Sometimes you may have heard someone say that, in the end, ‘everything is energy’. ‘Einstein said himself that mass equals energy, we are energy ourselves, light is energy, and everything in this universe is energy.’ Often, it is represented as the fundamental substance everything is made out of. And energy is conserved. Both statements are incorrect.

To get straight to the point: energy is a mathematical concept. It is not a substance and it is not a mysterious ‘elusive something’. Nothing is flowing from one object to the other. It is a number, very useful and ingenious to perform calculations with and base predictions on for the state of a system. It is clever mathematical bookkeeping originating from the seventeenth century polymath Gottfriend Wilhelm von Leibniz.

Here, we must differentiate between physical objects (so, not energy) and properties of those physical objects. Think of properties like position, volume, mass, velocity, and energy. These five proporties are numbers. Mathematical quantities. In high school, we were taught to express quantities in numbers of units, which signified physical phenomena of physical objects, such as, respectively, location, size, inertia, motion, and… what energy signifies, you will read after this.

Let us take a rolling cannonball A as an example. This physical ball has two measurable properties: a mass A and a velocity A. Suppose, there is another rolling cannonball: mass B, velocity B, only in the opposite direction. They will collide. You may assume that their speeds and the direction of their speeds will have changed after the collision.

Leibniz noticed that, for each ball its mass multiplied by its squared velocity and then added all together, this total sum before the collision is equal to the total sum after the collision. \[ \begin{split} \Big[\big(\text{mass A}\big) &\times \big(\text{velocity A}\big)^2\Big] \\ &+ \Big[\big(\text{mass B}\big) \times \big(\text{velocity B}\big)^2\Big] \\ &\text{before the collision} \\ &= \\ \Big[\big(\text{mass A}\big) &\times \big(\text{velocity A}\big)^2\Big] \\ &+ \Big[\big(\text{mass B}\big) \times \big(\text{velocity B}\big)^2\Big] \\ &\text{after the collision} \end{split} \]

Both the product and the sum are nothing more than a number. The mathematical result of the product of mass and velocity (squared), we call energy. Leibniz, however, did not, but used, rather poetically, the Latin term vis viva, ‘living force’.

Many years of refinement and extension of the mathematical concept followed. Leibniz’s formulation appeared to be missing a factor of one half, an extension of the vis viva-concept to heat was necessary, and it experienced heavy competition from the conservation of momentum from rival Newton.

Ultimately, at the beginning of the 19th century, the polymath Thomas Young became the first to use the term energy in written form in his book A Course of Lectures on Natural Philosophy and the Mechanical Arts: In Two Volumes—even though it would still undergo several evolutions. In the end, the concept was not only useful in mechanical and thermal calculations, but also in electrical, magnetic, chemical, and nuclear interactions, for instance.

### Conservation of energy

Emmy Noether, a mathematical genius, laid the mathematical foundation for the conservation law of energy, among others, which had been formulated a couple of decades before. Thanks to Noether’s theorem, we know why energy is conserved in an isolated system: the laws of nature are so-called time invariant. In other words, for a law of nature it does not matter if it is applied at ten o’clock in the morning or two hours earlier. Whether at four o’clock at night or fifteen minutes later, cannonballs will not collide any differently. Their operations are invariant. If a time-translated, isolated system, such as our cannonballs, works in the exact same fashion as it did before the time translation, we say we have a symmetric situation. And if laws of nature are time symmetric, we can, thanks to Noether’s theorem, derive the law of conservation of energy mathematically.

### Einstein and expansion

Something many people unfortunately do not know, is that, since Einstein’s general relativity—a little over a hundred years ago, practically at the same time as Noether’s proof of her theorem—the law of conservation of energy does not apply to the observable universe we inhabit, after all.

That is to say, the law operates just fine at the scale on which we, humans, live our lives on a daily basis. The high school exams are still valid. Architects and engineers can still rely on it. However, at the scale of the observable universe, the one at which cosmologists work, the law does not hold. Spacetime itself is dynamical: it changes over time. Moreover, in 1998, Nobel prize-winning research showed that the observable universe is expanding exponentially. This, too, demonstrates that space itself is not symmetrical over the passage of time.

The law is thus untenable for the whole observable universe. However, when taken a piece of space and a piece of time small enough, the law works just fine. At this smaller scale, systems appear to be near-isolated from the rest of the universe. Noether’s theorem applies here, and, thus, the law of conservation of energy, which rests on her theorem.

### Not fundamental, but important

For two reasons, energy cannot be fundamental in a theory of our universe: the concept is a mathematical tool to quantify measurable properties such as mass and velocity and its law of conservation rests on another theorem, while, at the same time, it has been proven not to be conserved, about a hundred years ago.

Even though not an invisible, flowing substance or some other mysterious fundamental quantity, it is, nevertheless, highly useful in diverging areas such as fluid dynamics, statistical mechanics, astrophysics, nuclear physics, and quantum physics, even just to simply replace an intricate formulation such as \[ \frac{mc^2}{\sqrt{1-\dfrac{v^2}{c^2}}}, \] by \[ E \] for energy. Eh, ‘energy’.

Photo by ESA/Hubble/NASA. A Hubble Space Telescope image of Galaxy cluster Abell 2537. The amount of gravity, that is, warping of spacetime, caused by this galaxy is visible through the bending of the light of stars and galaxies behind Abell. The galaxy works as a lens. All is predicted by Einstein’s General Relativity.

*@kjrunia** is reading for a joint honours degree in mathematics and theoretical physics (final year) in England, at the School of Mathematics and Statistics and the School of Physical Sciences at The Open University, Walton Hall, Milton Keynes.*