Matt Strassler [March 27, 2012]
A number of puzzling features of the world — including a number that my readers have asked about in comments — have everything to do with the nature of mass and energy (and also momentum.) We’ve all heard these words and many of us have a vague idea of what they mean. Of course the notions of “mass” and “energy” exist in English (and in other languages) with multiple definitions. Unfortunately, none of these definitions matches what physicists mean. So you have to leave those other definitions at the door and try to deal with these notions in the precise way that physicists do. Otherwise you’ll end up very confused.
Even before I start, there’s the potential for confusion. In bringing up “mass and energy”, I do not mean to call to your mind a different pairing of words: “matter and energy”, which you will often hear people refer to as though they are opposites, or partners, or mutually exclusive categories, or something meaningful in some other way as a two-some. Well, they’re not. Matter and energy don’t belong to the same categories; putting them together is like referring to apples and orangutans, or to heaven and earthworms, or to birds and beach balls. [Here’s an article going into more detail about why matter and energy are a false dichotomy.] Matter (no matter how you define it — and there are different definitions in different contexts) is a class of objects that you will find in the universe, while mass and energy are not objects; they are properties that every object in the universe can have. Mass and energy are in deep interplay, and they deserve to be discussed together.
To understand mass and energy, we need to put momentum into the mix, and discuss the differences and relations among these quantities.
The word “energy” appears in English with many different meanings. When we are ill, we speak of feeling as though we have no energy, meaning strength and motivation; we speak of someone as showing lots of energy, meaning high activity; we complain about the rising price of energy, meaning fuel; we speak of spiritual energy as something unmeasurable but profound, perhaps a form of charisma. All of the various definitions are loosely related, which is why we choose the same word in describing these concepts. But in physics, energy is none of these things. You’d be making a big physics error if you mix and match one of the English definitions with the physics definition! Within physics, you must stick with the physics term, or you’ll get wrong answers and end up very confused.
Unfortunately, the meaning of “energy” in physics is very difficult to put into the form of a dictionary definition, with a short phrase that characterizes what it is. But do not be misled! This is a failure of language, and does not mean that physicists’ notion of energy is vague. Not at all! In any given physical system, it is very clear what the energy of the system is, both in terms of how to measure it experimentally and (if there are equations that can predict the system’s behavior) how to calculate it.
Part of what makes energy complicated to describe is that it can take many forms, not all of which are conceptually simple. Here are the three most commonly encountered:
- First, energy can be locked away in an object’s mass; on this website I call this mass-energy (which is the famous E=mc2 energy associated with mass, and also called “rest-energy”, since it is the energy that an object has when it is at rest, i.e. not moving.)
- Second, energy is associated with the motion of an object. On this website I call this motion-energy, whose technical name is “kinetic energy”; this kind of energy is rather intuitive, in that faster objects have more energy than slower ones, which is more or less what we would colloquially expect. Also, a heavier object has more motion-energy than a light one, if the two are traveling at the same speed.
- Third, and most confusing, energy can be stored in the relationships among objects (and is typically called “potential energy”). It can be stored in a stretched spring, or in the water behind a dam, or in the gravitational interaction of the earth with the sun, or in the relationship among atoms in a molecule. In fact there are lots and lots of ways to store energy. This sounds very vague, but again, it’s just a failure of words; in every one of these cases, there are precise formulas for what the stored energy of system is, and there are clear and well-defined ways to measure it.
There’s one more thing about this third kind of energy, interaction-energy as I will call it, that is especially confusing at first. Unlike mass-energy and motion-energy, which are always greater than or equal to zero, interaction-energy can be either positive or negative (and often is negative in interesting situations). I’m not going to deal just yet with this fascinating and subtle issue. We’ll get back to it later. [In a previous version of this article I called this `relationship-energy’, but decided against this choice more recently.]
Energy is a very special quantity, of great importance in physics. The reason it is so essential [along with momentum, see below] is that it is “conserved” — read this as physics-language for preserved, or for maintained without change. What precisely does this mean?
