Jerzy Plebäski (1928-2005) was a Polish theoretical physicist best known for his extensive research into general relativity, nonlinear electrodynamics and mathematical physics. He split his time between Warsaw, Poland, and Mexico, his permanent residence from the mid-1970s onwards. He is remembered, among other things, for defining the algebraic classification of the tensor of matter, for finding new solutions of the Einstein equations (for example, the Plebäski-Demiäski metric), formulation of the heavenly equations and the effective field theory relating GR and supergravity, known as Plebäski action. The first part of the book is developed from Plebäski's lecture notes.

The scope of this text; Preface to the second edition; Acknowledgements; 1. How the theory of relativity came into being (a brief historical sketch); Part I. Elements of Differential Geometry: 2. A short sketch of 2-dimensional differential geometry; 3. Tensors, tensor densities; 4. Covariant derivatives; 5. Parallel transport and geodesic lines; 6. The curvature of a manifold; flat manifolds; 7. Riemannian geometry; 8. Symmetries of Riemann spaces, invariance of tensors; 9. Methods to calculate the curvature quickly: differential forms and algebraic computer programs; 10. The spatially homogeneous Bianchi-type spacetimes; 11. The Petrov classification by the spinor method; Part II. The Theory of Gravitation: 12. The Einstein equations and the sources of a gravitational field; 13. The Maxwell and Einstein-Maxwell equations and the Kaluza-Klein theory; 14. Spherically symmetric gravitational fields of isolated objects; 15. Relativistic hydrodynamics and thermodynamics; 16. Relativistic cosmology I: general geometry; 17. Relativistic cosmology II: the Robertson-Walker geometry; 18. Relativistic cosmology III: the Lemaître-Tolman geometry; 19. Relativistic cosmology IV: Simple generalisations of L-T and related geometries; 20. Relativistic cosmology V: the Szekeres geometries; 21. The Kerr metric; 22 Relativity enters technology: the Global Positioning System; 23. Subjects omitted from this book; 24. Comments to selected exercises and calculations; References; Index.