Emmy Noether: The Mathematician Who Revolutionized Physics

The Mystery of the Drifting Rock in Space

Imagine you're an astronaut floating in the vast expanse of deep space. You decide to throw a rock as hard as you can. What happens next? According to Newton's first law, you might expect the rock to continue moving in a straight line at a constant velocity. But in reality, something unexpected occurs - the rock gradually slows down and eventually comes to a stop. This puzzling scenario raises an important question: Where did all the rock's energy go?

At the dawn of the 20th century, this problem of energy conservation perplexed some of the greatest scientific minds, including Albert Einstein himself. Einstein proposed a potential solution, but it was a lesser-known, unpaid mathematician named Emmy Noether who proved him wrong. In doing so, she established an entirely new paradigm for physics - one that forms the foundation of particle physics and explains why anything is conserved at all.

The Birth of a Revolutionary Idea

The story begins in 1915 at the University of Göttingen, where Einstein was delivering a series of six lectures on his new theory of gravity, which would later become the general theory of relativity. While the lectures were well-received, Einstein had not yet finalized the form of his field equations. One of the key challenges he faced was demonstrating that total energy was conserved within his new theory.

Einstein suggested that the long-established principle of conservation of energy might hold the key to deriving the correct field equations. This idea caught the attention of the legendary mathematician David Hilbert, who was in the audience. Intrigued, Hilbert began searching for energy conservation equations within Einstein's new theory.

The best Hilbert could find was a set of equations known as the Bianchi identities. These equations showed that energy was conserved, but only in a completely empty universe. For a universe like ours, filled with matter and energy, they seemed useless. Hilbert was stumped.

Fortunately, Hilbert knew just the person for the job - his new assistant, Emmy Noether.

Emmy Noether: A Mathematical Prodigy

From an early age, Emmy Noether dreamed of following in her father's footsteps as a mathematics professor at the University of Erlangen. Despite facing significant obstacles due to her gender, Noether persevered. She obtained special permission to attend lectures at the university, even though they refused to admit her as an official student. The Erlangen Academic Senate held that the admission of women "would overthrow all academic order."

Undeterred, Noether spent a semester at Göttingen in 1903. There, she encountered a novel approach to geometry using symmetry - a concept that would become central to her groundbreaking work.

The Power of Symmetry

Symmetry is a concept that's easy to recognize but challenging to describe precisely. In mathematics, symmetry is generalized to mean any action that leaves an object unchanged. For example, an equilateral triangle has six symmetries - three reflections and three rotations.

But symmetries can also be more abstract. Consider a mathematical function. If you shift this function up or down by a constant amount, its derivative (slope) remains unchanged. This translation symmetry is continuous, meaning you can shift it by any amount.

Over the next 12 years, Noether became a leading expert on symmetry. She earned her PhD in mathematics, becoming only the second woman in Germany to do so. It was this expertise that she brought to bear on Hilbert and Einstein's problem of energy conservation.

Einstein's Mistake and Noether's Insight

Einstein had proposed a new conservation equation stating that the sum of the energy of matter and the energy of the gravitational field remains constant. However, when Noether examined this equation, she realized Einstein had made a fundamental error. His equation disregarded the core principle upon which general relativity was built - general covariance.

General covariance, a cornerstone of general relativity, states that the laws of physics must remain the same in all frames of reference, including those that are accelerating or rotating. This principle is itself a kind of symmetry, precisely the type of problem Noether had spent her career studying.

Noether's Breakthrough: Symmetry and Conservation Laws

Noether began by considering the symmetries of the simplest possible case - an empty, static universe. In such a universe, there's nothing special about any particular point in space. This spatial symmetry leads directly to the conservation of momentum. Similarly, the universe's rotational symmetry results in the conservation of angular momentum.

Perhaps most importantly, Noether considered time symmetry. In a static universe, the laws of physics don't change over time. An experiment performed today would yield the same results as the same experiment performed tomorrow. Noether proved that this time translation symmetry leads directly to the conservation of energy.

In a groundbreaking theorem, Noether demonstrated that these examples were not coincidental. She proved that whenever there's a continuous symmetry in nature, there's a corresponding conservation law. This insight provided, for the first time, an explanation for where conservation laws come from.

The Real Universe: Symmetry Breaking and Energy Loss

However, the universe we inhabit is far from the static, empty one Noether initially considered. In the 1920s, astronomers discovered that the universe is expanding, with distant galaxies moving away from us at speeds proportional to their distance. This expansion implies that the universe was much more compact in the distant past.

Further observations in the 1990s revealed that this expansion is actually accelerating. This means that over large timescales, our universe is not symmetric in time. It was very different billions of years ago and will be different billions of years from now.

