The Theorem That Preserves the Universe’s Laws
Noether’s Symmetries: The Mathematician who Changed Our Perspective on Conservation Laws
Speckled across the history of science, there are countless hidden heroes; a hidden hero of general relativity was Emmy Noether. This is the story of her theorem — the theorem that changed our perspective on conservation in the universe.
Einstein himself said in the New York Times, “In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.” To understand why he said this, we need to build up our understanding of mechanics and relativity through the centuries.
Einstein is understandably the face of relativity. In 1905, he published “On the Electrodynamics of Moving Bodies” — the seminal work of special relativity. Einstein demonstrated two things in this paper: 1) that the laws of physics are invariant in all reference frames — which is to say, they don’t change depending on your perspective — and 2) that the speed of light is similarly invariant across reference frames.
This is crucially different than normal objects going normal speeds, which can appear faster or slower if you are moving in the same or opposite direction of the object’s motion. Light always travels at 300,000,000 meters per second: c is c regardless of your perspective.
Special relativity was a continuation on the work done by Galileo in the 17th century; Einstein took Galilean relativity to its extreme — beyond what people thought was reasonable or logical — and it worked.
But as much as special relativity worked, it was also strange. The propositions might seem simple enough, but they lead to a series of peculiar and counterintuitive consequences: simultaneity can be relative, length and mass can change when moving close to the speed of light, and time itself can be dilated at high speeds.
These conclusions were strange, but they were not unsound. More pressing was the fact that special relativity was incomplete: it failed to consider acceleration. Special relativity only concerned flat space and constant velocity. But Einstein’s happiest thought, as it is called, was when it occurred to him that a person in free fall — constant acceleration — would not feel their own weight. Acceleration is the same as gravity, always. This would develop into what is known as the Equivalence Principle.
In the coming years, Einstein would develop his theory of gravity around the Equivalence Principle: general relativity. This theory would come to replace the Newtonian gravity that had dominated our understanding of the weakest force for centuries. The universe is no longer absolutely static and flat; it can be warped and curved by mass.
Under Einstein’s Equivalence Principle, light should be curved (accelerated, in physics lingo) by gravity because it has inertia, just like everything else. Einstein modeled this by considering space and time to be one four-dimensional object — spacetime — which can itself be curved by gravity. We can think about this like a topographic map: light, just like a hiker, doesn’t want to go into a valley that it will have to climb back out of again, so it might curve to stay on flat space.
This is due to the Principle of Least Action: things in the universe want to do as little effort as possible. If you were to walk from one part of town to another, you would take the most direct path between those two locations; everything in the universe wants to do the same. Light is not going to do any more work than it needs to.
This concept of “action” has a specific definition in physics, and it can be understood through something called the Lagrangian:
L ≡ T — V
T is the kinetic energy of the system and V is the potential energy of the system. So, the Lagrangian is the difference between how much energy something is using versus how much it could be using, and as such, it tells us the dynamics inherent to a system. Integrating this value over time tells us the action of a system. The principle of least action states that the change in action will always be as little as possible.
This is where Emmy Noether enters the picture.
Noether had studied mathematics at the University of Erlangen in a time when women were only allowed to audit courses with permission of the instructor. She was fascinated by algebraic invariance and would ultimately apply this understanding to the physics of action.
Beginning in 1903, she attended the University of Göttingen where she audited classes by the mathematicians David Hilbert and Hermann Minkowski — both incredibly influential to Einstein’s relativity — as well as the astronomer Karl Schwarzschild, who would go on to be the first to give an exact solution to general relativity.
Seeds were planted that would take over a decade to sprout.
In the meantime, Noether returned to Erlangen where she was now granted full student status, and she completed her PhD on invariants in 1907. Invariants were her specialty, and that specialty was needed back in Göttingen. So, in 1915, and a 33-year-old Emmy Noether was invited back to the University of Göttingen by the best mathematicians in the world at the time. They needed help solving general relativity’s greatest challenge: the conservation of energy.
Energy — like momentum, angular momentum, charge, etc. — is supposed to be universally conserved in our cosmos; they aren’t created or destroyed — just moved around. Relativity warps our understanding of this.
In general relativity, spacetime is curved, and as a result, we need to develop two different coordinate systems: local and global. Local is like what we experience here on Earth, standing on the ground with our own two feet, whereas global is like what an astronaut on the ISS would look down on Earth to see.
In general relativity, gravity itself can contribute to the energy of a system, so there is no way to universally state the total energy of a system; the local and global perspectives might not agree. Thus, gravitational energy did not appear to be conserved in the traditional sense.
This is what Emmy Noether set out to understand, but her invitation to Göttingen was not without controversy. Faculty at Göttingen, disproportionately loud within the philosophy department, objected to Noether’s invitation on the basis that she was a woman. David Hilbert — the mathematician working on general relativity — backed Noether by yelling across the quad to the philosophy department: this is a university, not a bathhouse.
Despite this treatment, Noether got to work. During her first few years at Göttingen, she was not offered a teaching position, and she was not paid. But individuals like Hilbert appreciated her brilliance and supported her through the prejudice of the time. For example, because Noether was not allowed to teach, Hilbert would advertise classes as being taught by himself with Noether as an assistant, but then he would leave it up to her to lecture.
