### Mass-Energy Equivalence

In 1905, the same year Einstein published his first theory of relativity (special relativity), he also published three other groundbreaking papers.The four papers he published in 1905 have been dubbed his Annus Mirabilis (miracle year) papers. They include “On a Heuristic Point of View about the Creation and Conversion of Light,” “On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular Kinetic Theory of Heat,” “On the Electrodynamics of Moving Bodies,” and “Does the Inertia of a Body Depend Upon Its Energy Content? One of these papers showed that his theory has surprising implications about the nature of matter and energy.The paper, entitled “Does the Inertia of a Body Depend upon its Energy-Content?”, was originally written in German. An English translation is available here. Special relativity implies that when an object gains or loses energy, for example by absorbing or emitting light, the object’s mass must increase or decrease in proportion to the energy it gained or lost. Specifically, the object’s mass changes by an amount equal to the change in its energy divided by the speed of light squared. (See appendix F [link coming soon] for an explanation of how this result can be derived from the foundational assumptions of special relativity.)

If that is true, Einstein realized, then perhaps the total mass of an object is directly related to how much energy it contains! This relation between mass and energy is expressed in Einstein’s most famous equation, which says that the energy contained within an object is equal to the object’s mass times the speed of light squared:E = mc2

The meaning of this equation is easily misunderstood. It is often characterized as implying that mass is a form of energy, or that mass can be converted into energy and vice versa. Those characterizations are somewhat misleading, however. Mass isn’t a separate form of energy, to be included in the list alongside kinetic energy, potential energy, heat, and so on. Rather, the mass of an object just consists of the various forms of energy (heat, potential energy, etc.) the object contains. In other words, mass itself isn’t a form of energy at all; it’s a quantity that depends on the total amount of energy that an object contains in various forms.

Moreover, mass can’t be converted into energy, because it already consists of energy. But the energy trapped inside a piece of matter can be released—that is, it can be converted from its previous form (e.g. potential energy) into a form that can escape (e.g. light). And matter contains a lot of energy! In the above equation, E is energy, m is the object’s mass, and c is the speed of light: approximately 300,000,000 m/s. That’s already a big number, so the square of the speed of light is huge. This means that even an object with little mass contains an enormous amount of energy; and conversely, a large amount of energy yields only a small amount of mass.

Let’s see how many joules of energy are in a milligram of matter, that is, in one millionth of a kilogram.
E = mc2
E = 0.000001 kg × (300,000,000 m/s)2
E = 10-6 kg × (3 × 108 m/s)2
E = 10-6 kg × 9 × 1016 m2/s2
E = 9 × 1010 kg m2/s2
E = 9 × 1010 J
Ninety billion joules of energy (9 × 1010 J) are contained in just one milligram of matter! That’s enough energy to lift more than 9,000 metric tons a kilometer upward against the pull of Earth’s gravity!

If the energy in a whole kilogram of matter were released, it would be enough to replace all of the energy used in the entire United States—including the energy from fossil fuels, nuclear power, and renewable energy sources—for nearly 8 hours!According to recent estimates by the U.S. Energy Information Administration, the United States used nearly 1017 Btu of energy in 2014, which equates to about 1020 joules in a year, or 2.7 × 1017 J per day, or 1.14 × 1016 J per hour. Unfortunately, the only known way to release all of the energy in matter is by mixing matter with antimatter, and antimatter is in short supply. (See chapter 5.)

Where does all this energy come from? As you may recall from chapter 5, some elementary particles get their mass by interacting with the Higgs field. Such interactions involve energy, and that’s why the particles have mass. However, the mass produced by the Higgs interaction accounts for only a tiny fraction of the mass found in composite particles like protons and neutrons. In fact, if you add up the masses of the elementary particles in a proton (2 up quarks and 1 down quark), you only get about 1% of the total mass of the proton. (Try it!)

Ok, so where does the rest of the mass come from? The majority of the proton’s mass comes from the kinetic and potential energy it contains due to the interactions between quarks. A lot of potential energy is packed inside a proton, because those zippy little quarks are pushing and pulling on each other with the aptly-named strong force! That’s where most of the mass of protons and neutrons—and thus most of the mass of an atom—comes from. Indeed, most of your own mass consists of potential energy from the nuclear strong force.