The lack of time symmetry has profound implications for energy conservation. In our expanding universe, energy as we typically conceive it is not conserved. A photon of visible light emitted shortly after the Big Bang arrives at our telescopes today as a microwave, having lost 99.9% of its energy. This energy doesn't go anywhere - it simply disappears.

This phenomenon explains what happens to the rock thrown by our hypothetical astronaut. As it travels through the expanding universe, it gradually slows down and stops. Its energy doesn't transfer elsewhere; it vanishes.

Importantly, this doesn't violate any laws of physics. Noether's theorem shows us that without time or spatial symmetry, we shouldn't expect energy or momentum to be conserved.

Reconciling Theory with Observation

If energy isn't conserved in our universe, why does it usually seem to be? The answer lies in the timescales we typically observe. Over short periods, time translation symmetry approximately holds. An experiment performed today will yield essentially the same results tomorrow. Thus, for all practical purposes, energy is conserved in our day-to-day experiences.

It's only over vast timescales - millions of years or more - that the universe's expansion becomes significant enough to break this symmetry noticeably. Only then do we observe energy failing to be conserved.

Noether's Second Theorem: Local Symmetries

While Noether's first theorem explained the origin of conservation laws in classical physics, it didn't fully resolve the problem of energy conservation in general relativity. In a universe with varying curvature, you can't simply shift everything and expect the laws of physics to remain unchanged.

However, Noether realized that other symmetries still exist. General covariance ensures that the laws of physics look the same regardless of how you're moving. This principle allows us to change our frame of reference in any small region of space-time. These transformations, which apply locally rather than globally, are called local symmetries.

In a second theorem, Noether proved that these local symmetries don't yield the familiar conservation laws of classical physics. Instead, they produce something that only works locally: a continuity equation.

The Continuity Equation: A New Understanding of Energy Flow

To understand the continuity equation, consider the flow of water through a pipe. The equation describes how the amount of water changes in a section of the pipe and the difference between the water flowing in and out. Together, these terms ensure that no water is created or destroyed.

In general relativity, Noether found a similar continuity equation, but with a crucial difference. Imagine space-time as a collection of small patches, with energy flowing between them. In any individual patch, energy is conserved, just like in the water pipe analogy.

However, when we connect these patches, we must account for the curvature of space-time. This curvature introduces additional terms to the equation, as if there were small cracks between the patches through which energy can leak.

This new equation allows us to track energy as it moves through space-time. As energy decreases in one term, it increases in others that describe the curvature of space-time. This accounts for the energy we attribute to the changing gravitational field as the universe expands.

Remarkably, Noether's continuity equation turned out to be equivalent to the Bianchi identities that Hilbert had initially dismissed. Noether had proven that this was the best possible solution for energy conservation in general relativity.

Noether's Legacy: Reshaping Modern Physics

Noether's work fundamentally changed how physicists think about the universe. Instead of focusing solely on particles and forces, they began to consider the underlying symmetries of nature.

These ideas were soon applied to the quantum world, revealing that symmetries in the behavior of charged particles like electrons lead to the conservation of electric charge. In the 1960s and 1970s, Noether's insights directly contributed to the discovery of new fundamental particles like quarks and the Higgs boson.

Noether's theorems helped explain the origin of the forces of nature and even shed light on the source of all mass in the universe. Today, her work remains central to our efforts to develop a unified theory of everything.

Emmy Noether: A Life of Perseverance and Discovery

Despite her groundbreaking contributions, Noether faced significant challenges throughout her career. As a woman in early 20th century Germany, she was initially barred from official university positions. Even after gaining recognition for her work, she faced discrimination and was forced to flee Nazi Germany in 1933.

Noether found refuge at Bryn Mawr College in the United States, where she continued to teach and conduct research until her death. In an obituary, Einstein praised her as "the most significant creative mathematical genius thus far produced since the higher education of women began."

The Ongoing Impact of Noether's Work

Noether's theorems continue to shape our understanding of the universe. They've been applied to fields as diverse as particle physics, cosmology, and even string theory. Her work demonstrates the power of abstract mathematical thinking to reveal profound truths about the nature of reality.

As we continue to explore the frontiers of physics, from the smallest subatomic particles to the largest structures in the cosmos, Noether's insights remain as relevant as ever. Her legacy serves as an inspiration to scientists and mathematicians worldwide, reminding us of the transformative power of curiosity, perseverance, and innovative thinking.

In conclusion, Emmy Noether's work on symmetry and conservation laws represents one of the most significant contributions to 20th-century physics. By uncovering the deep connection between symmetries in nature and the laws that govern the universe, she provided a framework that continues to guide our understanding of the cosmos. As we face new challenges in physics and cosmology, Noether's theorems remain a cornerstone of modern scientific thought, a testament to the enduring power of mathematical insight to illuminate the mysteries of the universe.

Article created from: https://www.youtube.com/watch?v=lcjdwSY2AzM