If you start with an object or collection of objects — let’s call it a “system of objects” — that has a certain amount of energy at this moment (make sure you count all of it though — all the mass-energy, all the motion-energy, the stored energy of all types, etc.), and the parts of the system interact with each other but with nothing else, then at the end of the day the amount of energy those objects will have is the same as the amount they have now. Total energy is conserved — the total amount does not change. It can change from one form to another, but if you keep track of all the forms, you’ll find at the end just what you had at the start.
This is true even if some of the objects disappear and others take their place, as will happen if, say, one particle in the system decays to two other particles, which then join the system.
[Just to warn you, in Einstein’s theory of gravity this can get a little hairy; the statement is still basically true, but when you try to keep track of energy over long distances it can become more subtle. Don’t worry about this now, but someday you may need to know it, so that’s why I mention it.]
Why is energy conserved? Because of a mathematical principle that relates the apparent fact that the laws of nature do not change with time to the existence of a conserved quantity in nature, which by definition we call `energy’.
The most famous and general statement of this principle is due to Emmy Noether, one of the great mathematical physicists of the last century and a contemporary of Einstein’s. (Just yesterday there was a nice article about Noether in the New York Times, perhaps the first time her name had appeared in the paper since Einstein’s moving tribute to her following her untimely death.) Though highly respected among some members of the physics and mathematics communities, she suffered both gender and ethnic discrimination in her home country of Germany (where attempts by mathematics colleagues to get her a professorship at Goettingen were blocked, and from which she fled when the Nazis came to power). Then, after less than two years in the United States, where she taught at Bryn Mawr (a famous college, near Philadelphia, that even today admits only women among its students), she died of an illness in middle age.
Noether’s famous theorem (actually they are two closely related theorems) tells us that whenever there is a symmetry of the laws of nature — in this case, that the laws of nature are the same at all times — then as a direct consequence there is a conserved quantity in nature — in this case, energy.
Even better, the theorem tells us precisely what that quantity is — what all the different forms of energy are, for a given system of objects, that have to be added up to give the total energy. (Actually methods for doing this predate Noether, but her theorem brings many concepts together in one place.) This is why physicists always know precisely what energy is, and why it is easier to obtain it through equations than to define it through words.
The situation with momentum is similar to that of energy. The laws of nature are the same in all places; crudely speaking, experiments [when properly defined and controlled] give the same answers whether you do them to the north or south of here, the east or west of here, or whether you do them at the top of a building or deep in a mine. Pick any direction in space; then, according to Noether, momentum along that direction is conserved. And since there are three dimensions of space, with three independent directions you could go, there are three independent conservation laws. You can pick whichever three directions you like, as long as they are different. For instance, you can choose the three conservation laws to be momentum in a north-south direction, momentum in an east-west direction, and momentum in an up-down direction. Or you can pick three others, such as toward-and-away from the sun, along-and-opposite the earth’s orbit, and up-and-down out of the plane of the solar system. Your choice doesn’t matter; momentum along any direction is conserved.
The most common form of momentum is just that due to simple motion of objects, and it’s more or less what you might think intuitively: if an object is moving in a certain direction, then it has momentum in that direction, and the faster it moves, the more momentum it has. And a heavy object has more momentum than a light object if the two are traveling at the same speed.
One interesting consequence of the conservation of momentum is that if you have a system of objects sitting stationary in front of you (that is, the system as a whole isn’t moving, if you average appropriately over the motions of all its constituent objects) then it will continue to remain stationary unless something from outside the system pushes on it. The reason is that if it is stationary its total momentum is zero, and since momentum is conserved, it will remain zero forever, as long as nothing from outside the system affects it. This fact is going to very important later, so keep it in mind.
Mass, and its Relation to Energy and Momentum
Now it’s time to turn to mass.
Unfortunately, there is a lot of confusion about mass, because shortly after Einstein’s work on relativity there were two notions of mass that coexisted for a time. Only one (the one which Einstein himself settled on, and which was sometimes called “invariant mass” to distinguish it from the now-archaic term “relativistic mass”) is still used in particle physics today. I’ll explain why in a separate article.
