# 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.

Lectures posted so far:

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.

# Other lectures

A brief course on probability & statistics

Course contents:
• Introduction to probability. Sample space and definition of a probability space.
• Conditional probability. Bayes theorem.
• Random variable and probability distributions. Continuous and discrete probability distributions.
• The binomial distribution.
• The Poisson distribution.
• The hypergeometric distribution.
• The normal distribution
• The central limit theorem.
• Notion of sample statistic.
• Estimators.
• Confidence intervals.

A course about the xAct system for tensor analysis

Course being taught at the graduate level. The course slides can be found here.

Course contents:
• Package xTensor: coordinate-free tensor analysis.
• Package xCoba: tensor analysis in coordinates.
• Package xTerior: exterior calculus and its applications.
• Package Spinors: Penrose's spinor analysis in General Relativity.