General Relativity

1. Mathematical Formulation
   1. Tensor Calculus
      1. Introduction to Tensors
         1. Definition and basic properties
         2. Differences from vectors and scalars
         3. Types of tensors: covariant, contravariant, mixed
         4. Tensor algebra: addition, multiplication, contraction
      2. Metric Tensor
         1. Role in describing spacetime
         2. Relationship with distance and angles in curved space
         3. Metric signature and its implications for general relativity
         4. Examples of common metrics: Minkowski, Schwarzschild
      3. Riemann Curvature Tensor
         1. Definition and interpretation
         2. Relation to the notion of curvature in differential geometry
         3. Calculation via the Christoffel symbols
         4. Properties and symmetries, such as Bianchi identities
         5. Role in determining manifold curvature
      4. Ricci Curvature Tensor
         1. Definition as a contraction of the Riemann tensor
         2. Role in Einstein's Field Equations
         3. Geometrical significance in the context of volume change
      5. Einstein Tensor
         1. Definition and relation to the Ricci tensor
         2. Conservation law connections
         3. Importance in forming the left side of Einstein's Field Equations
   2. Einstein's Field Equations
      1. Origin and derivation from the Einstein-Hilbert action
      2. Explanation of the equation and its components
      3. Physical interpretation: relationship between matter and spacetime curvature
      4. The cosmological constant and its implications for cosmology
      5. Linearized field equations and weak-field approximations
   3. Equations of Motion
      1. Geodesic Equation
         1. Derivation using the principle of extremal aging
         2. Role in defining freely falling paths in a gravitational field
         3. Effects of spacetime curvature on the motion of particles
   4. Solutions to the Field Equations
      1. Schwarzschild Solution
         1. Characteristics of the Schwarzschild metric
         2. Application to non-rotating, spherically symmetric masses
         3. Implications for black holes and event horizons
         4. Connection with Birkhoff’s theorem
      2. Kerr Solution
         1. Description of rotating black holes
         2. Introduction to the Kerr metric and its properties
         3. Effects of angular momentum on spacetime: frame dragging
      3. Reissner-Nordström Solution
         1. Extension of Schwarzschild solution to include charge
         2. Characteristics of charged black holes
         3. Role in theoretical studies of charged black hole stability
      4. Friedmann-Lemaître-Robertson-Walker (FLRW) Metrics
         1. Application in cosmological models of the universe
         2. Description of homogeneous and isotropic universes
         3. Derivation and significance of the Friedmann equations
         4. Role in big-bang cosmology and cosmological constant discussions