General Relativity is a subject where one aims to understand the geometric structure of spacetime, defined by a partial differential equation, called the Einstein equation. In this course, we aim to introduce some bacis topics related to the Einstein equation from the viewpoint of differential geometry. Given that it has been one hundred years since the discovery of the Einstein equation, and that the subject has grown immense since, we try to keep the background knowledge minimal, and the approach concise.
One should master various methods to formulate the geometric structures of 4 dimensional spacetimes satisfying the Einstein equation, utilizing the languages of differential geometry as well as partial differential equations. Especially through investigating the geometric characteristics of the static and startionary solutions to the Einstein, equation, each student should grasp the importance of the Cauchy problem underlying the solution to the Einstein equation. With time permitting, one aims to understand the Hamiltonian formulation of the Einstein equation and its subsequent consequences fomulated by Penrose.
Einstein equation, calculus of variations, Lorentzian geometry, Cauchy problem, Hamiltonian system
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||Minkowski spacetime and Maxwell equation Maxwell Equation as a Cauchy problem Levi-Civita connection and geodesics Derivation of the Einstein eqaution Post-Newtonian approximation Einstein equation as a Cauchy problem Geometry of Schwarzschild spacetime Positive mass theorem and Penrose inequality||Exercises will be presented during the lectures, and the collection of the exercises should constitute a report.|
Noel Hicks, Notes on Differential Geometry Robert Wald, Introduction to General Relativity P.Dirac, General Relativity 佐々木節,一般相対論