Relativistic Mass of Sub-Atomic Particles: What Does it Mean?

[Dale]

Will the mass of an intact nucleus be found to be less than the sum of the protons and neutrons ?

Yes

 

[1977ub]

Also I guess I forgot gravitational potential energy is negative - in other words, you open the box, move things from the center out to the edges of the box, then close it. It is now found to have less inertial and gravitational mass.

 

[Dale]

in other words, you open the box, move things from the center out to the edges of the box, then close it. It is now found to have less inertial and gravitational mass.

You have that backwards.

 

[1977ub]

If the solar system is found to have less mass than the individual bodies, is this not due to the gravitational potential energy of their being separated in space?

 

[PeterDonis]

If the solar system is found to have less mass than the individual bodies, is this not due to the gravitational potential energy of their being separated in space?

No, it's due to the (negative) gravitational potential energy because they are not separated a lot more in space--i.e., because they are in a bound system with a finite size that is small on cosmological scales, instead of being spread out very far away from each other.

you open the box, move things from the center out to the edges of the box, then close it. It is now found to have less inertial and gravitational mass.

No, it will have more, because the things in the box are separated by a larger distance than they were before. That means you added energy to the box in order to move them apart.

If you want to decrease the inertial and gravitational mass of the box, without removing any of the objects inside, you would need to open the box and remove energy from it, bringing the objects inside closer together (i.e., making them more tightly bound) in the process.

 

[1977ub]

No, it's due to the (negative) gravitational potential energy because they are not separated a lot more in space--i.e., because they are in a bound system with a finite size that is small on cosmological scales, instead of being spread out very far away from each other.
No, it will have more, because the things in the box are separated by a larger distance than they were before. That means you added energy to the box in order to move them apart.

If you want to decrease the inertial and gravitational mass of the box, without removing any of the objects inside, you would need to open the box and remove energy from it, bringing the objects inside closer together (i.e., making them more tightly bound) in the process.

Ok then I'm back with my nagging biases. To make things more tightly bound, does this not imply perhaps that electromagnetic energy was contributed?

Aren't all forms of potential energy related to fundamental forces?

 

[PeterDonis]

To make things more tightly bound, does this not imply perhaps that electromagnetic energy was contributed?

You don't "contribute" energy to make things more tightly bound. You take away energy. As for what form the energy taken away can take, that depends on the details of how you do it. In natural processes like the formation of stars or planetary systems, electromagnetic radiation is typically the way that energy is taken away from the system to allow it to become more tightly bound.

 

[Dale]

make things more tightly bound, does this not imply perhaps that electromagnetic energy was contributed?

It doesn't matter! It can be any form.

Aren't all forms of potential energy related to fundamental forces?

It doesn't matter.

The form of the energy is an irrelevant detail, all that matters is that it is a component of the conserved four-momentum. I feel like you are just ignoring this key fact. The whole purpose of establishing conservation laws is because once you do so you don't need to worry about the details.

 

[1977ub]

You don't "contribute" energy to make things more tightly bound. You take away energy. As for what form the energy taken away can take, that depends on the details of how you do it. In natural processes like the formation of stars or planetary systems, electromagnetic radiation is typically the way that energy is taken away from the system to allow it to become more tightly bound.

Ok. So my impression seems correct. You send electromagnetic energy into the box and it becomes warmer and heavier, or more spread out and heavier, or starts moving around and gets heavier. What ways are there to remove energy / mass from a system without particles (massive or photons) being removed?

 

[PeterDonis]

You send electromagnetic energy into the box and it becomes warmer and heavier, or more spread out and heavier, or starts moving around and gets heavier.

Yes.

What ways are there to remove energy / mass from a system without particles (massive or photons) being removed?

At the fundamental level, there are only four interactions known (electromagnetic, weak, strong, and gravitational), and only two of them give rise to radiation that can travel long distances (the weak and strong interactions are short range and don't give rise to radiation). So the only other option besides the ones you list is gravitational radiation. But gravitational radiation from almost any system will be so much weaker than electromagnetic radiation that it can be ignored for all practical purposes. (The only exceptions are the kinds of systems we have detected gravitational waves from--black hole mergers, neutron stars merging to form black holes, etc.)

 

[Nugatory]

Also I guess I forgot gravitational potential energy is negative - in other words, you open the box, move things from the center out to the edges of the box, then close it. It is now found to have less inertial and gravitational mass.

That's not what negative potential energy means. The negative potential energy just means that we're using the arbitrary (but very convenient, which is why we do it) convention that the potential energy at infinite separation is zero. Whether the potential energy is negative or positive is irrelevant. What matters is whether the change in potential energy is positive or negative as you move things away from the center; and that change is positive.

 

[Nugatory]

What ways are there to remove energy / mass from a system without particles (massive or photons) being removed?

One of the most common ways is for the system to emit electromagnetic radiation. This process cannot be described as photons "being removed".

 

[1977ub]

One of the most common ways is for the system to emit electromagnetic radiation. This process cannot be described as photons "being removed".

The EM radiation can't leave without going somewhere, right? There have to be photons elsewhere which take this energy on then, right?

 

[SiennaTheGr8]

That's not what negative potential energy means. The negative potential energy just means that we're using the arbitrary (but very convenient, which is why we do it) convention that the potential energy at infinite separation is zero. Whether the potential energy is negative or positive is irrelevant. What matters is whether the change in potential energy is positive or negative as you move things away from the center; and that change is positive.

