## Course Information

**Lecturers:**

Dr Andy Hammerlindl & Associate Professor Todd Oliynyk, Monash University

**Synopsis:**

General Relativity is currently our most accurate theory of gravity. It applies across a huge range of physical scales describing the motion of small bodies such as satellites orbiting Earth to the dynamics of supermassive black holes and even our Universe. General Relativity is formulated in the language of Differential Geometry. In this course, we will introduce the background in Differential Geometry needed to understand the fundamental concepts and field equations of General Relativity. Applications of the theory to static black holes, the perihelion precession of Mercury’s orbit and gravitational waves will be discussed as time permits.

**Course Overview:**

*Differential Geometry:*

- Manifolds, manifolds with boundary, smooth maps, submanifolds, partitions of unity
- Tangent vectors and spaces, tangent bundles, tangent maps, vector bundles, vector fields
- Multilinear algebra, tensors, tensor bundle, tensor fields
- Contractions, index manipulation, tensor derivations, Lie derivative
- Metrics, covariant derivatives, Curvature

*Mathematical Relativity: * * *

- Fundamental concepts
- Einstein Field equations
- Schwarzschild solution
- Applications: perihelion precession of Mercury’s orbit, linearized Einstein equations and gravitational waves, etc.

**Contact Hours:**

28 hours.

**Prerequisites:**

The required background for this course is Multivariable Calculus and Linear Algebra. Previous experience with any of the following would be an advantage, but is certainly not necessary: Real Analysis, Topology, and Differential Equations.

**Assessment:**

- Assignments: 40%
- Final examination: 60%

**Resources:**

*Lecture Notes:*

- Lecture notes will be provided.

*Texts:*

- Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd ed., Springer, 1988
- M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013
- O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983
- M. Wald, General Relativity, University of Chicago Press, 1984

## Lecturer Biographies

**Dr Andy Hammerlindl, Monash University**

Andy is a Lecturer at Monash University who is originally from Canada. He has held previous positions in Sydney, Australia and Rio de Janeiro, Brazil. His research is in Dynamical Systems, which studies the long-term behaviour of systems which change in time. A large part of his research involves determining the types of chaotic behaviour which can occur in 3-dimensional manifolds. His work uses a combination of topology, geometry, foliations, and measure theory to establish new results.

**Associate Professor Todd Oliynyk, Monash University**

Todd is an Associate Professor at Monash University who is originally from Canada. He first came to Australia in 2002 as a postdoc at the University of Canberra and returned in 2007 to take up a Lecturer position at Monash University after an intervening positon as a Junior Scientist at the Max Planck Institute for Gravitational Physics in Potsdam, Germany. His main research areas are in General Relativity, Geometric Analysis and Partial Differential Equations.