What qualifies as “Energy” in the Einsteinian sense of the word?

Question

As an absolute beginner to special relativity (and all the 1900s Einstein stuff), I find it hard to grasp the real meaning of the term energy used in the popular equations. I’ve heard it is possible to account for all the forms of energy as a combination of kinetic and potential energies of the system.

But how exactly are these two,“fundamental” energies? I (like many others) find the concept of kinetic energy to be more intuitive than potential energy and often see potential energy itself as a kind of accounting trick to get a neat energy conservation equation. Since potential energy can be negative, it goes against intuition to see it contribute to some “negative mass”, say in a Hydrogen atom (which weighs less than the sum of its constituents)

So, yes. What I am looking for is a simple definition of what could be called energy and how that relates to mass in the case of mass-energy equivalence. And what allows us to consider potential energy while solving for the mass of a Hydrogen atom. Hope I made at least a little sense :)

Answer 1

In the classical sense, energy is most easily defined as "money in the bank": stored work which can be put to later use at will, to perform useful work.

In the case of gravitational potential energy, the only reason it can be negative is because of how we choose the elevation at which we define the origin of our height accounting scheme. For other energy calculations (for example, kinetic energy = 1/2mv^2) the squared term can never be negative so the energies are always positive and the book keeping is simpler).

Prior to Einstein, the interconvertibility of energy and mass was unknown but was since experimentally verified to high accuracy. This interconvertibility has no "classical" (i.e. newtonian) explanation and this makes it perhaps harder to grasp, but here is how I think of it:

It requires work to squeeze together hard enough all the protons and neutrons that form a uranium nucleus against the electrostatic repulsion of the constituent protons and get the strong nuclear force to stick them together. The work required shows up as a very slight increase in the mass of the resulting nucleus, relative to the masses of all those individual protons and neutrons. Fissioning that nucleus releases that stored energy and can be used to perform useful work (generating electrical power) or in some cases useless work (frightening a political adversary) and the resulting mass deficit represents the energy release.