I Is the mass defect still considered with invariant mass?

[albertrichardf]

Hello.
Suppose you were to assemble a sphere of negative charges. When you are done, the rest mass of the sphere is larger than that of the negative charges because they gain energy in forming the sphere. But the invariant mass of the electrons can't change and apparently gaining energy doesn't increase their mass but their inertia. So does the concept of mass defect still hold without relativistic mass? And does the sphere have higher invariant mass than the sum of the invariant masses of the electrons?
Thank you.

 

[jbriggs444]

does the sphere have higher invariant mass than the sum of the invariant masses of the electrons?

Yes, it does. Invariant mass is not an additive quantity. Another simple example is a pair of photons (or pulses of light) traveling in opposite directions. Separately, each has an invariant mass of zero. Taken together the combined system has a non-zero invariant mass.

 

[albertrichardf]

So the energy of the internal parts have some mass of their own?

 

[jbriggs444]

So the energy of the internal parts have some mass of their own?

In my opinion, you should not try to localize the extra mass. It is not associated with any specific part. It is an attribute of the system as a whole. If you insist on assigning the extra mass to something specific, you could say that the extra mass in an assembly of electrons is carried by the electrostatic field that they produce.

If one had that pair of photons (or light pulses) moving in opposite directions, it would not be correct to divide the invariant mass equally and assign it to the individual photons. That would work for the center-of-momentum frame. But in other inertial frames of reference, the system energy would be divided in different proportions. There is no invariant fact of the matter governing the assignment of invariant mass between the two photons. If there is no invariant fact of the matter, there is no grounds to consider the assignment physically meaningful.

 

[Herman Trivilino]

Suppose you were to assemble a sphere of negative charges. When you are done, the rest mass of the sphere is larger than that of the negative charges because they gain energy in forming the sphere.

I would say it slightly differently. The negative charges don't gain energy. The collection of negative charges gains energy. But otherwise, yes. This is the true meaning of Einstein's mass-energy equivalence.

But the invariant mass of the electrons can't change and apparently gaining energy doesn't increase their mass but their inertia.

Invariant mass and rest mass are two names for the same thing. Mass is a third name for that same thing. Having more than one name for the same thing is a source of confusion.

So does the concept of mass defect still hold without relativistic mass?

Of course there's a mass defect. And of course it exists when you expunge relativistic mass from your vocabulary. Changing terminology doesn't change the physics.

And does the sphere have higher invariant mass than the sum of the invariant masses of the electrons?

Yes.

Confusion arises when people start talking about different kinds of mass. That is the reason for expunging relativistic mass.

So the energy of the internal parts have some mass of their own?

No, the energy contribution of the internal parts is some of the mass of the sphere. The notion that you can add up the mass of the constituents to get the mass of the composite body is part of the Newtonian approximation. Einstein's mass-energy equivalence tells us that such a notion is false.

 

[albertrichardf]

In my opinion, you should not try to localize the extra mass. It is not associated with any specific part. It is an attribute of the system as a whole. If you insist on assigning the extra mass to something specific, you could say that the extra mass in an assembly of electrons is carried by the electrostatic field that they produce.

If one had that pair of photons (or light pulses) moving in opposite directions, it would not be correct to divide the invariant mass equally and assign it to the individual photons. That would work for the center-of-momentum frame. But in other inertial frames of reference, the system energy would be divided in different proportions. There is no invariant fact of the matter governing the assignment of invariant mass between the two photons. If there is no invariant fact of the matter, there is no grounds to consider the assignment physically meaningful.

Thank you for the answer. Just defining the inertial mass as non-additive does resolve a lot more issues than I thought. It also changes radically from the way I used to think of the mass, and its quite an interesting concept. Guess that's what happens when you use slightly older textbooks.

Though I am still wondering: the argument holds for individual photons but if you were to calculate the difference in the mass of the system now minus the sum of the invariant masses of its components, couldn't you use it as the culprit for the mass defect? Say you had the electron system this time, because that's easier to work with. In assembling the electrons they must gain some energy, and even though this energy gain could differ from electron to electron, the overall energy difference should the same in every frame right? Because if it wasn't then in one system, the sphere could do more work than in another. And that would violate energy conservation because by changing frame you can gain more energy for free. So if the overall energy difference is an invariant quantity, couldn't you just call that the extra mass?

