I have a question regarding the famous equation E = mc[SUP]2[/SUP].
I have always taken it to mean that mass could be viewed to simply be a type of energy and under the right conditions other energy could be converted to mass just as mass could be converted to other energy. Most books and articles about Special Relativity seem to say exactly the same thing.
In another physics thread, I noticed that this interpretation is not shared by everyone, but I did not want to hijack things there any further. So now my question: is mass just another form of energy? If not, I would appreciate a cite, because I simply cannot find anything that contradicts my interpretation, but I am aware that I may be subconsciously preselecting what I look at.
Oh, and I decided to celebrate my 100th post by creating my very first thread.
The mass of a system is a portion of its total energy, but it’s not actually possible to convert the non-mass energy of a closed system into mass.
OK, that was a mouthful. Let’s break it down. First of all, abandon any notion you might have that mass is additive. If a system is made up of multiple subsystems, the mass of the whole system is not, in general, the sums of the masses of the individual subsystems. That is, if I have a system consisting of two rocks, one with a mass of 1 kg and the other with a mass of 2 kg, the whole system does not necessarily have a mass of 3 kg.
Now, just what is mass? First, we have to look at how the energy of a system can change, when you look at it from different reference frames. If I have a rock sitting on the ground, I can say that it’s at rest. I can also, though, consider it from a reference frame where it’s moving very quickly. In a reference frame where it’s moving quickly, it’s going to have more total energy than it does in a reference frame where it’s moving slowly, or one where it’s not moving at all.
Now, for that rock, it’s never going to have an energy lower than the energy it has in the reference frame where it’s not moving. A more complicated system (one with multiple moving parts) might not have any reference frame where it’s not moving, but it will always have some reference frame where the energy is at a minimum (incidentally, this reference frame will always be the one where the total momentum of the system is zero). The energy it has in this minimal, zero-momentum reference frame is what we call the mass of the system.
And that is convertable 1:1 into energy. So there is extra energy/momentum provided by that mass being (inevitably, right?) accelerated relative to another reference frame.
Right?
So, if a rock of plutonium is converted 100% into energy (or whatever), there’s energy always left over? In the form of particles going about their business?
Thinking like this, I get the picture that every mass (everything) gets a little extra gravy, the same way a satellite can get a swing acceleration from a loop around a massive object. Analogously–and here’s where I lose it, I’m sure–since there are so many (some figure must be used in the math for “how many”) reference frames, it sounds like any mass can take advantage, so to speak, of reference frames and “have” extra energy all the time. That can’t be right.
I’m not sure how this will go over with the specialists, but I think it might give a useful perspective.
It sort of depends on what you picture mass as being. A proton for example has mass of a little under 1 billion electron volts (see link anon). The constituents of the proton are 3 quarks, 3 gluons, a bunch of fleeting quantum fluctuations known as sea quarks, and probably some fairy dust.
The quarks only account for about 1% of the resting mass. Most of the balance is from the gluons. Unfortunately, as force carrying bosons, gluons can’t have mass. So the whole question gets very interesting.
Mass is not convertable into energy, it already is, and always will be, energy. And in a closed system, it’ll also remain mass.
Let’s take a simple example where one might casually say that “mass is converted into energy”. A neutral pion is an unstable subatomic particle. When it’s just sitting there, it has mass. And let’s say for the sake of simplicity that we’re in the zero-momentum reference frame for the particle, so that its mass is all the energy it has. Well, after some amount of time, that pion will decay, and turn into a pair of photons. A photon has no mass, and so it’s tempting to say that “the mass of the pion was converted to the energy of the photons”. But remember what I said about mass not being additive: One photon doesn’t have mass, but two photons together can. And in this particular case, the pair of photons produced by the decay of the pion have exactly the same mass as the original pion.
What do I mean when I say that the two photons have mass? The same thing I meant when I said the pion has mass. If I look at those two photons in different reference frames, their total energy will be different. I can pick a reference frame that’s chasing after one of the two photons, and depending on how fast my reference frame is moving, I can get the energy of that one photon arbitrarily small. But by doing so, I’m also increasing the energy of the other photon, the one that’s going the other way. The net effect is that if I try to chase either of the photons, the total energy of the pair of photons increases. There’s an absolute minimum energy I can find for the total system, and that’s in the same frame of reference as the one where the original pion was at rest. But by definition, an absolute minimum achievable energy is a mass, so the pair of photons has a mass.