The definition of mass m that I will be using throughout this website is the one that has a particular relation between energy and momentum. For an object that is moving on its own (that is, not interacting in any significant way with other objects), Einstein proposed (and countless experiments confirm) that its energy E, momentum p and mass m satisfy a simple Pythagorean relation
- E2 = (p c)2 + (mc2)2 (equation #1)
[Remember Pythagoras, who said that a right-angle triangle with sides of length A and B and hypotenuse of length C has to satisfy C2 = A2 + B2? It’s the same type of relation: see Figure 1.] Here c is a constant speed that, as we will see in a moment, is the universal speed limit. We’ll also see in a minute why it is called “the speed of light.”
According to Einstein’s equations, the velocity of an object, divided by the speed limit c, is just the ratio of p c to E,
- v/c = (p c) / E (equation #2)
i.e. the ratio of the length of the horizontal side of our triangle to the length of its hypotenuse. (This ratio is also equal to the sine of the angle α shown in Figure 1.) Wow! There it is, folks. Since the sides of a right-angle triangle are always shorter than its hypotenuse (i.e. the sine of any angle is always less than or equal to 1), no object’s velocity can be faster than c, the universal speed limit. As the velocity of the object increases (for fixed mass), both p and E become very large (Figure 2), but E is always bigger than p c, and so v is always less than c!
Next, notice that if the object is not moving, so that its momentum p is zero, then the relation in equation #1 simplifies to
- E2 = (mc2)2 , or in other words E=mc2 ,
Einstein’s famous relation that mass is associated with a fixed amount of energy (which is what I call mass-energy on this website) is just the statement that when the triangle becomes a vertical line, as in Figure 3 (left), its hypotenuse becomes the same length as its vertical side. But let me say that again, because it is so important: this relation E=mc2 does not mean that energy is always equal to mass times c2; only for an object that is not moving (and therefore has zero momentum) is this true.
Another interesting thing to note is that for a massless particle, the vertical side of the triangle is zero and the hypotenuse and horizontal side have the same length, as in Figure 3 (right). In such a case, E is inevitably equal to pc, which in turn means that v/c = 1, or in other words, v=c . Thus we see that a massless particle (such as a photon [a particle of light]) inevitably travels at the speed c. And so the speed of light is the same as the universal speed limit, c.
Meanwhile, for a massive particle, as shown in Figure 4, no matter how big you make the momentum and the energy, E is always a little bit bigger than p c, and so the velocity is always less than c. Massless particles must travel at the speed limit; massive particles must travel below it.
At the other extreme, consider a slow massive object, moving very slowly compared to the speed of light, as in the case of a car. Then since its velocity v is much less than c, its momentum p times c is much less than E, and (as you can see from Figure 5) E is just a little bit bigger than m c2. Thus a slow object’s motion-energy E – mc2 is much smaller than its mass-energy mc2, while a fast object’s motion-energy can be made arbitrarily large, as we saw in Figure 4.
One tricky point I should mention: momentum is not just a number, it is a “vector”. That is, it has a size and a direction; it points in the direction the particle is moving. When I write “p” I’m just referring to its size. In many cases we have to keep track of the direction of the momentum too, but we don’t have to in equation #1 that relates momentum to energy and mass.
A final tricky point: I’ve used triangles and a bit of ordinary trigonometry because everyone knows them from high school. But experts-to-be should beware: the right way to understand Einstein’s equations is using hyperbolic trigonometry, which most laypeople never encounter, but which is essential for understanding the structure of the theory, and makes important details such as how two velocities add, why lengths contract, etc., far more transparent. Non-experts can safely ignore this, though someday I might write a page explaining it, as it involves lovely mathematics.
But Velocity is Relative…?
Now if you’ve been paying attention, you’re wondering what the heck I’m talking about. You know that the speed of a particle — or of anything that travels slower than the speed of light itself — depends on your point of view.
