VILLANOVA UNIVERSITY Dr. A. J. PHARES

VILLANOVA UNIVERSITY Dr. A. J. PHARES GENERAL RELATIVITY SPRING 2014

SYLLABUS

The main references are Gravitation and Cosmology by S. Weinberg, and Gravitation by Misner, Thorne & Wheeler. There will be take-home exercises (40%), a written term project (25%) including its oral presentation (15%), and the formal submission of lecture notes collected twice during the semester for feedback, before final submission at the end of the semester (20%). This final report should follow the format of the course outline:

Part I. TENSOR CALCULUS Chapter 1. An Introduction to Vector Spaces 1. Conditions for a Space to be a Vector Space 2. Linear Independence, Dimensionality, and Base Vectors 3. Euclidean Vector Spaces – Norm – Proper Euclidean Vector Spaces 3.1. Metric, Covariant and Contravariant Components of a Vector 3.2. Matrix Notation 3.3. Change of Base 4. n-Dimensional Geometry (Affine Space) Chapter 2. Tensor Algebra 1. Dual Vector Spaces 2. Tensor Product 3. Tensors and Change of Base Chapter 3. Differential Geometry 1. Euclidean Space in Curvilinear Coordinates 2. Tensors in Differential Geometry 3. Arclengths, Volume Elements Chapter 4. Riemann Geometry 1. Christoffel Symbol and Coordinate Transformations 2. Covariant Derivatives 3. Vector Operations in n-dimensions 4. Manifolds and Geodesics 4. Riemann Tensor, Bianchi’s Identity, Curvature of Space Part II. SPECIAL THEORY OF RELATIVITY Chapter 1. Lorentz Transformation 1. The 4-dimensional manifold and arclength in that manifold 2. Boost along a given axis 3. Boost along an arbitrary axis: General case of the Lorentz Transformation 4. Velocity Transformation Chapter 2. Relativistic Dynamics 1. Relativistic Force, Momentum 2. Work and Energy Chapter 3. Covariant Formulation of Electrodynamics 1. Current density 4-vector 2. Energy-Momentum Density Tensor 3. Maxwell’s Equations Part III. GENERAL RELATIVITY Chapter 1. Principle of Equivalence 1. Inertial and gravitational mass 2. Eötvös Experiment Chapter 2. The Newtonian Limit 1. Covariant derivative of the 4-velocity vector 2. Geodesics and change of variables 3. Weak and Stationary Gravitational Fields Chapter 3. Spherically Symmetric and Static Gravitational Field: The Fundamental Three Predictions