I would say it slightly differently. The negative charges don't gain energy. The collection of negative charges gains energy. But otherwise, yes. This is the true meaning of Einstein's mass-energy equivalence.
Invariant mass and rest mass are two names for the same thing. Mass is a third name for that same thing. Having more than one name for the same thing is a source of confusion.

It feels a bit strange to refer to a collection as gaining energy, but I guess that's because I'm not used to it.
And having different names for the same thing is definitely bothersome, though I suppose it's less annoying than having to invent ten different names for all the kinds of masses you need when you use relativistic mass.

Of course there's a mass defect. And of course it exists when you expunge relativistic mass from your vocabulary. Changing terminology doesn't change the physics.

Of course it won't but changing the terminology does mean changing the definitions. I was unsure of whether the invariant mass of the sphere would be defined as its mass in its rest frame or the sum of the constituents, even if the latter has no real use.
And thank you for answering

 

[Vitro]

Say you had the electron system this time, because that's easier to work with. In assembling the electrons they must gain some energy, and even though this energy gain could differ from electron to electron, the overall energy difference should the same in every frame right? Because if it wasn't then in one system, the sphere could do more work than in another. And that would violate energy conservation because by changing frame you can gain more energy for free.

You have a misunderstanding here, energy is conserved but it's not invariant. Conserved means that in the same reference frame it is the same at any instant in time, say before and after some process. Invariant means the same in different reference frames, which energy is not. A bowling ball can do more work in a frame where it is moving than in a frame where it is at rest.

 

[albertrichardf]

Good point with the balling ball. I realized that the work done by or on an open system is not invariant -in a collision with two bowling balls one can gain energy in one frame but lose energy in the other. Thank you for pointing it out.

 

[jartsa]

Suppose you were to assemble a sphere of negative charges. When you are done, the rest mass of the sphere is larger than that of the negative charges because they gain energy in forming the sphere. But the invariant mass of the electrons can't change and apparently gaining energy doesn't increase their mass but their inertia. So does the concept of mass defect still hold without relativistic mass? And does the sphere have higher invariant mass than the sum of the invariant masses of the electrons?
Thank you.

I disagree with everybody else.

Suppose we were to assemble a sphere of negative charges. When we are done, the rest mass of the sphere can be measured to be larger than that of all the original negative charges. Also the rest mass of one electron can be measured, that rest mass will be larger than the rest mass of one original electron.

So the invariant mass of an electron increases when the electron gains energy, only exception is the kinetic energy. The coordinate dependent kinetic energy does not have any effect on the coordinate independent mass, and the invariant kinetic energy of an electron is always zero, so it has zero effect on invariant mass.

 

[DrGreg]

Also the rest mass of one electron can be measured, that rest mass will be larger than the rest mass of one original electron.

Why do you think that?

 

[Herman Trivilino]

It feels a bit strange to refer to a collection as gaining energy, but I guess that's because I'm not used to it.

That is in fact what happens when something as simple as lifting a book from the floor to the table top happens. It is common to state that the gravitational potential energy of the book increased, but in fact what has happened is that the potential energy of the book-Earth system increased. This system is a collection of two particles, and the mass of that collection increased when the book was lifted.

The true meaning of the Einstein mass-energy equivalence can be seen only by looking at composite bodies. That is, bodies that are a collection of particles. This is a notion obscured by the introduction of a relativistic mass.

And having different names for the same thing is definitely bothersome, though I suppose it's less annoying than having to invent ten different names for all the kinds of masses you need when you use relativistic mass.

There is no need to have either the bothersome arrangement or the annoying one. There is only one kind of mass, the ordinary mass, and there's no need to make mention or use of any other kind of mass. That's the way almost all high energy physicists do it and it works perfectly well. You just have to realize that the role of mass is different from its role in the Newtonian approximation. It is equivalent to rest energy.

There are many physicists, some of them prominent, who advocate for having more than one kind of mass. Usually they call them rest mass and relativistic mass. Many but not all are unaware that the latter has fallen out of usage, but to become aware all they have to do is grab a college-level introductory physics textbook that was written in the last 25 years or so and look at it. Once you remove relativistic mass from your vocabulary there is no need to distinguish the two by using the adjective "rest" to refer to the other. Moreover, there is no need, for the same reason, to attach a subscript "o" to the symbol $m$.