If you’re sitting at home reading a book, you would claim the book has zero velocity (and relative to you, it is indeed stationary) and therefore it has no momentum and no motion-energy, only mass-energy. But if I were standing on the moon, I would remind you that the earth is spinning, so you’re actually being carried along by that spin, and moving (relative to me), at hundreds of miles (or kilometers, if you prefer) per hour. So you, and your book, do have momentum, according to me.
Who is right?
Galileo’s version of relativity — the first relativity principle — argues that you and I are both right. Einstein’s version of relativity agrees with Galileo on this point — that both you and I are correct — but makes important adjustments to what followers of Galileo’s relativity would have claimed for the energy and momentum and mass of the book, by setting them into the Pythagorean relationship of equation #1.
But if everyone is right, which E and which p should I put into the energy/momentum/mass relationship, E2 = (p c)2 + (mc2)2 , our equation #1, for the book? Should I put the E and p that you measure as you read the book, namely E = mc2 and p=0? Or should I put the E and p that I think the book has, watching you move with the earth?
The answer is the whole point of Einstein’s equation #1. Every observer will measure a different E and a different p for the book, depending on how fast the book is moving relative to them; but for every one of these observers, the equation E2 = (p c)2 + (mc2)2 will be correct!!!
Magic! Actually no, just genius — the recognition in 1905 of how to replace the set of relations proposed by Newton and his descendants with a profound new set of relations which were still consistent with earlier experiments but do turn out to be a more accurate representation of reality. You can’t fully appreciate how much of a mind-bend this requires until you look at all of the ways that this could have gone wrong, and how many other equations you could have proposed (and some people did) for which inconsistencies internal to the mathematics, or disagreements with previous experiments, can arise. (Indeed, amateur physicists send me their attempts to “correct” Einstein’s equations all the time, but I’ve never seen any of them thoroughly check whether their own equations are fully self-consistent… a very tall order, and the downfall of most theoretical ideas.)
But then, How can Energy and Momentum Be Conserved?
Now wait! you say, with your head about to split open and brains about to spill onto the floor [and yes, I do remember how this feels] — energy and momentum are supposed to be conserved — so how can different observers disagree about what they are?
More magic — actually predating Einstein. Trust me, the universe is a very, very clever bookkeeper, and even though different observers will disagree about how much energy an object has, or a system of objects has, they will all agree that this energy does not change with time. The same is true for momentum. I promise we’ll see examples of this later.
Mass, however, is completely different from energy and momentum. First, unlike energy and momentum, mass is simply not conserved. There are all sorts of processes in nature in which the total mass of the objects in a system change; for instance, a massive Higgs particle (if such things exist, and they probably do) can decay into two massless photons. There’s no symmetry associated with mass, and Noether provides us with no conservation law. But second, as I mentioned before, unlike the energy and momentum of an object (or system of objects), whose values depend on the observer (in particular on his or her speed relative to what he or she is measuring), all observers, no matter what they are doing, will agree on the mass m of an object (or system of objects). This is not obvious at all; it is true because of the devilishly clever way that Einstein’s equations work.
Where We Are So Far
So — now you know a bunch of perhaps seemingly conflicting things. You know
- Energy and momentum of an isolated physical system are conserved (the total energy and the total momentum of an isolated system doesn’t change over time) from every observer’s point of view
- But different observers, if they are moving relative to one another, will assign a different amount of energy and momentum to the system!
- The sum of the masses of the objects that make up a physical system is not conserved; it may change.
- But the mass of any object is something that all observers will agree on.
To this list of facts we will add two additional facts and two conclusions:
- The mass of a physical system of objects is not the sum of the masses of the objects that make up that system.
Instead, the mass of a physical system, which all observers agree on, is determined by its energy and momentum, and satisfies its own version of equation #1:
- (Esystem)2 = (psystem c)2 + (msystemc2)2 (equation #1′)
It follows then, without the need for an additional conservation law, that even though the sum of the masses of the objects that make up a system is not conserved, the mass of the system is conserved, because it is related by equation #1′ for a system to the energy and momentum of the system, which are conserved.
- the mass of a system of objects is the only thing on our list that is both conserved and agreed by all observers.