Is this convention really arbitrary in the context of reckoning a system's rest energy?

 

[Nugatory]

The EM radiation can't leave without going somewhere, right? There have to be photons elsewhere which take this energy on then, right?

Photons aren't what you think they are; and you should reread post #39 of this thread.

 

[1977ub]

Photons aren't what you think they are; and you should reread post #39 of this thread.

I understand plenty about photons, I think. I reread post #39. I don't think I have any further questions about this.

 

[Herman Trivilino]

If the solar system is found to have less mass than the individual bodies, is this not due to the gravitational potential energy of their being separated in space?

Not separated, collected.

 

[SiennaTheGr8]

@1977ub

Mass is just a measure of how much energy a system has in its rest frame: $E_0 = mc^2$, where $E_0$ is the system's rest energy.

What contributes to a system's rest energy? Answer: all of the energy "internal" to the system.

Say that the system is full of gas molecules. To calculate the system's rest energy (i.e., its mass), we must add up all of the "internal" energy contributions: the kinetic energies of the gas molecules (as measured in the system's rest frame, of course), the potential energy associated with their relative positions, and also the rest energies (masses) of the individual particles.

Then we could "zoom in" on a single molecule and itemize its rest energy as the sum of the kinetic, potential, and rest energies associated with its constituent atoms. We could "zoom in" on a single atom in the molecule and itemize its rest energy as the sum of the kinetic, potential, and rest energies associated with its subatomic particles.

Etc.

The potential-energy contributions associated with the relative positions of a system's constituents can be positive or negative, depending on whether the force in question is repulsive (positive) or attractive (negative). The potential-energy contributions approach zero in the limit that the system's constituents are infinitely far apart.

 

[SiennaTheGr8]

@1977ub

To calculate the system's rest energy (i.e., its mass), we must add up all of the "internal" energy contributions ...

(or we can just weigh it)

 

[Nugatory]

Is this convention really arbitrary in the context of reckoning a system's rest energy?

You're right, it's not arbitrary for that purpose. I was trying/hoping to avoid that subtlety because much of the recent discussion has been about how the rest energy changes as energy enters and leaves the system; and because (as you pointed out while I was writing this) we can jus weigh the system to establish its rest energy.

 

[PeterDonis]

What contributes to a system's rest energy? Answer: all of the energy "internal" to the system.

The potential-energy contributions associated with the relative positions of a system's constituents can be positive or negative, depending on whether the force in question is repulsive (positive) or attractive (negative). The potential-energy contributions approach zero in the limit that the system's constituents are infinitely far apart.

(or we can just weigh it)

Since we're having this discussion in the relativity forum (as opposed to the classical physics forum), it's worth pointing out that, in GR, all of these statements have limitations.

First, in relativity, the idea of determining a system's rest energy by adding up all of the "internal" contributions is formalized in what is called the Komar energy (more usually called "Komar mass", because in relativity energy and mass are just different units for the same quantity, and the term "mass" is more usual in the GR literature--as opposed to, say, the QFT literature, where the term "energy" or even "momentum" is more usual). The basic idea is that you integrate the stress-energy tensor over all of space, paying appropriate attention to the fact that spacetime is curved.

However, the Komar energy integral is only well-defined in a limited class of spacetimes, the stationary spacetimes--which are basically the ones in which there is a notion of "space" that is independent of "time" (I use the quotes because for precision these terms should be formalized and made precise, and they can be, but there are a lot of pitfalls lurking for the unwary in doing so). It turns out that these are also the only spacetimes in which there is a well-defined concept of "gravitational potential energy"; and it turns out that, in these spacetimes, the Komar energy integral is what you would expect it to be for a bound system, taking the (negative) contribution of gravitational potential energy into account (basically because taking proper account of spacetime curvature, in such spacetimes, is taking the contribution of gravitational potential energy into account).

Interestingly, the other notion of total energy you mention--just weigh the system--corresponds, if we take "weigh" to mean "determine by measurements purely external to the system", to a different concept of energy in relativity--actually, to two of them, called the ADM energy and the Bondi energy. These are well-defined for a different limited class of spacetimes, the asymptotically flat spacetimes--which are basically the ones which describe an isolated system surrounded by empty space. The difference between them is that the ADM energy never changes--even for a system that emits radiation that escapes to infinity. (The reason is that, at any finite time, the radiation has only traveled some finite distance from the system, because of the finite speed of light, so it is always present somewhere in the spacetime, and the ADM energy will therefore include it.) The Bondi energy was developed in order to make rigorous the idea that systems which emit radiation away to infinity lose energy: basically, the Bondi energy is the ADM energy minus whatever energy is carried away to infinity by radiation. This means that the Bondi energy, unlike the ADM energy, can change with "time" (again, this term needs to be properly formalized and made precise) as a system radiates and becomes more tightly bound.

For the even more limited class of spacetimes which have both properties--stationary and asymptotically flat--the Komar energy and the ADM energy are the same, so everything fits together consistently. But there are also important spacetimes--such as the FRW spacetimes used in cosmology--where none of these concepts of energy are well-defined, and so none of the ideas we have been talking about in this discussion apply.

 

[SiennaTheGr8]

Thanks, @PeterDonis. Very informative write-up, there.