A follow up question to In $E=mc^2,$ does it not matter what constitutes the mass?
Do the different physical properties of chemical elements and compounds corresponds with the different sources of energy which compose of the mass of those elements and compounds?
In answers to my previous question it was clarified that different forms of energy make up the total energy in a gram of sugar vs. a gram of water vs. a gram of lead, but ultimately a 1 gram mass of any of those objects has the same total energy. That has me thinking: the differences in properties between these objects must be accounted for in the differences in forms of energy composing each object. For example, the forms of energy which compose a gram of water (hydrogen bonding, covalent bonding, and more) are different than the forms of energy which compose a gram of salt (ionic bonding and other forces).
Aside from the extreme complexity of the various forms of energy interacting to compose water from elementary particles, is the relationship between energy and mass/matter really this simple? Or are differences in the properties of objects dependant on more than the forms of energy of which they are made of, in which case, what other influences on properties of physical objects are there at play?
Answer
You are confusing things some. Chemical energy and nuclear energy, and other kinds of energy are indeed different forms of energy, due to different forces, or equivalently different kinds of interactions (In physics there are four forms of interactions, the so called 4 forces). They will form different looking pieces of matter, and have different effects on on other matter when say they are mixed, but if the total matter for each case is the same, they have the same total energy.
But this is the important thing. If you somehow take two equal amounts of energy and are able to convert them each totally into matter, their mass will be identical. Exactly given by E = m$c^2$. You have to be careful that you don't add other uncounted energy in. For instance, all that matter in each case has to be at rest, or else you have to add their kinetic energies in. Also, the matter 'things' in each case need to be one clump, or you'd have to account for gravitational and other forces between clumps. So you have to be careful and make sure you didn't miss something (such as neutrinos which are hard to detect).
There is more. The strong equivalence principle says that that mass, from whatever energy, in each clump, say 1 gram of sugar and 1 gram of sand, will exert exactly the same gravitational force. Of course their inertial masses are also identical.
An interesting way to see that the specifics of the forces makes no difference regarding total mass, consider two black holes at rest at some very large distance at rest with masses m and m (to make it easy). They will be attracted to each other (or equivalently in general relativity, GR, moves in geodesics in the spacetime towards each other) and start moving towards each other, accelerating as they get closer. They'll get very close, and if their trajectories are not perfectly in line, rotate around each other a number of orbits, and merge with each other into one larger black hole. The mass of the new black hole will be M, with
$M = 2m - \mu$, and amazingly gravitational waves will be radiated and its total energy at infinity (or far enough away that you can consider it as no longer interacting with the black hole left behind), will be e, and the kinetic energy of the resultant black hole KE, with
$\mu c^2$ = e + KE
The changes in mass and energy were multiple, but the translations between them always follow Einstein's mass energy equation. The mass of the remaining black hole is exactly what some body (small enough to call it a test body, i.e. Ignore its own gravitational field), some astronomical distance away will conclude, from the black hole's gravitational attraction, is the mass of that black hole. The energy that escaped in the gravitational wave and the kinetic energy of the final black hole, make it balance out. Notice that you don't need to know the black hole's internal binding energy (which does not exist, but it would in any other body) or its rotational energy (which exists because their trajectories were not exactly aligned and so there needed to be some angular momentum) as it all goes to define its total mass M.
Yes, mass and energy are numerically equivalent, with the $c^2$ being the translation factor. In advanced physics one uses natural units where c is set to equal 1, so mass and energy are indistinguishable in terms of their total values. We just call it mass energy, sometimes use either term for the other.
Note that the mass or energy may have different forms, and do different things, even when the total value is always the same. So a gram of sand or sugar may have different amounts of nuclear mass and chemical or electromagnetic energy, that will always depends on differences between the two substances, but total masss for both are the same, and if you somehow converted each into total energy, of whatever kind, the total for each would be the same. And their gravitational effects on other matter are exactly equal (even though 1 gram is a small gravitational force).
So for total mass or energy the constituent parts contributions to total mass, or total gravitational field created (assuming they both are point masses as an idealization) by each, are the same. Regardless of internal details. Stil a gram of sugar you may be able to eat ok, you wouldn't a gram ((or say a kg) of sand. Same mass, but Lots and of other differences.
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