Caption: Shown is the only physics formula everyone knows: E=mc**2, where E is energy, m is mass and the vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s =3*10**5 km/s ≅ 1 ft/ns.
E=mc**2 is, of course, a basic result of special relativity and is obtained in the physicsy derivation of special relativity from the special relativity postulates. A physicsy derivation is one where you start with basic axioms, but introduce extra ones has you go along as seems reasonable to physical intuition (i.e., educated guessing) or by clairvoyance (i.e., you believe you know where you have to arrive). Pure mathematics it isn't.
In fact, the derivation of E=mc**2 is remarkably simple though well beyond the scope of this file: i.e., e_mc2.html.
But what does E=mc**2 mean?
Actually, E=mc**2 means two things:
All energy has mass: i.e., exerts a gravitational field and has a resistance to acceleration.
To emphasize this sameness, one often uses the unified term mass-energy in relativity speak rather than either of energy or mass. In fact, in the appropriate contexts, mass-energy, energy, and mass are all synonyms. However, one often uses energy or mass depending on which aspect of mass-energy is being emphasized.
As an example of Meaning 1, if you add/subtract kinetic energy or heat energy E to/from a system, you add/subtract mass E/c**2 to/from the system.
Such changes in mass are NOT noticed in everyday life NOR were they observed experimentally prior to the advent of special relativity in 1905 (when people started looking for them) because ΔE/c**2 is so small compared to a system's rest mass (see the Meaning 2 of E=mc**2 below). Nowadays, they have been super well verified at least by indirect means.
Some further points:
In modern formulation, the laws of conservation of mass and conservation of energy are fundamentally the same thing and are considered as separate laws only as emergent laws in cases where changes in rest mass are below notice (see the Meaning 2 of E=mc**2 below and Wikipedia: Relativistic mechanics: Rest mass and relativistic mass). However, it many applications scientific and technological those changes are below notice.
Actually, yours truly believes that a measurement of mass of an object is the only direct measurement of energy and it is only a measurement of total energy. All other energy measurements are indirect measurements since you measure some other quantities and calculate energy from a formula. For example, kinetic energy KE=(1/2)mv**2 (the energy of motion in the classical limit) is measured by measuring m and v and using the just given formula.
It does NOT strictly hold for the energy of the gravitational field.
In fact, the gravitational field is NOT like other force fields (the electromagnetic field and the force fields of the strong nuclear force and weak nuclear force). It is an emergent field from the curvature of space and it does NOT have localizable energy that itself gravitates. Albert Einstein (1879--1955) ruled that out in developing general relativity since it led to the paradox of mass-energy creating a gravitational field that had its own mass-energy that created its own gravitational field, and so on ad infinitum.
The curvature of space is what creates gravity itself and that curvature of space encodes energy implicitly in a non-local fashion that does NOT in itself have anymore gravitating effect.
The encoded implicit energy does NOT appear in the energy-momentum tensor T_ij which dictates the curvature of space in the Einstein field equations (which along with the geodesic equation are the core formulae of general relativity). Hence, general relativity avoids the ad infinitum paradox discussed above.
Note by "non-local", we mean there is NOT so much energy here and so much energy there and there is NO density of energy. This is unlike the other force fields.
Consider gravitational waves again. When they are emitted from a physical system (e.g., slowly from a binary pulsar or in sudden blast from a binary black hole merger) energy (or, if you prefer, mass-energy) is lost from the physical system, but as the gravitational waves propagate they propagate they make NO contribution to the energy-momentum tensor T_ij of the space they propagate through. In fact, the energy-momentum tensor T_ij would be all zeros if there were nothing else in space. Yet when the gravitational waves interact with other physical systems they deposit energy. Where was that energy between emission and interaction of the gravitational waves? It wasn't locally in the energy-momentum tensor T_ij. As mentioned above, general relativity does NOT guarantee that the amount of energy emitted will be recovered ever from the gravitational waves (see Roger Penrose, The Road to Reality, 2004, p. 467--468).
To give an imperfect analogy, say you wanted to get a car to Reno, Nevada. You could drive it and then it was a car all the time. Or you could disassemble the car and send the parts separately to Reno and then reassemble the car. Did the car exist when in parts? Non-locally. In fact, general relativity does NOT guarantee the "car" will be the same size when reassembled: it could be smaller or bigger.
For more on the tricky point of non-local energy, see Roger Penrose, The Road to Reality, 2004, p. 464--469.
There is such a thing as rest mass energy (usually just called just rest mass for simplicity) since massive particles have intrinsic mass NOT due to any other energy form.
This is the energy calculated from E=mc**2 for an object of mass m when the object is at rest in an inertial frame: hence the name rest mass.
Since any energy form can be converted into other all other energy forms, rest mass can be converted into all other energy forms. But note conversions are NOT always easy to do in practice.
Some further points:
Exotic dark matter particles (if they actually exist) also have rest mass and are massive particles, but they are NOT baryonic matter.
Massive particles can be observed when at rest.
Note that massless particles always move at vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s =3*10**5 km/s ≅ 1 ft/ns relative to any local inertial frame, and so are NEVER at rest in a local inertial frame. But could they still have rest mass in some sense? NO. If they had rest mass, then the formula
But E=mc**2 means that massless particles do have mass since they have energy (finite energy). To be super precise, we should call massless particles rest-massless particles, but that's NOT the convention.
But it can be done in principle. For example, just rapidly collide 0.5 kg of matter with 0.5 kg of antimatter and both will annihilate to form gamma rays (and maybe other stuff ???) which will give explosive heating with the above calculated amount of 25 megatons TNT. But there is NO practical way to accumulate any macroscopic amounts of antimatter.
Pair creation is actually inverse annihilation since you create massive particles, NOT destroy them.
At the microscopic level, the mutual annihilation of matter and antimatter goes on all the time at a very low level including wherever you are. For example, positrons (the antiparticles of electrons) are products of certain radioactive decay processes (specifically some beta decay processes) that occur at a low level just about everywhere. But the rate of positron production is very low usually. The produced positrons run into electrons pretty quickly and mutually annihilate to create gamma rays. The rate of heating from this natural and common process is relatively low.
Ever since the advent of special relativity in 1905 people have been mesmerized by how much energy is available in principle in rest mass. But the practical ways of getting a significant fraction of it from matter are limited to nuclear reactors and nuclear bombs.