Vittorio N. An Overview of General Relativity and Space-Time 2023

Textbook in PDF format

This textbook equips Masters’ students studying Physics and Astronomy with the necessary mathematical tools to understand the basics of General Relativity and its applications. It begins by reviewing classical mechanics with a more geometrically oriented language, continues with Special Relativity and, then onto a discussion on the pseudo-Riemannian space-times. Applications span from the inner and outer Schwarzschild solutions to gravitational wave, black holes, spherical relativistic hydrodynamics, and Cosmology. The goal is to limit the abstract formalization of the problems, to favor a hands-on approach with a number of exercises, without renouncing to a pedagogical derivation of the main mathematical tools and findings.
Part I From Forces to Curvature
Space and Time: The Classical View
From Space and Time to Space-Time
From Inertial to Non-Inertial Reference Frames
Pseudo-Riemannian Spaces
The Riemann-Christoffel Curvature Tensor
From Non-inertial Frames to Gravity: The Equivalence Principle
Part II From Curvature to Observations
Observational Test of the Equivalence Principle
Field Equations in the “vacuum”
Test-Particles in the Schwarzschild Space-Time
The Classical Tests of General Relativity
Gravitational Waves in the “Vacuum”
Part III From Singularities to Cosmological Scales
Black Holes
Field Equations in Non-“Empty” Space-Times
Further Applications of the Field Equations
Theoretical Cosmology
The Hot Big-Bang
Appendix A Exercises