###### Photo source: groovypost.com

It came as a tremendous surprise, because in his student days, Einstein had been a lazy dog. He never bothered about mathematics at all

Hermann Minkowski (from Carl Zelig’s book «Albert Einstein»)

**According to scientific lore, Albert Einstein formulated E = mc² in 1905 and explained with divine ease how energy can be released in stars and nuclear explosions. So says the legend. In fact, Einstein was not the first person to consider the equivalence of mass and energy.**

The famous formula for the energy E hidden in any material body of mass M: E = mc

^{2}(C = 300 thousand km/sec — the speed of light) became a meme of the Atomic Age.

With the development of atomic physics, it became clear what this huge mysterious energy was. To extract it as radiation, take half a gram of matter and half a gram of antimatter and bring them into contact.

You get an explosion of 21.5 kilotons of TNT. Which is roughly the same power as the Fat Man atomic bomb dropped by the Americans on August 9, 1945, on Nagasaki, Japan. Some of this energy is released in, for example, nuclear reactions within stars, heating up both our Sun and our Earth.

Albert Einstein analyzed this formula in his famous article of the year 1905 «Does the inertia of a body depend on the energy contained in it?»

In the same year, Einstein published a paper devoted to explaining the photoelectric effect on the basis of the quantum theory of light. He received the Nobel Prize for it (1921). Einstein was not an opponent of quantum theory. He was only an opponent of its probabilistic interpretation.

The formula is so iconic that you might be tempted to think that Einstein wrote it on inspiration

But he didn’t. He is not its author. In 1864, the eminent Scottish physicist Maxwell published a paper called «The Dynamical Theory of the Electromagnetic Field.»

Taking into account the known laws of electricity and magnetism, using a mechanical model of wave propagation, he presented equations having the same meaning for electric forces as the law of gravitational interaction of celestial bodies published by Newton almost 200 years before.

One remarkable consequence was Maxwell’s conclusion that light is what electromagnetic waves are.

Where everyday intuition can’t help, scientists are guided by a mysterious, irrational sense of beauty. They rely on a sense of logical perfection, ideas of symmetry, philosophical doctrines, and things outside of physics

Symmetry in mathematics is characterized by a set of transformations that translate a picture, figure, or equations into themselves. For example, rotating a cube around independent axes.

After each 90-degree rotation, it is aligned with itself. Since two consecutive rotations are again some rotation, all such transformations together form a single object — a symmetry group.

As for the cube and what its symmetry group looks like, we understand it intuitively. For space and time, it results from a fundamental physical principle: all laws of nature look the same in reference frames moving uniformly and rectilinearly relative to each other.

In other words, sailing in a large ocean ship and not looking out the porthole, you could not determine by any experiments whether it is moving or standing in the port.

Metaphorically speaking, physics «aligns with itself» when moving between equivalent reference frames. The group of symmetries in physics, called «Galilean transformations,» was known for Newton’s equations of motion — the key scientific product of the pre-relativistic era.

And now, it turned out (by H. A. Lorentz and A. Poincaré) that Maxwell’s equations possessed another, more complex symmetry group (Poincaré called its turns «Lorentz transformations» in honor of the discoverer).

According to Newton, physics is symmetric this way, and according to Maxwell, it is symmetric that way. One of the two was wrong! And it looks like Newton’s own theory of gravity had to be corrected

The new fundamental symmetry of the universe combined, mixed (as rotations mix the faces of a symmetrical cube) previously seemed to be dissimilar objects — electricity and magnetism, energy and momentum, space and time. It was incomprehensible and fascinating.

It was from the formula for mixing energy and momentum that the relationship E = mc^{2} was derived.

The geometrical logic of this new symmetry was well understood by Einstein’s math teacher, Hermann Minkowski.

He (following A. Poincare) belongs to a mathematically transparent description of the fact that the special theory of relativity changes the theater of action of physical laws.

Three-dimensional space, plus eternal equally flowing time, is replaced by four-dimensional space-time (Minkowski space), the ingredients of which are mixed when moving from one uniformly moving frame of reference to another:

The views I want to develop before you arise on an experimental-physical basis. That is their strength. Their tendency is radical. From now on, space in itself and time in itself are reduced to the role of shadows, and only a certain kind of their connection retains a chance for reality

From a 1908 paper by H. Minkowski, from M. Born’s book «Reflections and Reminiscences of a Physicist»

For Einstein, the main lesson was this reading of that formula: M = E/C², meaning the equivalence of energy and inert mass (light gravitates!).

This thought, Minkowski’s geometric ideas, Henri Poincaré’s work, and the help of his friend Marcel Grossmann in issues of Riemannian geometry — led Einstein to a new theory of gravity 10 years later.