Albert Einstein is the undisputed face of scientific genius. In his magnum opus, the General Theory of Relativity (1916), Einstein stunned the world by proving mathematically that gravity isn’t a force as Newton imagined but instead is the curvature of space-time caused by the uneven motion of mass along the fabric (geodesics) of the universe. It was a magnificent piece of work but one that needed advanced mathematical insights that even Einstein did not possess at the time he was writing the first drafts of his paper. Without those mathematical insights, Einstein’s work was not only incomplete, it was inconsistent with existing physical laws of conservation. Luckily for him and for science, a woman mathematician named Emmy Noether appeared on the scene and “saved” Einstein. Here is their story.

In the summer of 1915, Einstein was preparing the final touches to his theory of general relativity which he was to present at the Prussian Academy of Sciences in Berlin in November. Excitement was in the air as he was about to unveil a theory that presents a radically new vision of gravity, one in which gravity is not a force but a phenomenon arising from interactions between mass, energy and the curvature of space-time (the so-called fabric of the universe).

Around that time, two great mathematicians at Gottingen: Felix Klein and David Hilbert expressed concern that Einstein’s theory has a “hole” because it violated energy conservation, a sacred principle in physics. One of the many paradoxical consequences of this failure of energy conservation was that an object could gain speed as it lost energy (by emitting gravity waves) whereas clearly it should slow down. In an attempt to fix this hole, Klein and Hilbert sought help from Emily Noether, perhaps the most brilliant woman mathematician ever.

Noether was a mathematician who achieved high distinction despite the tremendous social barriers she faced as a woman academic. Her masterpiece was a theory named after her that establishes a connection between conserved quantities (such as energy or angular momentum) and symmetry.

In theoretical physics, symmetry is tied to the Lagrangian, a mathematical concept used to describe the dynamics of a system. If a change of a basic variable such as the position of the system or a shift of the time of origin leaves the Lagrangian unchanged, the system is symmetric. For example, a cylinder has symmetry, though not as much as a sphere: it doesn’t change if you rotate it around its axis of symmetry, though if you rotate it around any other axis, it looks different. These are examples of continuous symmetries, because the angle can be as large or small as you like. Noether’s theorem is about continuous symmetries [1]. It can be stated a bit more precisely as follows: If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

A more sophisticated statement of the theorem is:

Every differentiable symmetry of the action of a physical system has a corresponding conservation law.

Noether’s theorem implied that energy wasn’t conserved in Einstein’s theory because symmetry was broken. This is because the concept of space and time is dynamic and hence changeable. Since the very nature of space can change over time, continuous time symmetry is broken. That’s the case with the expanding universe where energy can be lost in the event of cosmological redshifts, or it can be created from nowhere through dark energy. The bottomline is that Einstein’s theory is valid only under the special case where space is unchanging over time, which goes against reality. This was the “hole” in Einstein’s theory that needed fixing. Fortunately, Noether’s theorem doesn’t just tell us why symmetry is broken in Einstein’s theory, it also gives an explicit formula of a quantity that is conserved. That quantity goes by an esoteric-sounding name: Landau-Lifshitz pseudotensor [2]. This quantity saves energy conservation by incorporating the entire universe’s gravitational potential energy to offset the gains or losses due to red shifts and dark energy. By a stroke of luck, this insight was exactly what Einstein needed to patch the hole in his theory. The rest, as they say, is history.

Notes:

[1] The theorem was proved in 1915, and published three years later as Noether, E.. “Invariante Variationsprobleme.” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1918 (1918): 235-257.

[2] For a more precise discussion of the Landu-Lifshitz pseudotensor in the context of the General Theory of Relativity, see https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Landau%E2%80%93Lifshitz_pseudotensor.html