Ok through this part; it’s how I already understood things.
And this is the part I did not grasp; if I’m honest, I’m still grappling with it.
If I take your example, then I can sort of see mass representing the lowest possible energy value that a closed system can have. I suppose when we are talking about mass, we are always talking about the mass at rest, unless we specify otherwise. This “mass at rest” is invariant across all inertial frames, correct? Incidentally, I keep coming across physics comments around the internet that claim that this rest mass is not conserved, although invariant. Is that correct?
But where does this come from? Is it something that is observed and therefore assumed true, or does it somehow result from another part of Special Relativity?
It’s quite straightforward (as in, a single one-line calculation) to derive it from special relativity, provided that you formulate it in terms of tensors. The catch to this, of course, is that you have to understand tensors. To put it succinctly, the mass is defined as the norm of the energy-momentum vector, and the norm of a vector (or any other tensor) is always invariant.
And yes, whenever we refer to “mass”, we mean the same thing that’s sometimes called “rest mass”. You can, of course, define “mass” to mean what’s sometimes called “relativistic mass” (after all, you can define terms to mean whatever you want), but that quantity isn’t particularly interesting in any sort of problem, and doesn’t make anything any easier to understand, so there’s very little reason to use that definition.
The people who are saying that it is not conserved are probably talking about how the sum of the masses of the products of a reaction is typically less than the sum of the masses of the reactants. For example with Chronos’ “pion -> two photons” example:
mass(pion) > mass(photon #1) + mass(photon #2)
since the mass of the pion is positive, and the mass of photon #1 is zero and the mass of photon #2 is zero.
and we can casually say that “the mass of the pion was converted into energy that was carried away by the photons”.
On the other hand we can also write:
mass(pion) = mass(photon #1 and photon #2)
in which the big idea is that the mass of the *system *of photon #1 and photon #2together is non-zero (and equal to the mass of the original pion). This is allowed because in SR mass is not additive, and a system with mass can be made of up massless components.
A slight extension of what Chronos has said is this:
The mass (or rest mass) of a body can be defined as the norm of its energy-momentum vector (which is invariant), its energy in a given frame will be the time component of the vector in that frame (which is not invariant). When a body is in its rest frame the time component of the energy-momentum vector is equal to the norm of the energy-momentum vector or in other words the mass and the energy are equal.
If we use P to represent the energy-momentum vector, then for a system consisting of several bodies labelled 1,2,3…N the total energy-momentum vector P[sub]TOT[/sub] can be defined as:
So if we define |P[sub]TOT[/sub]| as the mass of the system, then it is not necessarily equal to the sum of the masses that make up the bodies in the system and whilst if the system is closed then P[sub]TOT[/sub] and hence |P[sub]TOT[/sub]| are conserved; P[sub]1[/sub], P[sub]2[/sub], P[sub]3[/sub], etc are not necessarily conserved and neither is |P[sub]1[/sub]| + |P[sub]2[/sub]| + |P[sub]3[/sub]| + … + |P[sub]N[/sub]|.
Sorry for the late thanks. I’ve been off trying to relearn tensors. It’s been awhile, so I’m discovering just how much rust can gather on concepts you haven’t worked with in awhile.
Anyone have any good links for picking up tensors so that it’s intuitive? Otherwise I’ll just plod on with whatever sites the google overlords see fit to throw me.
It’s really important to take the modern approach as there are some very awful old books on tensors out there that describe all the things that tensors do, but don’t describe what they are!
Look for an approach that defines tensors in terms of functionals on vector spaces and isn’t a slave to index notation.
While you shouldn’t be enslaved to index notation, and it’s important to realize that the tensors themselves fundamentally exist independently of any coordinate system, it’s also important to note that the use of indices and the Einstein summation convention makes a lot of calculations a lot easier in practice.
and thanks (I think)