Veritasium - The Problem In Relativity Einstein Couldn't Fix
Emmy Noether's work in the early 20th century addressed a fundamental problem in physics: the conservation of energy in Einstein's general relativity. Noether's theorem established that every continuous symmetry in the laws of physics corresponds to a conservation law. For example, translational symmetry leads to the conservation of momentum, rotational symmetry to angular momentum, and time symmetry to energy conservation. This insight was groundbreaking because it provided a mathematical foundation for understanding why certain quantities are conserved in physics. Noether's theorem also explained why energy is not conserved in an expanding universe, as the lack of time symmetry in such a universe means energy conservation does not hold globally. Her work laid the groundwork for modern physics, influencing the development of quantum mechanics and the discovery of fundamental particles like quarks and the Higgs boson. Despite facing significant gender-based obstacles, Noether's contributions have been recognized as some of the most important in 20th-century physics.
Key Points:
- Noether's theorem links symmetries to conservation laws, providing a foundation for understanding energy conservation.
- In an expanding universe, energy is not conserved due to the lack of time symmetry.
- Noether's insights influenced the development of quantum mechanics and the discovery of fundamental particles.
- Her work solved a major problem in Einstein's general relativity regarding energy conservation.
- Despite gender-based challenges, Noether's contributions are pivotal in modern physics.
Details:
1. 🌌 The Curious Case of the Space Rock
- Throwing a rock in space illustrates Newton's first law, but contrary to the expectation of constant velocity, it eventually slows and stops, raising questions about energy conservation.
- In the early 20th century, Einstein and other physicists grappled with the problem of energy conservation, leading to Einstein proposing a solution that was ultimately incorrect.
- Emmy Noether developed a revolutionary framework for understanding conservation laws, which are foundational in particle physics today.
- In 1915, Einstein's lectures in Gottingen on gravity were groundbreaking, but he faced challenges in proving energy conservation within his theory of relativity.
- David Hilbert, captivated by Einstein's theory, sought to establish energy conservation but encountered limitations with existing mathematical identities, which only applied to an empty universe.
- Hilbert recruited Emmy Noether, who despite facing gender-based educational barriers, had mastered the concept of symmetry at Gottingen.
- Noether's theorem demonstrated that symmetries in physical systems correspond to conservation laws, a principle that is a cornerstone of modern physics.
- Mathematical symmetries, evident in geometric shapes or functions, show how certain properties remain unchanged under transformations, which is central to Noether's theorem.
2. 🔍 Noether's Breakthrough in Physics
2.1. Noether's Contribution to Physics
2.2. Einstein's Theory of Relativity
3. 🌠 Einstein and the Quest for Relativity
3.1. Einstein's Use of Tensors in General Relativity
3.2. Noether's Problem with Energy Conservation
4. ⏳ Time Symmetry and Energy Conservation
- The principle of least action determines that systems follow paths minimizing a quantity called action, derived from the integral of the Lagrangian L over time.
- The Lagrangian, typically defined as kinetic energy minus potential energy, plays a crucial role in formulating mechanics.
- Noether's theorem highlights that symmetries in action, particularly time translation symmetry, lead to conservation laws, including energy conservation.
- A shift in time by a small amount, resulting in a change in the Lagrangian, does not affect motion equations due to symmetry, illustrating the conservation principle.
- The derivative of the Lagrangian concerning velocity, multiplied by velocity and evaluated as zero, signifies a conserved quantity, often energy.
- In the simplest form, this conserved quantity is identified as total energy, reaffirming that time symmetry ensures energy conservation.
- For example, in a closed system where time symmetry holds, the total energy remains constant, illustrating practical implications in physics.
5. 🔄 Noether's Theorem: The Symmetry Connection
- Noether's Theorem establishes that every continuous symmetry in physics corresponds to a conservation law, such as time translation symmetry leading to energy conservation and translational symmetry leading to momentum conservation.
- On cosmic scales, the universe's expansion, highlighted by the observation that distant galaxies move away from us, challenges traditional time symmetry, implying energy conservation does not hold over vast distances.
- The discovery of the universe's accelerating expansion in the 1990s further indicates a lack of time symmetry, suggesting energy is not conserved on cosmic timescales.
- A practical example is a photon emitted after the Big Bang arriving as a microwave, losing 99.9% of its energy, illustrating the non-conservation of energy over cosmic distances.
- These insights show that in an expanding universe, traditional conservation laws are not applicable, underscoring that energy loss does not violate physical laws but reflects the universe's dynamic nature.
- This understanding allows physicists to align theories with observable phenomena, moving beyond forcing fundamental concepts when they do not fit observed cosmic behavior.
6. 🌍 Cosmic Scale Symmetries and Energy Dynamics
- Time translation symmetry ensures consistent experimental results on short timescales, implying energy conservation. However, over millions of years, the universe's expansion breaks this symmetry, indicating energy non-conservation on cosmic scales.
- Noether's first theorem highlights energy loss in particles but doesn't fully address energy conservation in general relativity, where global symmetries are challenged by variable space-time curvature.
- General covariance allows changing frames of reference without altering physical laws, maintaining local symmetries.
- Noether's second theorem introduces continuity equations for local symmetries rather than global conservation laws, similar to fluid dynamics in pipes. In general relativity, energy conservation appears as these local continuity equations, with space-time curvature causing energy 'leakage' between regions.
- Curvature terms in these equations account for energy attributed to the gravitational field as the universe expands, with Noether proving continuity equations' equivalence to Bianchi identities, solving general relativity's conservation issues comprehensively.
- Her work, considered foundational, resolved conservation issues in general relativity that challenged other physicists, with her theorems among the most significant in 20th-century physics.
7. 📚 Emmy Noether: Legacy and Adversity
- In 1933, after Hitler became chancellor, Noether was banned from teaching at universities due to her Jewish heritage, but she continued teaching in her home kitchen.
- Despite the discrimination, Noether's dedication to math was evident as she taught anyone willing to learn, regardless of their affiliations, including a student in a Nazi stormtrooper uniform.
- Noether secured a teaching position at Bryn Mawr in America, where she taught until her death, highlighting her resilience and passion for education.
- Albert Einstein praised Noether in a New York Times obituary as the most significant creative mathematical genius since the higher education of women.
- Noether's theorem revolutionized the approach of physicists, encouraging them to consider physics through the lens of symmetries.
8. 🔬 From Symmetry to Quantum Discoveries
- Physicists applied symmetry concepts to quantum physics, realizing charged particles like electrons possess symmetries.
- Electrons have a phase that can be offset by any arbitrary amount, maintaining symmetry if done uniformly, leading to the conservation of electric charge.
- In the 1960s and '70s, Noether's insights contributed to the discovery of fundamental particles such as quarks and the Higgs boson.
- These discoveries explained the origin of forces and mass in the universe, bringing us closer to a theory of everything.
- Emmy Noether's work on mathematics laid the groundwork for these advancements, emphasizing the importance of independent inquiry and experimentation.