Introduction into General Theory of Relativity

We start with the definition of the Penrose-Carter diagram for flat space-time. On this example we explain the uses of such diagrams. Then we continue with the definition of the Kruskal-Szekeres coordinates which cover the entire black hole space-time. With the use of these coordinates we define Penrose-Carter diagram for the Schwarzschild black hole. This diagram allows us to qualitatively understand the fundamental properties of the black hole.

We start with the definition of Killing vectors and integrals of motion, which allow one to provide conserving quantities for a particle motion in Schwarzschild space-time. We derive the explicit geodesic equation for this space-time. This equation provides a quantitative explanation of some basic properties of black holes. We use the geodesic equation to explain the precession of the Mercury perihelion and of the light deviation in curved space-time.

We start with the definition of the so called perfect fluid energy-momentum tensor and with the description of its properties. We use this tensor to derive the so called interior solution of the Einstein equations, which provides a simple model of a star in the General Theory of Relativity. Then we continue with a brief description of the Kerr solution, which corresponds to the rotating black hole. We end up this module with a brief description of the Cosmic Censorship hypothesis and of the black hole No Hair Theorem.

We start with the derivation of the Oppenheimer-Snyder solution of the Einstein equations, which describes the collapse of a star into black hole. We derive the Penrose-Carter diagram for this solution. We end up this module with a brief description of the origin of the Hawking radiation and of the basic properties of the black hole formation.

With this module we start our study of gravitational waves. We explain the important difference between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity. We define the gravitational energy-momentum pseudo-tensor. Then we continue with the linearized approximation to the Einstein equations which allows us to clarify the meaning of the pseudo-tensor. We end up this module with the derivation of the free monochromatic gravitational waves and of their energy-momentum pseudo-tensor. These waves are solutions of the Einstein equations in the linearized approximation.

In this module we show how moving massive bodies create gravitational waves in the linearized approximation. Then we continue with the derivation of the exact shock gravitational wave solutions of the Einstein equations. We describe their properties. To help to understand this module we provide two complementary videos. One with the explanations how to perform the averaging over directions in space. And the other video is with the derivation of the retarded Green function.

With this module we start our discussion of the cosmological solutions. We define constant curvature three-dimensional homogeneous spaces. Then we derive Friedman-Robertson-Walker cosmological solutions of the Einstein equations. We describe their properties. We end up this module with the derivation of the vacuum homogeneous but anisotropic cosmological Kasner solution.

In this module we derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant. We describe the geometric and causal properties of such space-times and provide their Penrose-Carter diagrams. We provide coordinate systems which cover various patches of these space-times.