The Biggest Misconception in Physics

At the turn of the 20th century, brilliant minds, including Albert Einstein, struggled with the problem of energy conservation. Einstein sought to prove that total energy was conserved in his new theory of gravity, later known as the general theory of relativity. However, his initial attempts and proposed solutions, which involved pseudotensors, were flawed because they violated the principle of general covariance, a core tenet of his theory stating that the laws of physics should have the same form in all frames of reference. Legendary mathematician David Hilbert also found similar difficulties, leading him to seek assistance from Emmy Noether.

Emmy Noether, a brilliant mathematician who faced academic discrimination due to her gender, became a leading expert on symmetry. She learned that symmetry in mathematics refers to any action that leaves an object or system unchanged. These can be discrete symmetries, like the rotations and reflections of an equilateral triangle, or continuous symmetries, such as shifting a mathematical function without altering its derivative. Noether used her understanding of symmetry to tackle Einstein's conservation problem.

Noether's pivotal insight came from considering a static, empty universe. She demonstrated that continuous symmetries directly lead to conservation laws. For example, translational symmetry (the universe looking the same everywhere) leads to the conservation of momentum. Rotational symmetry (the universe looking the same regardless of orientation) leads to the conservation of angular momentum. Most significantly, time translation symmetry (the laws of physics remaining constant over time) leads to the conservation of energy. This theorem revealed the fundamental origin of all conservation laws.

The universe we live in is not static; it is expanding, and its expansion is accelerating. This means that over large timescales, our universe does not possess time translation symmetry. Consequently, as described by Noether's first theorem, energy is not conserved in such a universe. This explains phenomena like a rock slowing down and stopping in deep space, or photons losing energy as they travel across vast cosmic distances. The energy doesn't go anywhere; it simply disappears due to the lack of global time symmetry. However, for everyday short timescales, time translation symmetry approximately holds, making energy appear conserved.

Noether's first theorem dealt with global symmetries in an empty universe. However, in general relativity, where spacetime is curved and properties can change from point to point, global symmetries like translations and rotations don't apply. Instead, general relativity relies on local symmetries, specifically general covariance, which means the laws of physics are the same in any small region, regardless of how one changes the frame of reference. In her second theorem, Noether proved that these local symmetries do not lead to traditional conservation laws, but rather to continuity equations. These equations demonstrate that while energy is conserved locally within small patches of spacetime, the curvature of spacetime itself allows energy to effectively 'leak out' between these patches. This explained why Hilbert's Bianchi identities, which implied energy conservation only in an empty universe, were the best possible solution in general relativity.

Emmy Noether's two theorems fundamentally reshaped physics, providing the origin of all conservation laws and resolving a major outstanding problem in general relativity. Her work encouraged physicists to think about the universe in terms of symmetries, paving the way for discoveries in quantum mechanics, the standard model of particle physics, and understanding the origins of mass. Despite facing discrimination, Noether's mathematical genius, recognized by Einstein himself, laid critical groundwork for a more complete understanding of the universe.