Lectures in Differential Geometry and General Relativity
This is a set of on-line lectures dealing with classical General Relativity and Differential Geometry.
As of now the lectures are classified into topics. If you are interested in a topic which is not listed, please contact me
at alfonso AT math.uminho.pt and I will check if I can add a lecture about
the corresponding topic. Of course, I accept questions about the lectures posted here.
In these lectures I introduce the notion of conformal boundary and conformal compactification.
I explain in detail the conformal compactification of Minkowski, de Sitter and anti de Sitter solutions. I also introduce the
notion of Penrose diagram and construct the Penrose diagram for a number of exact solutions of the Einstein field equations.
To follow the lectures of this topic you are assumed to be familiar with the representation of the space-time as a
Lorentzian manifold. You also need to know that observers in the space-time are represented by (inextendible) causal curves.
Brief introduction of the notion of conformal boundary
Conformal compactification of the Minkoski solution
Lecture contents:
The Minkowski solution in standard Euclidean coordinates.
The conformal embedding of the Minkowski space-time into the Einstein static universe.
Conformal diagram of Minkowski space-time.
The conformal boundary of Minkowski space-time and its different regions.
The global causal properties of the Minkowski space-time.
Conformal compactification of de Sitter and anti-de Sitter.
Lecture contents:
Maximally symmetric spaces in pseudo-Riemannian geometry.
Maximally symmetric solutions of the Einstein equations: Minkowski, de Sitter and anti-de Sitter solutions.
Embedding of de Sitter solution in five dimensional flat space-time.
The conformal embedding of de Sitter solution into the Einstein static universe.
Construction of the conformal boundary and study of the global properties of the de Sitter solution.
Embedding of anti de Sitter solution in five dimensional flat space-time.
The conformal embedding of anti-de Sitter solution into the Einstein static universe.
Construction of the conformal boundary and study of the global properties of the anti-de Sitter solution.
Penrose diagram of the Schwarzschild solution.
Lecture contents:
Spherically symmetric space-times in dimension four.
The notion of Penrose diagram of a spherically symmetric space-time.
Example: Penrose diagram of the Minkowski space-time.
The Schwarzschild solution in Schwarzschild coordinates.
Penrose diagram of the Schwarzschild solution and construction of the Kruskal extension.
Global causal properties of the Schwarzschild solution.
Penrose diagram of the Reissner Nordström solution.
Lecture contents:
The Reissner-Nordström solution and its classification: m>q, m=q and m< q.
Penrose diagram of the case m>q: maximal extension and global causal properties.
Penrose diagram of the case m=q: maximal extension and global causal properties.
Penrose diagram of the case m< q: maximal extension and global causal properties.
Relativistic Cosmology
In this lecture I explain in detail the ΛCDM model and its implications to the global causal structure of the
universe. I also compute the Penrose diagram of the universe using the material introduced in the lectures about Penrose
diagrams. The lecture can be understood as a mini-course in Cosmology.
Familiarity with elementary General Relativity is assumed.
Lecture contents:
The cosmological principle and the Weyl postulate. Scales of the known Universe.
Construction of the FLRW space-time and its classification: open, flat and closed universes.
The cosmological redshift and blueshift. Relation to the expansion and contraction of the universe.
Matter-energy content of the universe. Review of geometrized units.
The FLRW cosmological models and their dynamics. Cosmological equations.
Cold dark matter hypothesis and its implications for the dynamics of the universe. Definitions
of ΩΛ, ΩM, ΩR and Ωk.
The ΛCDM model.
Construction of the Penrose diagram of the Universe and discussion of its global properties.