Lecture Notes from PHYS675 - Introduction to Relativity, Gravitation, and Cosmology, as taught by Dr. Jonathan McKinney in the Fall, 2016 semester.

Lecture 1 (08/30/16): Special Relativity

Lecture 2 (09/01/16): Special Relativity

Lecture 3 (09/06/16): Special Relativity

Lecture 4 (09/08/16): Special Relativity

Lecture 5 (09/13/16): Special Relativity

Lecture 6 (09/15/16): Special Relativity

Lecture 7 (09/20/16): Wrapping Up Special Relativity
- Postulates of Special Relativity
- Relativistic Energy
- Time Dilation
- Space Contraction
- The Twin Paradox

Lecture 8 (09/22/16): Introduction to Gravity
- Metric from Gravitational Fields
- Einstein's Equivalence Principle
- The Free-Fall Lab Frame
- Gravitational Time Dilation
- The Doppler Effect
- Masses as Sources of Gravity

Lecture 9 (09/27/16): Four-Vectors & One-Forms
- Lorentz Transformation Matrix
- The Invariant Metric Tensor
- Definition: Four-Vectors
    - ex: Four-velocity
- Basis Vectors
- Tensor Algebra
- Definition: One-Forms
- Contravariant vs. Covariant
- Basis One-Forms

Lecture 10 (09/29/16): Tensor Algebra
-----------------END OF MIDTERM I MATERIAL-------------------------

Lecture 11 (10/11/16): Tensor Algebra

Lecture 12 (10/13/16): Curved Spacetime Metrics
- Conservation of Number Flux
- Coordinate Transform
- Metrics Beyond Minkowski
- Manifestation of *Real* Gravity
- Definition:  Curvature
- Tensor Calculus in General Relativity
    - The fully covariant derivative

Lecture 13 (10/18/16): Connection Coefficients
- Covariant Derivatives of Vectors
- The Levi-Civita Connection
- The Connection Coefficients (Christoffel Symbols)
- Connection with Respect to the Metric
- One-Form Covariant Derivative
- Tensor Reconstruction

Lecture 14 (10/20/16): Parallel Transport & Geodesics 
- Generalizing the Metric
- Parallel Transport
- Affine Coordinates and "Relevant Derivatives"
    - Projection of the vector field along vector path
- Acceleration and Equations of Motion
- Time-Parameterization of Gravity
    - Euler-Lagrange equation and conservation of angular momentum
- Geodesics
-----------------END OF MIDTERM II MATERIAL-------------------------

Lecture 15 (11/01/16): Curvature 
- Recap: Geodesics
- Definition of Curvature: The Riemann Curvature Tensor
- Geometric Representation of Spacetime Curvature Paths
- Geodesic Deviation
    - Manifestation of tidal acceleration
- Riemann Symmetries
    - Ricci tensor & Ricci scalar
- Bianchi Identity
- Derivation of Einstein's Field Equations
    - Newtonian/weak-field limit

Lecture 16 (11/03/16): The Schwarzschild Metric 
- Deriving the Schwarzschild Metric
- Time-Independent Solutions
- Proper Time and the Stationary Solution

Lecture 17 (11/08/16): ABSENT

Lecture 18 (11/10/16): ABSENT

Lecture 19 (11/15/16): The Linear Regime of General Relativity
- Perturbation of Spacetime & Gauge Freedoms
- Einstein's Field Equations: The D'Alembertian Form
    - Newtonian/weak-field limit
- Normal Modes: The Gravitational Wave Solution
- Transverse-Traceless Gauge Condition

Lecture 20 (11/17/16): Photon Orbits
- The Schwarzschild Metric for Photons
- Circular Photon Orbits
- Deflection of Light
-----------------END OF MIDTERM III MATERIAL-------------------------

Lecture 21 (12/06/16): Black Holes
- The Schwarzschild Metric: r < R_{S}
- Time Evolution within a Black Hole
- Kruskal-Szekeres Coordinates
- Rotating Black Holes
    - The Kerr metric
- Frame Dragging

Lecture 22 (12/08/16): Cosmology