## PHY 430: Relativity: Special & General

3 credits | Prerequisites: PHY 222, 223, 301 for Major; PHY 230 for Minor

### Course rationale

General Relativity (GR) is one of the most beautiful theories in physics. Not only it generalizes Newton’s law of gravitation, but also gives us a geometric picture of space-time. It also magnificently connects Riemannian geometry to gravity. All these are succinctly expressed in terms of Einstein field equations. This elective course will focus on the basic concepts of GR, Einstein’s beautiful theory of gravitation.

### Course content

Special Theory of Relativity: Einstein’s postulates, inertial observer, space-time diagrams, Minkowski spacetime, Lorentz transformation, invariant hyperbolae, length contraction, time dilation, mass-energy relation, velocity and acceleration in special relativity; Tensors in Special Relativity: definition of a vector, 4-velocity, 4-momentum, scalar product, photon, metric tensor, one-forms, general tensor, transformation, raising and lowering index; Manifold: physical motivation-gravity as geometry, the definition of a manifold, vector, and tensor on manifold, Lie derivative; Riemannian Manifold: the metric tensor, an example-expanding universe, causality, tensor densities, introduction to differential forms, integration on a manifold; Analysis on Manifold: covariant differentiation, Christoffel’s symbol, parallel transport, geodesic, properties of geodesic, examples from expanding universe; Riemannian Geometry: the Riemann curvature tensor, Ricci tensor, and scalar, properties of Riemann tensor, Bianchi identities, Weyl tensor, Cotton tensor, symmetry and Killing vectors, maximally symmetric spaces, geodesic deviation, tetrad formalism, Cartan’s structure equation; Einstein’s Geometric Theory of Gravitation: physics in curved spacetime, Einstein’s field equation, Einstein-Hilbert action, variation of Einstein-Hilbert action to Einstein’s equation, properties of Einstein’s equation, the cosmological constant, energy conditions, equivalence principle; Some Simple Solutions of Einstein’s Equation: flat space, de Sitter space, Anti-de Sitter space, Birkhoff’s theorem, derivation of Schwarzschild metric, event horizon, singularities, geodesics in Schwarzschild spacetime, experimental tests, Schwarzschild black holes, an extension of Schwarzschild spacetime, stars and black holes; Gravitational Radiation: linearized gravity, gauge transformation, degrees of freedom, Newtonian fields, the trajectory of photon, bending of light, transverse-traceless gauge, gravitation waves, production of gravitation waves by the binary star system, observation of gravitational waves GW150914.

### Course objectives

- Understand the geometry of Minkowski spacetime.
- Analyze vector and tensor calculus on Minkowski spacetime.
- Understand the concept of manifold, and Riemannian geometry.
- Understand the relation between geometry and gravitation.
- Familiarize with Einstein’s field equation.
- Analyze different solutions of Einstein’s equation, and particle trajectories.
- Familiarize with the idea of black holes.
- Understand the concept of linearized gravity, and gravitational waves.

### References

- A First Course in General Relativity (3rd edition) by Bernard Schutz
- Spacetime and Geometry: An Introduction to General Relativity (1st edition) by Sean M. Carroll
- Gravitation by Charles W. Misner, Kip S. Thorne, and John A. Wheeler
- Gravitation: Foundations and Frontiers (1st edition) by T. Padmanabhan
- General Relativity by Robert Wald
- The Large Scale Structure of Space-time by Stephen Hawking, and G. F. R. Ellis
- Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg