Another way to get energy equivalent of a mass

[jmatt]

A thought experiment ...

You have a mass sitting in space far from any source of gravity. We know the energy it contains from e = inertial mass x speed of light squared.

Using a different approach, what would you have to sum to get this total energy from properties of the mass such as

1. Total binding energy or all chemical bonds.
2. Total binding energy of nuclear forces.

etc?

Is this an approach that would ever sum to inertial mass x speed of light squared?

 

[mfb]

You can look at the different energy components, this is fine. You will see that there are positive and negative contributions.

Most of the energy comes from the binding energy in protons and neutrons (positive, ~940 MeV per nucleon).
A smaller, positive contribution comes from the interaction of up- and down-quarks with the Higgs field, usually described as quark masses (~10 MeV per nucleon)
A small, negative contribution comes from the nuclear binding energy, unless you have pure hydrogen without any deuterium/tritium nuclei (some MeV per nucleon, depends on the nucleus)
A smaller positive contribution comes from the mass of the electrons (511keV per electron, in uncharged objects this corresponds to 511keV per proton as both have the same number).
An even smaller negative contribution comes from the binding energy of electrons to their atoms. This depends on the material, with ~10eV per electron for hydrogen and several keV for heavy metals.
Chemical bonds are in the range of some eV per atom, and negative, too.
Thermal energy at room temperature is some meV per atom (positive), and mechanical energy something like µeV per atom (positive).
If you have some charge separation, you can include this, too (again positive), and probably some gravitational binding energy (negative).

And if you add everything, you get the total energy of the object, which corresponds to an effective mass (usually just called "mass" if you do not care about the internal details).

 

[jmatt]

Thanks MFB!

Is the state of our knowledge and computaional ablities such that we can actually do this? Could we take a penny and compute all of the energies listed above and come near mcc with any accuracy?

 

[mfb]

You can calculate this, and get a value so close that probably no scale can measure a difference.
In penning traps, masses of atoms can be measured with an accuracy of some eV, that is enough to see the nuclear binding energy, electrons with their binding energy and probably even chemical bonds (if molecules can be stored there).

 

[jmatt]

Thanks again.

It is astounding how nature (and Mr. Einstein) has wrapped up all that complexity in mcc.

Gives me the chills!

 

[jmatt]

One more question please. Would it be possible to calculate the energy in the penny without any reference or need to know the inertial mass or gravitational weight? If the elemental composition is known is there a way to do this?

 

[mfb]

That is what I said:

You can calculate this, and get a value so close that probably no scale can measure a difference.

You have to know the number of atoms per isotope and the chemical structure of the penny, of course.

 

[jmatt]

And finally good sir; why is this possible? It seems to be an unreasonable leap from the the quantum world to the realm of relativity. Could mcc be predicted only from quantum mechanics? If one has a deep understanding of quantum mechanics is it natural that these energies add up to mcc?

As you might guess i find this to be both satifying and deeply mysterious.

 

[mfb]

That is not related to quantum mechanics. The energy levels of particles are determined via quantum mechanics, but apart from that it is just special relativity, which can be used on the scale of atoms and macroscopic objects at the same time without problems.

 

[jmatt]

Thanks for your answers.

At the risk of trying your patience may I re-cast the question? Was there ever a version of quantum mechanics where space and time were treated classically? If such a version can be imagined (pretend that relativity has not been discovered, but we are really good at QM) could this version calculate the total energy of the penny and would that energy add up to mcc?

 

[mfb]

Was there ever a version of quantum mechanics where space and time were treated classically?

Everything usually called "quantum mechanics" is non-relativistic. You get a decent description of electron energy levels, but you cannot use it for nuclear effects. And you have to add E=mc^2 by hand, of course.
Dirac found a relativistic equation for electrons, and the modern quantum field theory uses special relativity as fundamental property of spacetime.

 

[jmatt]

Mfb you are a patient man (or woman?).

I still am not sure i have asked the question i have correctly so please consider this. An alien scientist is sitting in an unaccelerated spaceship far from any gravity. This alien knows nothing of special relativity, does not know the special nature of the speed of light, but in every other respect is a brilliant physicist with access to any measuring equipment that he can imagine. On a table before him is a lump of coal (or something). Is it possible for him to arrive at the same energy content of the lump as an Earth physisct does who knows SR and simply uses e= mcc?

Thanks!

 

[mfb]

<- m

Without knowing about special relativity (or, better, general relativity), the scientists would not know how to calibrate the energy scale. He could use an unbound state as 0 or whatever, but he would not include all the energy contained in the mass of the particles.

As an analogy, look at potential energy in the gravitational field of earth: If you know the size and mass of earth, you can define "potential energy 0" as "in infinite distance" and calculate a value for an object on the floor (about -62.5MJ/kg) and 1m above the floor (g*h=9.81 J/kg more). If you do not know this, but want to use the concept of potential energy, you can define the floor as 0. In this case, an object 1m above the floor has a potential energy of g*h=9.81 J/kg. The difference is the same, but the 0 point is different.

 

[jmatt]

Ahh, light dawns over Marblehead. In the anaolgy you provided e=mc^2 provides a "floor" so to speak to begin the discussion of the energy content. Do all of the + and - contributions you mentioned above cancel one another so that the floor (mcc) is where we end up?

 

[mfb]

If you add all energy components, and divide the sum by c^2, you get the mass of the object, right.