GENERAL RELATIVITY

This course provides the theoretical and conceptual foundations of General Relativity (GR). Upon completion, students will be able to:

1. Master tensor calculus and the tools of differential geometry applied to spacetime.
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2. Understand variational principles and the derivation of Einstein's field equations.
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3. Analyze exact solutions of GR (Schwarzschild, FLRW) and extract their physical implications.
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4. Describe the physics of gravitational waves and the fundamentals of relativistic cosmology.

If the course is delivered in blended or remote mode, appropriate adjustments may

be made to the above, in order to ensure consistency with the syllabus.

Module 1: Introduction and Special Relativity (approx. 10 hours)

• Homogeneous and isotropic 3D space. motion of free-test particle

• Homogeneous  and isotropic space-time. non-linear transformations.

• Inertial observers, Lorentz transformations, and the Lorentz Group.
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• Minkowski metric, spacetime interval, and causal structure.
  

• 4-dimensional tensor formalism (4-vectors and 4-tensors).
  

• Relativistic dynamics: Free particle Action and Lagrangian.
  

• Covariant formulation of Electromagnetism: Faraday tensor and Maxwell's equations.

• Equivalence Principle: Weak and Strong Equivalence Principles; limitations of the Minkowski metric and the necessity of curvature.
  ‌Elements of Special Relativity:

Module 2: Elements of Differential Geometry

• Differentiable Manifolds: Charts, atlases, tangent and cotangent spaces.
  

• Tensor Analysis and Differential Forms: Vector fields, differential forms, and tensor algebra on manifolds, Lie bracket. 

• Parallel transport in 2D
  

• (Pseudo-)Riemannian Manifolds: Metric tensor and signature.
  

• Linear Connections: Covariant derivative and Christoffel symbols.
  

• Curvature: Geodesics, Riemann curvature tensor, Ricci tensor, and Ricci scalar.
  

• Geodesic Deviation:tidal forces.
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Module 3: Einstein's Field Equations

• Symmetries and Conserved Quantities: Killing vectors and Killing's equation.
• Lagrangian in space-time and Energy-momentum tensor‌

• Einstein's Equations:
  

  • Action principles: The Einstein-Hilbert action and variational derivation.

  • the Newtonian limit, linearization of Einstein eqs.
    

  • ‌Local Geodesic coordinate frame 

  • Energy-momentum tensor and conservation laws.
    

  • Bianchi identities and diffeomorphism invariance.
    

Module 4: The Schwarzschild Solution, Black Holes, and Classical Tests

• The Schwarzschild Solution: Birkhoff's theorem and metric derivation for static, spherically symmetric spacetimes.

• Geodesics and Orbits: Particle and photon trajectories in Schwarzschild spacetime.‌

• Classical Tests of General Relativity:
  

  • Perihelion precession of Mercury.
    

  • Light deflection by massive bodies.
    

  • Gravitational redshift and Shapiro time delay.
    

• Black Hole Physics:
  

  • Event horizons and singularities.
    

  • Analytic extensions beyond the horizon and Kruskal-Szekeres coordinates/diagrams. Kerr metric. 

  • Penrose compactification 
    

Module 5: Gravitational Waves

• Weak Field and Linearization: First-order Einstein equations and   TT - Transverse Traceless gauge.
  

• Wave Propagation: Gravitational waves in vacuum and polarization states

• Generation and Radiation:
  

  •  Gravitational wave emission from compact binary systems.

  • Astrophysical implications: Observations with interferometers (LIGO/Virgo/KAGRA) and Pulsar Timing Arrays.
    

Module 6: Introduction to Relativistic Cosmology

• Cosmological Principle: Homogeneity and isotropy on large scales.
  

• The FLRW Metric: Geometry of an expanding Universe (Friedmann-Lemaître-Robertson-Walker).
  

• Friedmann Equations: Dynamic evolution of the Universe and standard cosmological models (Matter-dominated, Radiation-dominated, and Dark Energy/Cosmological Constant).

• S. Weinberg, "Gravitation and Cosmology" 

• N. Vittorio, "An Overview of General Relativity and Space-Time", CRC press

• N. Vittorio, "General Relativity -Analytic and Symbolic Problems with Mathematica" , CRC press 

• S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Cambridge University Press.
  

• R. M. Wald, General Relativity, University of Chicago Press.
  

• B. Schutz, A First Course in General Relativity, Cambridge University Press.
  

• C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Freeman.