Trying to Fully Understand Mass Increase of Unstable Bonds

[Ascendant0]

In Modern Physics today, my professor covered "unstable bonds," and how when two masses bind together to form an unstable bond, the total mass $M$ is greater than $m_1 + m_2$ of the two masses. I tried to ask some questions to gain further understanding, but there were too many students asking too many questions for me to get a full answer on what I was wondering. So...

1) Where does the extra mass physically come from? I mean if I latch together two objects with a spring in between them, I could make a very unstable contraption with a sort of "stored energy" in the spring, but I know very well that won't increase the mass of that contraption at all whatsoever. I also know the energy the system gets is coming from the force of me pushing the system together and locking it into place. I'm wondering how it's different on this (molecular bond?) level, and why the mass does increase in that circumstance?

2) If two masses collided together in a vacuum to create an unstable bond, with no external forces causing it, just momentum and a collision, are they still able to acquire that extra mass even without anything external to draw it from? Does it have any effect on any equations to where you can see where that extra mass is coming from?

3) Tying into 2), if it doesn't have any impact on the equations (other than the ones that change due to the increase in mass $M$), assuming you didn't have the information for the total final mass $M$ to rely on, how else could you tell that two masses that came together formed an unstable bond? Does it take knowing in advance what molecules/particles cause unstable bonds with each other, or is there some kind of telltale sign(s) if you didn't have the information regarding the change in mass?

 

[Orodruin]

I mean if I latch together two objects with a spring in between them, I could make a very unstable contraption with a sort of "stored energy" in the spring, but I know very well that won't increase the mass of that contraption at all whatsoever.

You may want to reevaluate that knowledge. While it is true in a classical mechanics setting, it is no longer true in relativity. The mass increase is minuscule (stored energy/c^2) and likely way beyont the accuracy of any measurement. But theoretically, it is there. The reason classical mechanics works so well in this situation is that the negative binding energy is so small in comparison to the masses.

I also know the energy the system gets is coming from the force of me pushing the system together and locking it into place. I'm wondering how it's different on this (molecular bond?) level, and why the mass does increase in that circumstance?

It is no different. The additional energy must come from somewhere.

2) If two masses collided together in a vacuum to create an unstable bond, with no external forces causing it, just momentum and a collision, are they still able to acquire that extra mass even without anything external to draw it from?

They will not be able to form the unstable state - unless you bring in quantum tunneling.

Does it have any effect on any equations to where you can see where that extra mass is coming from?

In order to form the state, you need the extra energy. In the vacuum, with tunneling, that would have to come from the kinetic energy of the colliding particles.

3) Tying into 2), if it doesn't have any impact on the equations (other than the ones that change due to the increase in mass $M$), assuming you didn't have the information for the total final mass $M$ to rely on, how else could you tell that two masses that came together formed an unstable bond? Does it take knowing in advance what molecules/particles cause unstable bonds with each other, or is there some kind of telltale sign(s) if you didn't have the information regarding the change in mass?

Metastable states typically appear as resonances for colliding particles, meaning roughly that when you have just enough energy to produce them, the probability of the collision occurring increases significantly. With unstable states produced along with other stuff in reactions you can look at the energy and momentum of the end products in order to deduce the properties of the unstable state. This was the case, eg, when the Higgs boson was found (although note that it is fundamental rather than composite - it is still very very unstable).

 

[Ibix]

Where does the extra mass physically come from?

Same place the energy did. Your mass will decrease slightly because you gave away some energy. As Orodruin already noted, it's tiny on a human scale because $E=mc^2$, so if you burn a Joule your mass decreases by about $10^{-17}\mathrm{kg}$.

This is a first hint of a general thing in relativity, that mass is not additive. Just because I have a list of the masses of every one of your atoms, I do not know your total mass. I need to know about how the atoms bind together.

Energy and momentum remain additive and conserved, and are generally much more useful concepts in relativity.

2) If two masses collided together in a vacuum to create an unstable bond, with no external forces causing it, just momentum and a collision, are they still able to acquire that extra mass even without anything external to draw it from?

Assuming you are talking about two masses and a spring with a latch mechanism, this is one of the cases where mass just doesn't add. The mass of the bound system is not the same as the sum of the masses of the components measured separately. If you hide the components in a box and measure its mass, though, that will not change - roughly speaking that's because in that case you also measure some mass associated with the kinetic energy of the two masses, which you don't do when you measure their masses separately.

It's a surprisingly complex topic, but it gets a lot simpler once you introduce four vectors. It turns out that energy and momentum form a four vector, and mass is the modulus of that vector. You can add vectors, but the modulus of a sum is not yhe sum of the moduli.

