510, 999 electron-volts and the mass-energy of a proton is always 938,272,013 electron-volts, no matter what they are doing, so the mass-energy contribution to the total energy is the same for hydrogen as it is for a widely separated electron and proton. What must be the case is that
- the motion-energies of the particles inside hydrogen
- PLUS other sources of field-energy from non-ripple disturbances (really really small here)
- PLUS the interaction energies among the fields
- MUST EQUAL the binding energy of -14 electron volts.
In fact, if you do the calculation, the way the numbers work out is (approximately)
- the motion-energies of the particles = +14 electron volts
- other sources of field-energy from non-ripple disturbances = really really small
- the interaction energies among the fields = -28 electron volts.
and the sum of these things is -14 electron volts.
It’s not an accident that the interaction energy is -2 times the motion energy; roughly, that comes from having a 1/r2 force law for electrical forces. Experts: it follows from the virial theorem.
What is the mass of a hydrogen atom, then? It is
- the electron mass + the proton mass + (binding energy/c2 )
and since the binding energy is negative, thanks to the big negative interaction energy,
- mhydrogen < mproton + melectron
This is among the most important facts in the universe!
Why the hydrogen atom does not decay
I’m now going to say these same words back to you in a slightly different language, the language of a particle physicist.
Hydrogen is a stable composite object made from a proton and an electron, bound together by interacting with the electric field.
Why is it stable?
Any object that is not stable will decay; and a decay is only possible if the sum of the masses of the particles to which the initial object decays is less than the mass of the original object. This follows from the conservation of energy and momentum; for an explanation, click here.
The minimal things to which a hydrogen atom could decay are a proton and an electron. But the mass of the hydrogen atom is smaller (because of that minus 14 electron volts of binding energy) than the mass of the electron plus the mass of the proton, let me restate that really important equation.
- mhydrogen < mproton + melectron
There is nothing else in particle physics to which hydrogen can decay, so we’re done: hydrogen cannot decay at all.
[This is true until and unless the proton itself decays, which may in fact be possible but spectacularly rare --- so rare that we've never actually detected it happening. We already know it is so rare that not a single proton in your body will decay during your lifetime. So let's set that possibility aside as irrelevant for the moment.]
The same argument applies to all atoms. Atoms are stable because the interaction energy between electrons and an atomic nucleus is negative; the mass of the atom is consequently less than the sum of the masses of its constituents, so therefore the atom cannot simply fall apart into the electrons and nucleus that make it up.
The one caveat: the atom can fall apart in another way if its nucleus can itself decay. And while a proton cannot decay (or perhaps does so, but extraordinarily rarely), this is not true of most nuclei.
And this brings us to the big questions.
- Why is the neutron, which is unstable when on its own, stable inside some atomic nuclei?
- Why are some atomic nuclei stable while others are not?
- Why is the proton stable when it is heavier than the quarks that it contains?
To be continued…