Emmy Noether: The Mother of Modern Physics
Imagine floating in deep space and throwing a rock. Newton's first law suggests it should travel at a constant velocity in a straight line. However, it eventually slows down and stops, raising a fundamental question: where did the rock's energy go? This question, which baffled even Albert Einstein, was answered by Emmy Noether, a brilliant mathematician who revolutionized physics.
Einstein and the Problem of Energy Conservation
In 1915, Einstein presented six lectures at the University of Gottingen on his developing theory of general relativity. He grappled with demonstrating that total energy was conserved within this new framework. David Hilbert, a renowned mathematician, also sought energy conservation equations in Einstein's theory but found only the Bianchi identities, which were useful only in an empty universe.
Hilbert Turns to Noether
Faced with this challenge, Hilbert enlisted the help of his assistant, Emmy Noether. Noether, despite facing significant gender discrimination in academia, was a rising star in mathematics with expertise in symmetry.
Emmy Noether's Early Life and Education
Despite her passion for mathematics, Noether faced considerable obstacles due to her gender. She was initially denied formal admission to the University of Erlangen and had to seek special permission to attend lectures. She eventually earned a PhD and became a leading expert on symmetry.
Symmetry: A Fundamental Concept
Symmetry, broadly defined, is any action that leaves an object or system unchanged.
Types of Symmetry
- Geometric Symmetry: As exemplified by the symmetries of an equilateral triangle.
- Abstract Symmetry: Such as the translation symmetry of a mathematical function's derivative. If you shift a function up or down by a constant amount, its derivative remains unchanged.
- Continuous Symmetry: A symmetry that can be shifted by any amount.
Noether's Insight: Symmetries and Conservation Laws
Noether recognized that Einstein's proposed energy conservation equation contradicted the principle of general covariance, a cornerstone of general relativity. General covariance states that the laws of physics are independent of the frame of reference.
General Relativity and the Equivalence Principle
Einstein developed general relativity from his idea that the principle of relativity could also be applied to accelerating and rotating frames. This lead to his "happiest thought," when he imagined a window cleaner falling off a building. He realized that the window cleaner, while falling, would feel weightless. This thought experiment led Einstein to the equivalence principle. * The Equivalence Principle: This is the observation that gravitational mass and inertial mass are equal.
To ensure the laws of gravity held the same form in every frame of reference, Einstein employed tensors. Tensors are mathematical objects that remain invariant regardless of the coordinate system used.
The Problem with Einstein's Pseudotensor
Noether realized that Einstein's proposed equation contained a pseudotensor, which is not a true tensor and thus changes with different frames of reference. This led her to question the compatibility of general covariance and energy conservation.
Noether's Theorem: A Groundbreaking Discovery
Noether formulated a theorem linking symmetries to conservation laws. She began by considering an empty static universe.
Symmetries in an Empty Static Universe
- Translation Symmetry: The laws of physics are the same regardless of location, leading to the conservation of momentum.
- Rotational Symmetry: The laws of physics are the same regardless of orientation, leading to the conservation of angular momentum.
- Time Symmetry: The laws of physics are the same at all times, leading to the conservation of energy.
Deriving Conservation Laws Mathematically
Noether proved mathematically that time-translation symmetry is equivalent to energy conservation using the principle of least action and the Euler-Lagrange equations.
The Broader Implications of Noether's First Theorem
Noether proved that all conservation laws are connected to underlying symmetries. Translational symmetry leads to momentum conservation, rotational symmetry leads to angular momentum conservation, and time translation symmetry leads to energy conservation.
The Expanding Universe and the Breakdown of Time Symmetry
However, our universe is expanding. This expansion means that time translation symmetry doesn't hold true on large timescales. The universe was very different 13 billion years ago and will be different in the future.
Energy is Not Always Conserved
Because time symmetry is broken, energy, in the traditional sense, is not conserved on cosmological scales. For example, a photon emitted shortly after the Big Bang has lost a significant portion of its energy by the time it reaches us. This explains why the rock in space eventually stops – it loses energy as the universe expands.
Local vs. Global Symmetries
Noether's analysis extended beyond static universes. She recognized that while global symmetries (shifting the entire universe) may not hold in general relativity due to spacetime curvature, local symmetries (general covariance) do.
Noether's Second Theorem and Continuity Equations
Noether's second theorem deals with local symmetries and results in continuity equations rather than strict conservation laws. A continuity equation describes the flow of a conserved quantity, like water through a pipe. In the context of general relativity, it describes the flow of energy through patches of spacetime.
Energy Leaks and the Bianchi Identities
The curvature of spacetime introduces "cracks" between these patches, allowing energy to leak out. Noether demonstrated that this continuity equation was equivalent to the Bianchi identities, previously dismissed by Hilbert.
Noether's Legacy
Emmy Noether's work revolutionized physics by revealing the deep connection between symmetry and conservation laws. Despite facing discrimination and persecution, she persevered and made profound contributions to our understanding of the universe.
Noether's Influence on Particle Physics and Beyond
Noether's ideas extended to the quantum world, influencing the discovery of fundamental particles and forces. Her theorems are foundational to modern physics and offer the closest we've come to a theory of everything.
Overcoming Adversity
Despite facing discrimination as a Jewish woman in Germany during the rise of Nazism, Noether continued to teach and inspire. She eventually secured a position at Bryn Mawr College in the United States.
Einstein's Tribute
Einstein recognized Noether's genius, calling her the most significant creative mathematical genius since the beginning of higher education for women.