The Biggest Misconception in Physics

Key Concepts:

• Noether's Theorem (First and Second)
• Conservation Laws (Energy, Momentum, Angular Momentum, Electric Charge)
• Symmetry (Translational, Rotational, Time Translation, Gauge/Local)
• General Relativity
• General Covariance
• Tensors and Pseudotensors
• Equivalence Principle
• Lagrangian Mechanics
• Principle of Least Action
• Continuity Equation
• Expanding Universe

1. The Problem of Energy Conservation in General Relativity:

• At the beginning of the 20th century, physicists, including Einstein, struggled with energy conservation within the framework of general relativity.
• Einstein was giving six lectures on his new theory of gravity.
• Einstein proposed an equation suggesting that the total energy of matter and the gravitational field remains constant.
• However, Emmy Noether proved Einstein wrong, establishing a new paradigm for physics.
• Einstein's initial attempts to formulate field equations for general relativity faced challenges in demonstrating the conservation of total energy.
• David Hilbert, intrigued by Einstein's lectures, sought energy conservation equations but only found the Bianchi identities, which were only applicable to an empty universe.

2. Emmy Noether's Background and Expertise in Symmetry:

• Emmy Noether, initially denied formal admission to the University of Erlangen due to her gender, became a leading expert on symmetry.
• She earned a PhD in mathematics and used her expertise to assist Hilbert and Einstein with the energy conservation problem.
• Symmetry is defined as any action that leaves an object unchanged. Examples include reflections of an equilateral triangle and shifting a mathematical function.
• Noether's expertise in symmetry was crucial to her groundbreaking work.

3. Noether's First Theorem: Symmetry and Conservation Laws:

• Noether's first theorem establishes a direct relationship between continuous symmetries and conservation laws.
• She demonstrated that translational symmetry leads to the conservation of momentum, rotational symmetry to the conservation of angular momentum, and time translation symmetry to the conservation of energy.
• Example: In an empty, static universe, throwing a ball results in constant velocity due to translational symmetry. Rotating a metal rod results in constant rotation due to rotational symmetry.
• Mathematical Derivation: Noether used the principle of least action and Lagrangian mechanics to show that time translation symmetry implies energy conservation.
  • The action is the integral of the Lagrangian (kinetic minus potential energy) over time.
  • Using the Euler-Lagrange equation and the chain rule, she derived that the time derivative of a specific quantity is zero, implying a constant, which corresponds to total energy.
• Expanding Universe: The expansion of the universe breaks time symmetry, leading to the non-conservation of energy on large timescales.
  • Photons lose energy as the universe expands, and a rock thrown in deep space eventually slows down and stops.
  • Energy and momentum are not conserved if there is no time or spatial symmetry.

4. Noether's Second Theorem: Local Symmetries and Continuity Equations:

• Noether's second theorem addresses local symmetries, which are transformations that hold in small regions of space-time.
• Local symmetries lead to continuity equations rather than proper conservation laws.
• Example: The flow of water through a pipe illustrates a continuity equation, where the change in water amount in a section is balanced by the difference between inflow and outflow.
• In general relativity, the continuity equation describes the flow of energy between patches of space-time, accounting for the curvature of space-time.
• The continuity equation is equivalent to the Bianchi identities, providing the best possible description of energy conservation in general relativity.

5. General Covariance and Tensors:

• General covariance, a core tenet of general relativity, states that the laws of gravity must have the same form in every frame of reference.
• Einstein used tensors to satisfy general covariance. Tensors remain the same even when transformed from one coordinate system to another.
• Noether found that Einstein's proposed energy conservation equation contained a pseudotensor, which does not remain the same in different frames of reference.

6. Einstein's Equivalence Principle:

• Einstein's equivalence principle states that being in a rocket accelerating at 9.8 meters per second squared is equivalent to standing on the surface of the Earth.
• This principle allowed Einstein to develop a new theory of gravity by understanding accelerating frames.

7. Impact and Legacy of Noether's Theorems:

• Noether's theorems revolutionized physics by linking symmetries to conservation laws and providing a framework for understanding energy conservation in general relativity.
• Her work influenced the discovery of fundamental particles like quarks and the Higgs boson and helped explain the origin of mass in the universe.
• Einstein recognized Noether as a mathematical genius, and her theorems are considered among the most important in 20th-century physics.

8. Noether's Personal Life and Career:

• Noether faced discrimination as a woman in academia but persevered and made significant contributions to mathematics and physics.
• She was banned from working at universities by the Nazis due to her Jewish heritage but continued teaching in her home.
• She eventually obtained a teaching position at Bryn Mawr College in America, where she taught until her death.

9. Conclusion:

• Emmy Noether's theorems provide a fundamental understanding of conservation laws and their relationship to symmetries in the universe. Her work has had a profound impact on physics, shaping our understanding of fundamental particles, forces, and the nature of energy conservation in general relativity. While energy is not strictly conserved on cosmological scales due to the expansion of the universe, Noether's theorems provide the framework for understanding why it appears to be conserved on smaller scales and how to account for its behavior in the context of general relativity.