Noether’s primary focus was on algebraic invariance and relativity. To preserve the conservation of energy in general relativity, Noether turned her attention to symmetry and its relationship to action.
This is not the same type of symmetry that we are used to when we see a reflection in a lake or a butterfly’s wings; this is physical symmetry. Symmetry in physics means that we can transform a system without the laws changing: a feature of the system is preserved throughout the transformation.
We’ve actually already discussed an example of symmetry: the speed of light being the same in all reference frames. No matter how you look at it — no matter how you transform the system — c is always c. The speed of light is invariant because of this symmetry.
Action is similarly invariant in these symmetrical transformations, and Noether turned her attention to symmetrical time translations to better understand energy conservation. But what would it mean for a time translation to be symmetrical?
Take chess for example: The rules of chess are the same today as they were 100 years ago or as they will be 100 years from now. That is to say, the rules of chess are time independent. The same idea is true of the laws of physics: regardless of when we perform an experiment, the result of the experiment is the same.
This breaks down in the extremes — like the beginning of the universe — but it is generally true. If you perform an experiment today, the result would be the same if you had performed it in 1924 or if you perform it again in 2124. So, if you were to throw a frisbee into empty space, the properties of the frisbee in motion will remain the same; nothing about the frisbee changes over time. Specifically, the energy of the frisbee in motion is time-invariant. Thus, as a direct result of time translation symmetry, energy is conserved.
So, Noether solved energy conservation within general relativity, right? Not so fast. Noether showed that local time translation symmetries lead to the conservation of energy, but that conservation only applies locally.
Global conservation of energy would require a global time translation symmetry, which does not exist in curved spacetime. Global time symmetry cannot exist within general relativity because curved spacetime is dynamic — it can change — so energy conservation does not apply in the same way as it would in flat, static spacetime.
Noether’s work on time symmetry set the limitations of our classical understanding of energy conservation within general relativity: energy isn’t conserved as we once thought. Unsurprisingly, as with everything related to Einstein’s relativity, the conclusion is bizarre and counter-intuitive. But that does not mean that Noether’s work was without its merit.
On the contrary, Noether’s study on invariants and symmetry went far deeper than just the conservation of energy within time translation symmetries; her theorem is universal.
I don’t use that word lightly.
Noether found that for every symmetry there is a corresponding conservation law. Just as time translation symmetry leads to energy conservation, spatial and rotational translation symmetries lead to momentum and angular momentum conservation respectively.
Let’s consider the example of a frisbee being thrown in empty, flat space. In this empty universe, regardless of where we throw the frisbee, it will maintain its motion in the same way. This is a spatial translation symmetry, meaning nothing about the frisbee changes as its position changes; the frisbee moves with the same momentum no matter where it is. The system translates but momentum is invariant; it is conserved.
The same idea applies to rotational transformations. If you hold the frisbee in your hand, you can rotate it in any way you like without the frisbee changing in any way. The properties of the system and laws of physics are invariant under this symmetrical transformation. So, as the frisbee rotates in space, its angular momentum is similarly conserved.
These are just three symmetries and their corresponding conservation laws; however, Noether’s theorem says that for every symmetry, there is a conservation law — and vice versa. That is to say, Noether’s Theorem applies everywhere and everywhen, on the smallest scales of the universe as well as the largest.
Gauge symmetry of electromagnetism, for example, involves a change in the phase of the wave function without changing charge. From this, charge is conserved. These symmetries and conservations act as the scaffolding of quantum field theory — crucial to our understanding of physics on the smallest scales. Beyond, there may even be ways to use Noether’s Theorem to connect supersymmetry and string theory.
This universal theorem — for every symmetry there is a conservation law, and vice versa — is what she presented in her 1918 paper “Invariante Variationsprobleme” — except she didn’t present it; a mathematician named Felix Klein did. Despite her impact on the field of mathematics in the early 20th century, she still faced immense misogyny within the field. Klein presented her paper to the Royal Society of Sciences on her behalf because her membership to the society was not permitted.
This is the great irony of Noether’s story: she studied mathematical symmetry, but her academic life was profoundly asymmetrical. Due to the profound institutional asymmetries of the time, Noether faced intense misogyny and antisemitism throughout her career — constantly needing to readjust her pursuit because of changes in the rules regarding women in academia.
It was only on the individual level that Noether found symmetry and respect. David Hilbert, Felix Klein, Albert Einstein, and many others supported and were eager to work with Noether throughout her career. And such collaborations resulted in the most beautiful theorems and theories of physics. Noether set out to preserve energy conservation within general relativity, but she ended up discovering one of the most important and universal theorems in the history of physics.
When Einstein read Noether’s work, he told Hilbert: “Yesterday I received from Miss Noether a very interesting paper on invariants. I’m impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.”
As Lederman and Hill described it in their book Symmetry and the Beautiful Universe, Noether’s Theorem is “certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem.”
But unlike Pythagoras, Newton, and Einstein, Emmy Noether does not have the clout she deserves within the history of science. Noether’s Theorem is the sole common denominator linking general relativity and quantum mechanics; it applies across the whole universe — from quarks and electrons to black holes and the cosmic web.
If the universe demonstrates a symmetry, then there is always a corresponding conservation law: the balanced antidote to a universe dominated by Einstein’s wonderful and strange theories of relativity.
Written by Curran Collier
2024.11.08
