Department of Physics Department of Mathematics  

Duke University




    

Black Hole

General Relativity
(Math 527 and Physics 622)


Spring 2022

Instructor: Paul Aspinwall

Credits: 1.0, Hours: 3.0

Time: TuTh 10:15AM - 11:30AM

Location: Physics 235

Description

Introduction to the basic concepts and techniques of General Relativity. The course will cover the fundamentals of tensor calculus, Riemannian geometry, and Einstein's equations, as well as applications to cosmology and black holes.

This is a core course for students who want to work in general relativity, cosmology, gravitational lensing, theoretical astrophysics, string theory, or related subjects.

Homework

    will be given weekly.

Prerequisites

A sound knowledge of multivariable calculus (at least Math 212) and linear algebra (at least Math 218). A basic knowledge of classical mechanics and electromagnetism is desirable but the course will endevour to be self-contained.

Exams

A take-home final will be given due back at noon Wednesday, April 27.

Synopsis

A rough outline is as follows.
    0. Special Relativity
    • Minkowski Space
    • Lorentz Transformations
    I. Manifolds and Tensors
    • Tangent vectors and differentiable maps
    • Curves, vector fields, and one-forms
    • Tensor fields and the abstract index notation
    II. Riemannian Geometry
    • Covariant derivatives and parallel transport
    • Curvature and geodesics
    • Computing curvature
    III. The Einstein Field Equations
    • General and special covariance
    • Einstein's equation
    • The weak-field limit
    IV. Applications
    • Cosmology
      • Robertson-Walker universes
      • The cosmological constant (“dark energy”)
    • The Schwarzschild solution
      • Gravitational red shift
      • Black holes
      • Perihelion precession and bending of light
      • The Kruskal extension
    • Further Analysis of Black Holes
      • The Reissner-Nordström Solution
      • The Kerr Solution
      • The Ergosphere
      • Black Hole Thermodynamics
    • Gravitational Waves
    • Inflation
      • Slow Roll Inflation
      • Eternal Inflation and the Multiverse

Textbooks

The course will be based on the text:
  1. Robert M. Wald, General Relativity, University of Chicago Press, Chicago, 1984.
See also
  1. Sean M. Carroll, Spacetime and Geometry, An Introduction to General Relativity, Addison Wesley 2004. (See also gr-qc/9712019 for what might be considered to be an earlier form of this book online.)

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