Lev Landau has called GR “the most beautiful” of the scientific theories. The theory is first of all a description of the gravitational force. Nowadays it is very extensively supported by terrestrial and astronomical observations, and so far it has never been questioned by an empirical observation.
But GR is far more than that. It is a complete modification of our understanding of the basic grammar of nature. This modification does not apply solely to gravitational interaction: it applies to all aspects of physics. In fact, the extent to which Einstein's discovery of this theory has modified our understanding of the physical world and the full reach of its consequences have not yet been completely unraveled.
This chapter is not an introduction to GR, nor an exhaustive description of the theory. For this, I refer the reader to the classic textbooks on the subject. Here, I give a short presentation of the formalism in a compact and modern form, emphasizing the reading of the theory which is most useful for quantum gravity. I also discuss in detail the physical and conceptual basis of the theory, and the way it has modified our understanding of the physical world.
Let M be the “spacetime” four-dimensional manifold. Coordinates on M are written as x, x′, …, where x = (xμ) = (x0, x1, x2, x3). Indices μ, ν, … = 0, 1, 2, 3 are spacetime tangent indices.