 

[Herman Trivilino]

Also the rest mass of one electron can be measured, that rest mass will be larger than the rest mass of one original electron.

Reference books, including those published by organizations such as CODATA, NIST, and BIPM, list only one value for the mass of an electron. Textbooks, too. And they no longer call it the rest mass. It's simply the mass.

and the invariant kinetic energy of an electron is always zero

There is no such thing as invariant kinetic energy.

 

[jartsa]

Why do you think that?

Momentum of a system is the sum of momentums of its parts.

If a sphere has some extra mass, it needs some extra momentum to get it moving. The sphere can get some extra momentum by its parts getting some extra momentum. So when a part is moving with the sphere it has some extra momentum, so when the part is stopped the stopper must absorb some extra momentum. Now if we say that stopping the part and measuring the required impulse is a measurement of mass, then the measurement shows that there is some extra mass in the part.

 

[jartsa]

Reference books, including those published by organizations such as CODATA, NIST, and BIPM, list only one value for the mass of an electron. Textbooks, too. And they no longer call it the rest mass. It's simply the mass.

In the previous post I explain how one measurement of mass gives as a result an increased mass.

There is no such thing as invariant kinetic energy.

There is such kinetic energy that contributes to mass, and as mass is invariant, so is the kinetic energy that contributes to mass. So I called that kinetic energy invariant kinetic energy, because it's invariant.

 

[jbriggs444]

There is such kinetic energy that contributes to mass, and as mass is invariant, so is the kinetic energy that contributes to mass.

That is utterly wrong.

Mass is the magnitude of the energy-momentum four-vector. It is invariant because any change in energy that comes from a change in reference frame is accompanied by a change in momentum that comes with the change in reference frame. Neither the kinetic energy nor the momentum are invariant. But their norm, when combined as a four-vector is.

 

[Dale]

Also the rest mass of one electron can be measured, that rest mass will be larger than the rest mass of one original electron

I think that this is wrong. Please provide a reference.

Now if we say that stopping the part and measuring the required impulse is a measurement of mass

I don't think that this will measure mass in relativity. Do you have a reference for this?

So when a part is moving with the sphere it has some extra momentum, so when the part is stopped the stopper must absorb some extra momentum.

I am uncomfortable with this also, but consider those other points more problematic.

 

[albertrichardf]

Also the rest mass of one electron can be measured, that rest mass will be larger than the rest mass of one original electron.

Wouldn't this energy arise from the interaction of two electrons rather than just one electron? If you had two electrons in a certain configuration, you could have brought one from infinity while holding the other still, or you could have brought both from infinity and have them reach this arrangement, or you could have brought one a little from infinity and have the other move the rest of the way. There are an infinite number of ways an arrangement could be made, but the end result: that you need to give the system a certain amount of energy is the same. So the energy can be attributed to the arrangement.
Come to think of it, this idea of energy belonging to arrangements that fields bring with them as Mister T helpfully reminded me of (Never ever thought the A-team would help with physics) makes the mass defect being attributed to the arrangement seem like a natural extension, far more than relativistic mass anyway.

 

[PeterDonis]

This system is a collection of two particles, and the mass of that collection increased when the book was lifted.

Careful. The externally measured mass of the Earth-book system actually does not change at all in this process. All that changes is how that mass is distributed between the parts of the system. The energy that was used to raise the book came from somewhere, and that somewhere is also part of the system, so it counts in the externally measured mass.

 

[Herman Trivilino]

The energy that was used to raise the book came from somewhere, and that somewhere is also part of the system,

Why does it have to be part of the system? Can't it be external to the system?

 

[jtbell]

The energy that was used to raise the book came from somewhere, and that somewhere is also part of the system, so it counts in the externally measured mass.

If you lift the book in your hand, then the system no longer consists of only the Earth and the book. It consists of the earth, the book, and you. In lifting the book, you do work. Your internal energy decreases, the gravitational potential energy of the system increases, and the total energy of the system remains constant. The masses of the earth, the book, and the system remain constant, but your mass decreases.

Why does it have to be part of the system? Can't it be external to the system?

