General relativity gives a complete description of gravitation as it follows from conservation of energy–momentum, general gauge covariance and causality. While it has unprecedented predictive power towards cosmology, gravitational collapse and gravitational waves, only recently has gravitation become an experimental science beyond Newton's law of attraction [22] and beyond gravitational redshift [489], with the advance of LAGEOS, LAGEOS II [155] and Gravity Probe B [199]. These controlled experiments provide the first direct measurements of geodetic and frame-dragging precessions combined, from which we can gain confidence in astrophysical models involving strong gravitational fields and their radiation processes.

In this chapter, we summarize the most immediate aspects in a geometrical way to facilitate applications to astrophysics. We follow the general idea that the essential properties of general relativity derive from the Riemann tensor [485, 141]. We use the tensor notation of [631] with Latin indices and use geometrical units in which Newton's constant and the velocity of light are set equal to 1 unless otherwise specified.

Curved spacetime

General relativity describes the motion of particles in terms of world-lines xb(τ) in a curved spacetime with coordinates xb, with the eigentime τ commonly used as the parameter of the world-line family.