Lecture on Emmy Noether and Conservation Laws

Introduction to Energy Conservation Problem

• Newton's First Law suggests that an object in motion continues with constant velocity unless acted upon by an external force.
• In space, a rock thrown should continue indefinitely, but it slows down and stops.
• At the turn of the 20th century, energy conservation issues perplexed scientists like Einstein.
• Emmy Noether provided a solution, challenging Einstein's views and revolutionizing physics.

Background and Context

• 1915: Einstein delivers lectures on the General Theory of Relativity at the University of Gottingen.
• Einstein struggled with showing total energy conservation in his new gravitational theory.
• Mathematician David Hilbert and his assistant Emmy Noether investigated energy conservation.

Emmy Noether's Contributions

• Noether had a keen interest and expertise in symmetry, a concept she learned at Gottingen.
• Symmetry: Actions that leave an object unchanged (e.g., mirror reflection, rotation of a triangle).
• Noether became a leading expert on symmetry, earning a PhD in Mathematics.

Noether's Theorem and Symmetries

• Einstein faced issues with energy conservation in non-inertial frames.
• Noether identified a flaw in Einstein's conservation equation using pseudotensors, which don't remain constant across different frames.
• Noether's theorem established that every continuous symmetry leads to a conservation law:
  • Translational symmetry leads to conservation of momentum.
  • Rotational symmetry results in conservation of angular momentum.
  • Time symmetry corresponds to conservation of energy.

Implications in Physics

• In expanding universe models without time symmetry, energy isn't conserved.
• Example: A photon losing energy over time due to the universe's expansion.
• Over short timescales, energy conservation appears to hold.

General Relativity and Local Symmetries

• In General Relativity, global symmetries don't hold due to spacetime curvature.
• Noether's second theorem introduced local symmetries, resulting in continuity equations rather than traditional conservation laws.
• Energy conservation becomes local rather than global, with energy 'leaking' through spacetime curvature.

Legacy of Emmy Noether

• Proved the source of all conservation laws and solved energy conservation issues in General Relativity.
• Despite facing gender and political challenges, she continued her work and teaching.
• Influenced modern physics, including quantum mechanics and particle physics.
• Noether's theorems remain pivotal in understanding fundamental physics.

Conclusion

• Emmy Noether's work helped reshape physics through her exploration of symmetry and conservation laws.
• Her discoveries continue to be foundational in both classical and modern physics.

Additional Information

• Inspired future explorations and breakthroughs in physics and other sciences.
• The lecture emphasized the importance of continuous curiosity and experimentation in scientific discovery.