An overview of General Relativity is provided to a basic level. Its different nature with respect to the Newtonian Universal Gravitation is outlined. A cursory resume of the post-Newtonian approximation and its importance in testing Einstein’s theory is offered. A brief overview on the modified models of gravity that appeared in the last decades is outlined. A plan of the book is provided.

Newtonian gravity is reviewed and an attempt is made to combine it with special relativity, first by expanding the sources from mass to more general mass-energy, and then by considering relativistic force predictions. The gravito-electro-magnetic field equations are developed by analogy with Maxwell’s equations, and using dynamical source configurations familiar from the study of E&M. In addition to the fields, there are predicted particle interactions, like the bending of light, that go beyong Newtonain gravitational forces. Finally, it is clear that this attempt to combine gravity and special relativity lacks the necessary self-coupling of the gravitational field, which carries energy and therefore acts as its own source.

Newton's Universal Law of Gravitation is compared and contrasted to Coulomb’s Law and the differences highlighted. Tides are discussed, and the Equivalence Principle and how it leads to the notion of curved space-times is explained.

As an appendix, we can look briefly at the central ideas of General Relativity (though we are limited, since much of the maths is beyond our scope). We prepare the ground with a number of thought experiments, and then discuss, in outline, the geometrical ideas we have to use. We can get a sense of what Einstein's equation is doing, and we look at some solutions of Einstein's equation (including the Schwarzschild metric), describing possible spacetimes.

We will trace out some parts of the path that led Einstein to a new theory of gravity that is, unlike Newtonian gravity, consistent with the principle of relativity. The result will be general relativity, a theory that is qualitatively different from Newtonian gravity. In general relativity, gravitational phenomena arise not from forces and fields, but from the curvature of four-dimensional spacetime. The starting point for these considerations is the equality of gravitational and inertial mass, one of the most accurately tested principles in all physics. This leads to Einstein’s equivalence principle, the idea that there is no experiment that can distinguish a uniform acceleration from a uniform gravitational field – the two are fully equivalent.

The basic principles of general relativity are reviewed, in particular the different forms of the equivalence principle: the weak, Einstein, and strong equivalence principles. The concept of a metric is introduced within special relativity. The Einstein equations are derived in an heurisitic manner including the Christoffel symbols, the Ricci tensor, and the Ricci scalar. The Schwarzschild as the solution of Einstein‘s equation in vacuum are explicitly derived. The notion of the energy–momentum tensor, as the source term of the Einstein equations, is discussed in terms of the four-momentum of particles. For bulk matter, the definition of an ideal fluid is given. The conservation of the energy–momentum tensor in curved space-time is discussed. The Einstein equations are solved for a sphere of an ideal fluid to arrive at the Tolman–Oppenheimer–Volkoff equations, the central equations for the investigation of compact stars. Finally the analytically known solution for a sphere of an incompressible fluid, the Schwarzschild solution, is derived and used to set the Buchdahl limit on the compactness of a compact star.