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.