The energy needed to raise the book can indeed come from outside the system. For example, you might use solar energy to boil water, and use the steam to drive an engine that lifts the book. In that case the mass of the earth-book-engine system would increase according to the solar energy that entered the system.

 

[Herman Trivilino]

If you lift the book in your hand, then the system no longer consists of only the Earth and the book. It consists of the earth, the book, and you. In lifting the book, you do work.

I don't understand why the person doing the lifting has to be part of the system. I thought such things were a matter of choice.

The energy needed to raise the book can indeed come from outside the system. For example, you might use solar energy to boil water, and use the steam to drive an engine that lifts the book.

In this case it seems to me that the sun is playing the same role as the person did in the previous case.

 

[jtbell]

I don't understand why the person doing the lifting has to be part of the system. I thought such things were a matter of choice.

Sure, I agree that in principle, it's a matter of choice. In the case of a person lifting a book, I personally think that separating the person mentally from the system is a somewhat contrived way of thinking of it. How would one measure the mass of the earth-book (alone) system without including the person? I suppose you could have the person arrive in a spacecraft , lift the book, and then leave.

 

[jtbell]

I personally think that separating the person mentally from the system is a somewhat contrived way of thinking of it.

On the other hand, if you had two oppositely-charged objects stuck together in the lab, you could walk in, pull them apart with your hands, stick something in between to keep them separated, and then walk away. In that case it would be "natural" (IMO) to consider the two charged objects and the "something" as a system, without including you.

 

[Herman Trivilino]

Sure, I agree that in principle, it's a matter of choice. In the case of a person lifting a book, I personally think that separating the person mentally from the system is a somewhat contrived way of thinking of it.

To me it's more natural to model the book-Earth system as a two-particle system. Anything else is external to that system.

Note that a freshman student who uses $mgh$ to calculate the potential energy change is employing that model. When the book of mass $m$ is lifted to a height $h$ the potential energy of the system increases by $mgh$ and the mass of the system increases by $mgh/c^2$.

How would one measure the mass of the earth-book (alone) system without including the person?

One could determine it by looking up the mass of Earth in a reference.

Of course adding the mass of the book to that is insignificant as it's many orders of magnitude smaller, and the mass of lift many orders of magnitude smaller than that.

But one is usually only interested in the change in potential energy. One could calculate that as is I've described above, but it would be too small to measure.

 

[pervect]

Hello.
Suppose you were to assemble a sphere of negative charges. When you are done, the rest mass of the sphere is larger than that of the negative charges because they gain energy in forming the sphere. But the invariant mass of the electrons can't change and apparently gaining energy doesn't increase their mass but their inertia. So does the concept of mass defect still hold without relativistic mass? And does the sphere have higher invariant mass than the sum of the invariant masses of the electrons?
Thank you.

Let's talk about this in the context of special relativity (SR).

If the energy of the system increases when you assemble it (which it does), but the sum of the rest energies of the electrons doesn't change (which is also true), we can conclude that the rest energy of the system is more than just the sum of the rest energies of the electrons.

The explanation for this is that the total energy of the system also includes the energy in the electric field. So to keep energy as a conserved quantity, one must include the energy stored in the electric field, as well as the energy of each particle. The sum of the energies of the electrons is not the total energy of the system, and can't be, because the energy one puts into the system to assemble it has to go somewhere.

As I mentioned, the context in which this explanation works is the context of special (and not general) relativity. A full explanation of energy in GR gets very tricky, and if I'm reading your question right, you are interested in the SR case anyway. So I'll omit the complicated GR case, especially if there are still more questions about the SR case.

I've oversimplifed some thorny issues. For instance, a bare electrons has a self-energy in it's associated electric field, since we've just argued that it's necessary to include the energy in the field as a separate entity from the eneergy in the point particle. The issue that comes up is how to make this split work exactly. This is complicated by the fact that one wants to use point particle approximations. Rather than attempt to discuss this point, I'll mention Baez's PF insight article, "Struggles with the continuum", which talks about this at great length, in a much better manner than I could.

Some related notes on the language. In the technical language, "relativistic mass" is just another name for energy, with the units changed by dividing by c^2, which is a constant. So using the correct technical definitions of the terms, the only distinction between energy and relativistic mass is this constant factor of c^2.

