As one of the core chapters, the general properties of compact stars are discussed. Spheres of fluid in hydrostatic equilibrium are studied within general relativity. The concept of the mass–radius relation is introduced for the classic case of a gas of noninteracting neutrons. Landau‘s argument for a maximum mass of neutron stars and white dwarfs is delineated. Thereby, the Landau mass and radius is defined for studying scaling solutions of the Tolman–Oppenheimer–Volkoff equation. The power of scaling arguments is demonstrated for the case of a free Fermi gas with arbitrary particle mass, a relativistic gas of fermions with a vacuum term, and the limiting equation of state from causality. The concept of selfbound stars is put forward, giving rise to limits on the compactness and the maximum density achievable for compact stars in general. Generic interactions between fermions are studied and their implications for compact star properties are derived. The general properties of compact stars made of bosons with and without interactions are also investigated.

Hybrid stars are constructed by combining the neutron matter equation of state with the quark matter equation of state. Piecewise polytropes are utilized to interpolate between the limiting equations of state at low and high densities. The phase diagram of QCD is sketched and possible phase transitions are outlined. Connections to heavy-ion experiments and astrophysical systems are established. The two possible ways of constructing a phase transition, the Maxwell and the Gibbs construction, are developed. The possible existence of a pasta phase in the core is presented. The implications of a phase transition for the mass–radius relation of compact stars is discussed. The emergence of a new class of stable solutions of the Tolman–Oppenheimer–Volkoff equation, a third family besides white dwarfs and neutron stars, is put forward. The Seidov criterion for an unstable branch in the mass–radius relation is derived explicitly. Hybrid stars are classified according to the presence of the mixed phase in the mass–radius relation. The properties of hybrid stars are compared to those of ordinary neutron stars by their possible configurations in the mass–radius diagram.

The basic principles of general relativity are reviewed, in particular the different forms of the equivalence principle: the weak, Einstein, and strong equivalence principles. The concept of a metric is introduced within special relativity. The Einstein equations are derived in an heurisitic manner including the Christoffel symbols, the Ricci tensor, and the Ricci scalar. The Schwarzschild as the solution of Einstein‘s equation in vacuum are explicitly derived. The notion of the energy–momentum tensor, as the source term of the Einstein equations, is discussed in terms of the four-momentum of particles. For bulk matter, the definition of an ideal fluid is given. The conservation of the energy–momentum tensor in curved space-time is discussed. The Einstein equations are solved for a sphere of an ideal fluid to arrive at the Tolman–Oppenheimer–Volkoff equations, the central equations for the investigation of compact stars. Finally the analytically known solution for a sphere of an incompressible fluid, the Schwarzschild solution, is derived and used to set the Buchdahl limit on the compactness of a compact star.