Overview
This lecture explores how mathematics and physics are deeply interconnected, highlighting historical milestones, key theories, and the impact of advanced math on modern physics discoveries.
Historical Foundations of Math and Physics
- Physics and mathematics have been linked since the Renaissance with discoveries like Kepler's planetary laws.
- Newton developed calculus to formulate his gravity theory; Leibniz developed calculus independently.
- Maxwell unified electricity and magnetism with his electromagnetic theory.
- Einstein used non-Euclidean geometry (from Riemann) to develop general relativity.
- Quantum mechanics describes atomic behavior using mathematical frameworks, treating particles as waves and particles.
Mathematical Impact on Modern Physics
Black Holes and Singularities
- Black holes are so dense that not even light can escape; their properties are described primarily by mathematics.
- Hawking and Penrose proved the existence of singularities (points where physics breaks down) inside black holes.
- Hawking showed black holes emit "Hawking radiation" by combining math from quantum mechanics, general relativity, and thermodynamics.
Penrose Tiling and Quasi-Crystals
- Penrose demonstrated non-repeating tiling patterns using just two shapes; these patterns later appeared in naturally occurring quasi-crystals.
The Shape and Fate of the Universe
- The Big Bang theory suggests the universe began from a singularity and is expanding.
- Guth’s inflation theory predicts rapid early universe expansion, possibly making the universe "flat."
- Insufficient mass suggests eternal expansion unless "dark matter" or a "cosmological constant" slows or halts expansion.
- Negatively curved universes (hyperbolic geometry) could fold to form finite but unbounded spaces.
String Theory
- String theory models particles as 1D loops or strings rather than points.
- Superstring theory in 10 dimensions was advanced by Witten and others, aligning well with physical observations but remains experimentally untestable.
- The mathematics of string theory is highly advanced and not yet fully understood.
The Seiberg-Witten Equations
- Physics contributed new insights to topology when Seiberg and Witten showed physics equations could classify mathematical manifolds.
- These results impact how gravity and space are mathematically described.
Key Terms & Definitions
- Calculus — mathematics for studying rates of change; essential for Newton’s gravity.
- Non-Euclidean Geometry — geometry for curved spaces; used in general relativity.
- Singularity — a point in space where density and curvature become infinite.
- Penrose Tiles — non-repeating tiling patterns made from a small set of shapes.
- Quasi-crystal — a solid with an ordered, non-repeating molecular structure.
- Inflation Theory — early rapid expansion of the universe after the Big Bang.
- Dark Matter — unseen matter hypothesized to explain gravitational effects.
- Cosmological Constant — energy density of space, affecting universe expansion.
- String Theory — a theory modeling subatomic particles as vibrating strings.
- Manifold — a mathematical space that locally resembles Euclidean space.
Action Items / Next Steps
- Review examples of mathematical tools used in physics (calculus, non-Euclidean geometry).
- Read about the Big Bang, inflation theory, and string theory in more depth.
- Understand key contributions of Hawking, Penrose, Witten, and others to modern physics.
- Explore the mathematical definition and properties of manifolds and tiling patterns.