Using non-technical language, people may have various personal ideas that cause confusion. The only way I can think of to try to resolve this communicate obstacle is to offer the technical definitions so that we are (hopefully) both talking about the same thing when we use the same words. It's just not a good idea for several reasons to use language that isn't correct from the technical point of view of the technical community.

Thus, I try to write as clearly as possible using the correct technical language (I wouldn't say I always succeed at this goal, unfortunately). But I also try to remain aware that readers may not be using the same exact technical defintions that I am. This can lead to long and wandering discussions and misunderstandings. Sometimes I sense (or imagine, at least) impatience on the part of the reader by what seems like nit-picking, but I'm really not aware of any bettter way to communicate.

 

[PeterDonis]

Can't it be external to the system?

In principle it could be, but then "something as simple as lifting a book from the floor to the table top" would not be an apt description, since, as jtbell said, you would have to have a whole mechanism in place to collect the external energy and harness it to do the work of lifting the book. The simple way to do is is just pick it up, but that doesn't use external energy; you are part of the system.

 

[PeterDonis]

Note that a freshman student who uses $mgh$ to calculate the potential energy change is employing that model.

Which model? If you mean, a model in which $mgh$ represents the potential energy change of the entire Earth-book system, yes, this is true (at least in the approximation where $h$ is much smaller than the radius of the Earth). But it appears to contradict what you say later on in the same post:

One could calculate that as is I've described above

If you mean you could calculate the change in the Earth's potential energy by using $mgh$, where $m$ is the Earth's mass, $g$ is the acceleration due to gravity due to the book, and $h$ is the height change, this won't work. You can get a number out of it, but it won't mean what you are claiming it means. The number $mgh$, where $m$ is the book's mass, $g$ is the acceleration due to gravity of the Earth, and $h$ is the height change, is the change in potential energy of the Earth-book system (more precisely, it is the small $h$ approximation to the exact Newtonian formula, which is $GMm/R - GMm/(R+h)$, where $M$ is the mass of the Earth and $G$ is Newton's gravitational constant.) There is no way to separate it into "potential energy of the book" and "potential energy of the Earth".

 

[Herman Trivilino]

Which model? If you mean, a model in which $mgh$ represents the potential energy change of the entire Earth-book system, yes, this is true (at least in the approximation where $h$ is much smaller than the radius of the Earth).

Yes, that is precisely what I meant.

But it appears to contradict what you say later on in the same post:

If you mean you could calculate the change in the Earth's potential energy by using $mgh$, where $m$ is the Earth's mass, $g$ is the acceleration due to gravity due to the book, and $h$ is the height change, this won't work.

I don't recall saying that. I thought I said, and I certainly meant, the book-Earth system.

 

[Herman Trivilino]

In principle it could be, but then "something as simple as lifting a book from the floor to the table top" would not be an apt description, since, as jtbell said, you would have to have a whole mechanism in place to collect the external energy and harness it to do the work of lifting the book. The simple way to do is is just pick it up, but that doesn't use external energy; you are part of the system.

This is what I said ...

That is in fact what happens when something as simple as lifting a book from the floor to the table top happens. It is common to state that the gravitational potential energy of the book increased, but in fact what has happened is that the potential energy of the book-Earth system increased. This system is a collection of two particles, and the mass of that collection increased when the book was lifted.

 

[Herman Trivilino]

The energy needed to raise the book can indeed come from outside the system. For example, you might use solar energy to boil water, and use the steam to drive an engine that lifts the book. In that case the mass of the earth-book-engine system would increase according to the solar energy that entered the system.

You could also use solar energy to grow the food needed for a human to lift the book.

 

[PeterDonis]

You could also use solar energy to grow the food needed for a human to lift the book.

Yes, but that is just admitting that in fact the Earth (or the Earth plus the book plus the human) is not a closed system. There is energy flow across its boundary in both directions. What's more, if you are taking this energy flow into account, the system is not stationary either, so the notion of "potential energy" is not even well-defined. Obviously this is no longer "as simple as lifting a book", nor is it any good as an illustration of any point about potential energy or invariant mass.

So either you bring in complications that make the whole subject of this discussion pointless, or you accept a simple model in which the Earth-book system is closed, at least on the time scale required to lift the book (note that this time scale is much shorter than the time scale required to capture solar energy and grow the human's food), and in that model, the externally measured mass does not change when you lift the book.