Mario Livio Ph.D.


Symmetry and the Laws of Nature

Symmetry is one of the most useful tools in deciphering the cosmos.

Posted Oct 13, 2020

 NASA, ESA, and S. Beckwith (STScI) and the HUDF Team/Public Domain
The Hubble Ultra Deep Field. Almost 10,000 galaxies. Looking back to when the universe was less than a billion years old.
Source: Credit: NASA, ESA, and S. Beckwith (STScI) and the HUDF Team/Public Domain

Not until the 17th century did humans seriously think of the possibility that a body of laws exists that would explain all the observed physical reality. Galileo Galilei, René Descartes, and in particular Isaac Newton demonstrated for the first time that a handful of mathematical laws can explain a wealth of phenomena, ranging from falling apples and ocean tides to the motion of the planets. In the 19th century, Michael Faraday and James Clerk Maxwell were able to do the same for electricity and magnetism.

Then the 20th century witnessed the birth of not one but two scientific revolutions. First, Einstein’s theories of Special Relativity and General Relativity inextricably linked the concepts of space and time, and suggested that gravity is not some mysterious force that acts across distance, but rather a manifestation of the warping of the fabric of space-time by masses, a bit like a trampoline sagging under the weight of a person standing on it. Second, Quantum Mechanics asserted that we can only determine the probabilities of outcomes of experiments, not the outcomes themselves. To paraphrase Einstein, God does appear to play dice with the world.

With every step along this path of a deeper understanding of the universe, the role of symmetry as the foundation for the laws of nature has become increasingly appreciated. A symmetry of the laws means that when we observe phenomena from different points of view, we discover that they are governed precisely by the same laws. For example, the law of gravity takes precisely the same form whether we are here on Earth, on the Moon, or in a galaxy ten billion light-years away. This is an example of symmetry under translation (see my previous post). If the laws of physics were not symmetric under translation (that is, they were changing from place to place), it would have been impossible to understand the cosmos.

The laws of physics are also symmetric under rotations. That is, physics has no preferred direction in space. We discover the same laws whether we determine directions with respect to the north, south, or the nearest Starbucks.

A simple example can help clarify the difference between a symmetry of shapes and a symmetry of the laws. The ancient Greeks believed that the orbits of the planets are circular, because circles were considered perfect, being symmetric under rotation by any angle about an axis passing through the circle’s center and being perpendicular to the circle’s area. When astronomer Johannes Kepler discovered that the orbits are in fact ellipses, even Galileo didn’t believe it, since circles seemed more elegant. Newton later showed that elliptical orbits were a natural product of his universal law of gravitation. The fact that his law was symmetric under rotation simply meant that the orbits could have any orientation in space, not that the shape of the orbit had to be circular.

In 1915, German mathematical physicist Emmy Noether proved a remarkable theorem that now bears her name. She showed that to every continuous symmetry of the laws of physics there is a corresponding conservation law and vice versa. For example, the symmetry of the laws under translation corresponds to conservation of linear momentum (the product of the mass and the velocity), the symmetry with respect to the passing of time (the laws do not change with time) corresponds to the conservation of energy, and so on. Noether’s theorem therefore demonstrated that two of the pillars of physics, symmetries and conservation laws, are really two manifestations of the same fundamental property.

The symmetries I have described so far had to do with things that don’t change when we change our viewpoint in space and time. Many of the symmetries underlying the subatomic world and the basic forces of nature are associated with changing our perspective on the identity of elementary particles. For example, quantum mechanics allows for electrons to be in states that mix them with another elementary particle called a neutrino. In other words, particles can carry the label “electron,” “neutrino,” or a mixture of both. It turns out that the forces of nature are symmetric (take the same form) under any interchange between electrons, neutrinos, or a mixture of the two, and many other such so-called gauge symmetries exist.

Nobody knows yet whether symmetry is truly the most fundamental concept in the workings of the cosmos, but there is no doubt that symmetry principles have been extremely fruitful in our endeavors to decipher the universe.


Livio, M. (2005). The Equation That Couldn't Be Solved. New York: Simon & Schuster.

Livio, M. (2020). Galileo and the Science Deniers. New York: Simon & Schuster.