Compact objects (CO) are the endpoint of stellar evolution, more precisely, compact stars are equilibrium configurations where the pressure acting against gravity is due to the electron (white dwarfs) or neutron (neutron stars, NS) degeneracy pressure. When this pressure is not enough to counteract gravity, the dying star collapses to a black hole (BH). COs are described by their internal structure and their gravitational field. NS consist in equilibrium configurations of self-gravitating fluids. Instead BHs don’t have an internal structure, they are described on purely geometrical grounds, i.e. solely by their gravitational field. From this point of view, they are “GR’s elementary particles”. BH are by definition the most compact objects, with a compactness C = M/R = 0.5 (in units where G=c=1 and for static BH), M being the mass and R the (Schwarzschild) radius. Supermassive BH have masses of the order of 106 Msun and are believed to reside at center of galaxies. The supermassive BH candidate Sagitarius A* in the center of the Milky Way is the closest. Understanding the motion of surrounding stars is very important and their observation has led to the conclusion that the central object in the Milky Way has indeed the characteristic of a BH. The spin parameter of these BHs is estimated using spectroscopy methods, e.g. by fitting the deformation of the iron line spectrum emitted by the accretion disk and red-shifted due to the space-time geometry. While first measurements pointed to almost extremal BH, there are AGN candidates with slower rotational parameter. On another hand, astrophysical BH are much less massive, with mass ~ Msun. They are not directly observed but their presence is deduced inside binary system by studying the motion of the companion. NS are the second most compact objects, with typical compactness around C ~ 0.25-0.35. The typical mass is of the order of 1.5-2.5 Msun and the radius is ~7.5-14 km, depending on the EOS. The TOV equation describing them extends to the rotating case, and leads to a system of coupled PDEs, which can be solved numerically. In the case of slow rotation, a perturbative treatment has been introduced by Hartle and consists in expanding the model in terms of the rotational frequency (see the paper here). Most of observed NS are in a regime that is well described by the slow rotation approximation. As the most compact objects in nature, NS and BH are ideal laboratories to test fundamental interactions both on theoretical and observational aspects. Linear perturbations around static BHs have been described by the famous work of Zerilli, and Regge and Wheeler. The formal solutions to these equations are available in terms of series of congruent hyper-geometric functions and Coulomb’s wave functions. The method for treating perturbations of the Kerr BH is based on the Newman-Penrose formalism. The equation describing such perturbations is the celebrated Teukolsky equation, that also admits formal series solution. The mode spectrum of BH is given by the QNM spectrum, which consists in “normal” modes with damping; for NS it also consists in QNMs, since gravity is taking away energy in the form of gravitational wave. It must be stressed that in the case of NS the spectrum is somehow richer since the system now consists in a fluid, gravity and coupling between both. QNMs are similar to harmonic oscillators with friction, and are characterized by a frequency and a damping time.
Let us finally stress that the physics of NS have drawn a lot of attention recently from the discovery of invariant relations between Love numbers, the momentum of inertia and the quadrupole of slowly rotating NS. It was found in 2010 by Lau, Leung and Lin that the frequency of the f-mode oscillation, the mass and the moment of inertia are related by relations that do not depend on the NS equation of state (see also here). In 2013, the Science magazine published an article by Yagi and Yunes investigating the slow rotation approximation for NS in GR. They found that universal relations connecting the tidal Love number with the momentum of inertia and the rotation induced quadrupole hold. They further showed that it was in principle possible to distinguish GR from at least one particular AMG model assuming two independent measurements of the moment of inertia and of the Love number. The relations have been extended using an effective position dependent Love number. Their robustness have then been debated for the cases where magnetic field is present and for the case of fast rotation. I found with collaborators that the case of fast rotation still admits invariant relations, and further displays a new type of universality, involving the radius of the pulsar. This further opens the door on indirect radius measurements. The universal relations have also been investigated for some alternative models of gravity and have been shown to hold, but indistinguishable from those predicted by GR, in the fraction of the parameter space already constrained by observations. While for rotating NS, the exterior space-time is not known analytically, BH enjoy a no hair theorem; all rotational multipoles are related to each other and the unique (uncharged) solution is the Kerr black hole. One of the consequence of the universal relations and of following work is that NS in fact satisfy an approximate ‘few hair’ relation. This gives even more credit to the idea of using a slow rotation expansion of higher order, keeping the first few hairs only.