We just have to remember that the mass of a system is not the sum of the masses of the objects that make up that system, but something given by equation #1′.
Rather than my trying to explain this any further, let’s see how it works. An illustrative example is worth a thousand words. And I’m going to use the most current and famous example of them all: let us take, as our system, a Higgs particle (which I’ll assume to have mass of 126 GeV/c2), and see how the various statements made above play out when the Higgs particle decays to two photons.
One Higgs Particle, Two Photons, and Three Observers
So what we’re going to do is look at a Higgs particle decaying to two photons from the point of view of three different observers. These observers are shown in Figure 6, along with the Higgs particle that they are initially looking at. (Not that they can actually “look” at it with their eyes; it’s around for far too short a time, and is far too small, to see. It is their scientific instruments that will give them some opportunity to understand its properties.) One, whom we’ll call Peter, views the Higgs particle as not moving at all. Marie is moving down, relative to Peter. And Chris is moving to the left, relative to Peter. That means Marie sees the Higgs particle as moving up, while Chris sees it as moving to the right. How these three observers see the Higgs decay is sketched in Figure 7. Peter sees the Higgs decay to two photons of equal energy, one going up and one going down. Marie will see the Higgs decay to two photons of different energy, the upward-heading one with more energy than the downward-heading one. And Chris will see the Higgs decay to two photons that are heading up to the right and down to the right. What we’ll do next is figure out (a) what energies and momenta does each of these observers assign to the Higgs and two photons? and (b) how does each of these observers come to the conclusion, separately, that energy and momentum is conserved in this process?
A Stationary Higgs Particle Decays
Let’s first examine the Higgs particle from Peter’s point of view. Peter stares (or rather his measuring apparatus “stares”) at the Higgs, and sees what? (I’m going to put a bar over everything that Peter observes; we’ll compare this with what Marie and Chris observe later.) The Higgs isn’t moving, so its momentum p0 is zero, and by Einstein’s equation #1 the Higgs, with its mass of m = 126 GeV/c2, has energy E0 = m c2 = 126 GeV.
Now according to the conservation of energy and momentum, the system of a Higgs particle will retain all of its energy and momentum as it decays. (This is true as long as nothing external affects the Higgs particle during this process. You might wonder whether I should worry about the earth pulling on the Higgs particle gravitationally, which would be an external effect that could change its momentum. The answer is that on the short time that it takes for a Higgs to decay, the effect of gravity is so tiny that if I told you how small it is it would make you giggle. You can forget about it.) So when the Higgs decays, the energies of the particles that form its debris must add up to 126 GeV, and the momenta of the particles (remembering that momentum is something with size and direction — it is a “vector”, in technical parlance) will add up to zero.
The two massless photons to which the Higgs decays could go off in any direction, but just to keep things simple, let’s imagine they go off vertically, one heading up, the other recoiling from it and heading down. (Why must they go off in opposite directions? Hang on a second…)
How much momentum must these two photons have? Well, that’s easy. First, the total momentum of the system — the sum of the two photons’ momenta — must be equal to zero, since the Higgs before the decay has zero momentum (from Peter’s point of view). Now each photon has a momentum that has a size and a direction. The only way these two momenta could add to zero is if they have equal size and opposite directions. So if one goes up, the other has to go down, and they have to have equal size.
Second, the total energy of the system is just the sum of the two photons’ energies. (That’s because there is no interaction-energy between them [except an unbelievably minuscule gravitational energy which you can forget about]. Of course, since they have no masses, all their energy is in motion-energy.) Moreover, for a massless particle like a photon, Einstein’s equation #1 implies that E = p c, where p is the size of the particle’s momentum. Because of this, the two photons, which have momenta of equal size, must also have energies of equal size too. And since the two energies must sum to the energy of the Higgs particle, each photon must have energy equal to one half of the Higgs particle’s energy:
- E1 = E2 = 1/2 (126 GeV) = 63 GeV.
and since p = E/c for a massless particle,
- p1 = 63 GeV/c upward
- p2 = 63 GeV/c downward.
This is summarized in Figure 8.