 

[Orodruin]

decrease slightly because you gave away some energy

Increase slightly because they added some energy.
Their system is a metastable state with higher mass than the constituents.

 

[Ibix]

Increase slightly because they added some energy.
Their system is a metastable state with higher mass than the constituents.

I think the OP was describing a system of masses and springs whose system mass is higher than the sum of component masses because of supplied energy. The supplied energy comes from the OP's body's chemical potential, which decreases and therefore his mass decreases.

Are we talking at cross purposes, or am I missing something.

 

[Orodruin]

I think the OP was describing a system of masses and springs whose system mass is higher than the sum of component masses because of supplied energy. The supplied energy comes from the OP's body's chemical potential, which decreases and therefore his mass decreases.

A bit convoluted but fine. I assumed we were only talking aboutthe relevant system. The energy is obviously provided from outside, regardless of mechanism.

 

[pervect]

The I level explanation of your problem, which is slightly inaccurate, is that fields have a mass of their own. When you have an unstable bond, there is a field between the atoms, adding energy to that field has mass. Similarly for compressing a spring. Typically reference is made to E=mc^2. However, this isn't really a complete explanation :(.

Before special relativity, in classical mechanics, the problem with mass was noted, and people came up with the concept of "electromagnetic mass" which has been depreciated.

See for instance the wikipedia article on "electromagnetic mass". https://en.wikipedia.org/wiki/Electromagnetic_mass. A quote from the current (as of this posting) version:

Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for some time also considered as a dynamical explanation of inertial mass per se. Today, the relation of mass, momentum, velocity, and all forms of energy – including electromagnetic energy – is analyzed on the basis of Albert Einstein's special relativity...

The A level explanation, the one which uses relativity rather than classical mechanics, basically involves replacing the idea of mass, which is a scalar, with the stress-energy tensor, which is a rank 2 tensor. It's compoents involve energy, momentum, and pressure. For point particles, one can get away with the invariant mass, m, which is what most people on the forums use, but the notions of point particles aren't really enough to answer your questions.

In my experience, people learning relativity at the appropriate level (the issues I am talking about aaren't covered in introductory classes) are told to use the stress energy tensor but they typically don't understand why they need to do that and they resist. The original papers on the topic are mostly in German, which doesn't help if one tries to take a historical approach. Basically, though, it's a natural outgrowth of the principle of covariance.

Max Jammer has some books on the topic, "Concepts of Mass in classical and modern physics", also "Concepts of mass in contemporary physics". The books take a philosophical and historical approach, with some mathematics, rather than being a textbook approach, and talks about some of the general issues involving mass. Note that the Newtonian concept of mass as a "quantity of material" is not compatible with E=mc^2, E is not a quantity of material.

I would guess that you would most likely want to stick with the I level explanation, even though it's slightly wrong. However, some things about the I-level explanation may nag at you, unfortunately the real cure to these nagging issues turns out to be rather radical. For the most part, if you are not purusing a serious interest in physics, I'd recommend noting that the concept of mass turns out to be not as simple as it appears, but sticking with the consistent (but slightly wrong) Newtonian ideas. Unfortunately half-measures to try and "fix" the Newtonian ideas tend to also be wrong, and wind up confusing people over effects that really don't matter for most applications. It's useful to know the amount that the Newtonian ideas are wrong, though, and E=mc^2 is sufficient for that purposes.

 

[cianfa72]

Same place the energy did. Your mass will decrease slightly because you gave away some energy. As Orodruin already noted, it's tiny on a human scale because $E=mc^2$, so if you burn a Joule your mass decreases by about $10^{-17}\mathrm{kg}$.

This is a first hint of a general thing in relativity, that mass is not additive. Just because I have a list of the masses of every one of your atoms, I do not know your total mass. I need to know about how the atoms bind together.

Note that here we're talking about invariant mass, not "relativistic mass" which is actually a misleading term (basically a synonymous of Energy).

Energy and momentum remain additive and conserved, and are generally much more useful concepts in relativity.

They are the really important conserved physical quantities.

It's a surprisingly complex topic, but it gets a lot simpler once you introduce four vectors. It turns out that energy and momentum form a four vector, and mass is the modulus of that vector. You can add vectors, but the modulus of a sum is not the sum of the moduli.

As above mass is the invariant mass.

 

[Nugatory]

Note that here we're talking about invariant mass, not "relativistic mass" which is actually a misleading term (basically a synonymous of Energy).
….
As above mass is the invariant mass.

I’m not understanding your point here. You’re the only one to have mentioned relativistic mass in this thread, everyone else has been talking about invariant mass all along.
If you are uncertain about the resolution here, namely that masses are not additive, it might be best to start your own thread instead of hijacking this one.