Motion in curved spacetime, classical equations in covariant form, tidal forces, Einstein’s field equation in empty space, and weak (linearized) gravitation.

The relation between local spacetime curvature and matter energy density is given by the Einstein equation – it is the field equation of general relativity in the way that Maxwell’s equations are the field equations of electromagnetism. Maxwell’s equations relate the electromagnetic field to its sources – charges and currents. Einstein’s equation relates spacetime curvature to its source – the mass-energy of matter. This chapter gives a very brief introduction to the Einstein equation; we consider the equation in the absence of matter sources (the vacuum Einstein equation) and will include matter sources in Chapter 22. Even the vacuum Einstein equation has important implications. Just as the field of a static point charge and electromagnetic waves are solutions of the source-free Maxwell’s equations, the Schwarzschild geometry and gravitational waves are solutions of the vacuum Einstein equation.

The detection of gravitational waves from the merger of black holes and the merger of two neutron stars are discussed. The linearized theory of general relativity is introduced. The concept of the gauge invariance is put forward and the transverse-traceless gauge for gravitational waves is presented. Einstein‘s famous quadrupole formula for gravitational waves is developed. The principle of detecting gravitational waves is outlined. As applications, the emission of gravitational waves from a nonvanishing ellipticity of rotating neutron stars is derived. The chirp mass is introduced and the emission of gravitational waves from compact binary systems is obtained. The formula for the tidal deformability and the Love number is put forward and discussed with regard to the recent measurement of a neutron star merger by the LIGO–Virgo scientific collaboration.