Energy and momentum are conserved, but mass of the objects is not, since the photons are massless and the Higgs was not. What about the mass of the system? What is the mass of the system of two photons? It isn’t zero. In fact it is obvious what it is. Just as for the Higgs itself (which initially made up the entire system), the system of two photons has the same energy and momentum as the Higgs did to start with,
- Esystem = E1 + E2 = 63 GeV + 63 GeV = 126 GeV
- psystem = p1 (up) + p2 (down) = 63 GeV/c up + 63 GeV/c down = 0
And since psystem = 0 for Peter,
- msystem = Esystem / c2 = 126 GeV/c2
which is the Higgs mass; the system’s mass did not change during the decay, as we expected.
An Observer Who Views the Higgs As Moving Up
Marie is moving downward relative to Peter, so from her point of view, Peter and the Higgs are moving upward relative to her. Let’s say that the Higgs has a speed of v = 0.8 c, or 4/5 the speed of light, relative to her. Unlike Peter, she views the Higgs as having non-zero momentum, and similarly views the photons as having unequal, though still opposite, momenta, so that the sum of their momenta isn’t zero.
How are we going to figure out how much momentum and energy the Higgs has, and the two photons to which it decays, from Marie’s perspective? Well, to do this we need one more set of simple equations from Einstein. Suppose from one observer’s perspective an object has momentum p and energy E. Then, from the perspective of a second person who is moving with speed v along (or opposite) the same direction that the object is moving, the object will have momentum p and energy E given by the equations
- p = γ (p+ v E/c2) (equation #3)
- E = γ (E + v p) (equation #4)
where γ satisfies yet another Pythagorean relationship
- 1 = v2/c2 + 1/γ2 (equation #5)
according to Einstein. This allows us to convert between what Peter observes and what Marie (or any other observer moving at the speed v) observes. What we’re about to find is illustrated in Figure 9.
To compare Marie’s views to Peter’s views, we need v and γ. I claim
- if v=4/5 c,
- then γ = 5/3.
[Check: using equation #5, 1 = (4/5)2 + (3/5)2 = 16/25 + 9/25 = 25/25 .]
Peter says: the Higgs has p0=0, E0=126 GeV. What about Marie? She says: the Higgs has
- p0 = γ v E0 = (5/3) (4/5) E0 = 168 GeV/c upward, and
- E0 = γ E0 = (5/3) E0 = 210 GeV
Meanwhile Peter says the that two photons have E1 = E2 = 63 GeV and each has E = p c. Then we can figure out using equations #3 and #4 that Marie will observe
- E1= γ (1+v)E1 = 189 GeV , p1 = E1/c (moving upward)
- E2= γ (1-v) E2= 21 GeV , p2 =E2/c (moving downward).
It works! Energy is conserved, according to Marie, because
- E0 = 210 GeV
- E1 + E2 = (189 + 21) GeV = 210 GeV
and momentum is also conserved
- p0 = 168 GeV/c upward
- p1 + p2 = 189 GeV/c upward + 21 GeV/c downward = (189 – 21) GeV/c upward = 168 GeV/c upward
And the mass of the system is equal to the Higgs mass both before and after the decay, because both before and after the decay,
- Esystem = 210 GeV
- psystem = 168 GeV/c upward
which (from equation #1′) makes the mass of the system again 126 GeV/c2, as it was for Peter, since
- 2102 = 1682 + 1262
An Observer Who Views the Higgs As Moving To The Right
Ok, now what about Chris? Chris is moving to the left relative to Peter, also, to keep things simple, at speed v=4/5 c, so relative to Chris the Higgs (and Peter) are moving to the right at speed v = 4/5 c. Now the same calculation that we did for Marie tells us that the Higgs energy is E0 = 210 GeV and p0= 168 GeV, but unlike for Marie, for whom the Higgs is moving upward, for Chris the Higgs’ momentum is to the right. This is illustrated in Figure 10.
Now the Higgs decays to two photons. If from Peter’s perspective the two photons are moving upward and downward, for Chris, who sees the Higgs and Peter moving to the right, one of the photons will be moving up and to the right, while the other will be moving down and to the right, as shown in Figure 10. What will their momentum and energy be?
