Theory of Gravitational Waves
  The existence of gravitational radiation is a natural prediction of any
relativistic description of the gravitational interaction. In this chapter, we
focus on gravitational waves, as predicted by Einstein's general theory of
relativity. First, we introduce those mathematical concepts that are necessary
to properly formulate the physical theory, such as the notions of manifold,
vector, tensor, metric, connection and curvature. Second, we motivate,
formulate and then discuss Einstein's equation, which relates the geometry of
spacetime to its matter content. Gravitational waves are later introduced as
solutions of the linearized Einstein equation around flat spacetime. These
waves are shown to propagate at the speed of light and to possess two
polarization states. Gravitational waves can interact with matter, allowing for
their direct detection by means of laser interferometers. Finally, Einstein's
quadrupole formulas are derived and used to show that nonspherical compact
objects moving at relativistic speeds are powerful gravitational wave sources.