Well, we can’t answer this with equations #4 and #5, because those were appropriate for particles and observers moving in the same direction. Now we have an observer moving to the left and photons moving up and down. For this case, the equations are
- up-down part of p = up-down part of p
- right-left part of p = γ ([right-left part of p] + v E/c2)
- E = γ (E + v [right-left part of p])
And these equations are going to be simpler than they look, because from Peter’s point of view, p has no right-left part; all the momentum is either up or down. So Chris sees the Higgs as having
- up-down part of p0 = up-down part of p0 = 0
- right-left part of p0 = γ v E0/c2 = (5/3) (4/5)126 GeV/c = 168 GeV/c rightward
- E = γ E0 = (5/3) 126 GeV = 210 GeV
and sees the upward-going photon has having
- up-down part of p0 = up-down part of p1 = 63 GeV/c upward
- right-left part of p0 = γ v E1 = (5/3) (4/5) 63 GeV/c = 84 GeV/c rightward
- E = γ E0 = (5/3) 63 GeV = 105 GeV
with the formulas for the second photon being the same except that its up-down part points downward. Notice that E = p c for both photons, using the Pythagorean theorem for the size p of each photon’s momentum — see the inset in Figure 10 —
- p12 = (upward part of p1)2 + (rightward part of p1)2
- in other words: (105 GeV/c)2 = (63 GeV/c)2 + (84 GeV/c)2
which you can check with a calculator is correct (or you can divide by 212 first and verify it by hand.)
So again Chris observes completely different energies and momenta than Peter and Marie. But Chris still sees energy and momentum are conserved, as you can see from the summary of the numbers given in Figure 10. And Chris also sees that the system of two photons has a mass equal to the mass of the Higgs. Why? The total up-down part of the system’s momentum is zero; it cancels between the two photons. The left-right part of the system’s momentum is 168 GeV/c; the total energy of the system is 210 GeV; and that’s just what Marie saw, with the only difference that she had the system’s momentum going up instead of to the right. But the direction of the momentum doesn’t affect equation #1′ ; only the size of the momentum appears there. So, like Marie, Chris also sees that the mass of the system of two photons is 126 GeV/c2, the mass of the initial Higgs particle.
So — we see that the three different observers
- Disagree about how much energy and momentum the Higgs has
- Disagree about how much energy and momentum each of the two photons has
- Agree that energy and momentum are conserved in the decay
- Agree therefore that the mass of the system is conserved in the decay
- Agree also that the mass of the system is 126 GeV/c2
- Agree moreover that the sum of the masses of the objects in the system was not conserved; it has decreased to zero from 126 GeV/c2.
This is no accident. Einstein knew that energy and momentum were conserved according to previous experiments, so he sought (and found) equations that would preserve this feature of the world. And he also discovered along the way that the mass of a system would have to satisfy equation #1′.
Bonus: How This is Used to Seek the Higgs Particle
And now, as a bonus, you get to learn how particle physicists try to discovery the Higgs particle. What they do is
- look at proton-proton collisions that produce two photons
- compute the mass of the system of two photons (the invariant mass of the pair of photons, in technical jargon)
Whenever a Higgs particle is produced and decays to two photons, no matter how fast and in what direction the Higgs particle is moving relative to the laboratory, the system of two photons to which it decays will always have mass equal to the mass of the Higgs particle that produced the two photons! So unlike random processes that make two photons, which will form a system with a random mass, the Higgs particles will always produce a system of two photons with the same mass. And so if Higgs particles are being produced in the data, and if they are decaying sometimes to two photons, we expect to see a peak from the Higgs decays over a smooth background from other, random processes.
This is, in fact, what we see a hint of in current Large Hadron Collider data; and during 2012, if all goes well we will either see the peak in the data firm up and become convincing, or we will see it retreat and disappear. And if the peak is there, the location of the peak — the mass shared by a surprising number of two-photon events — tells us the mass of the Higgs particle. And we did!!!