Perturbation theory and approximate methods

Perturbation around compact objects

Linear perturbations around static black holes 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 hypergeometric 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 perturbation is the celebrated Teukolsky equation, that also admits formal series solution ( The mode spectrum is given by the QNM spectrum, which consists in “normal” modes with damping, see for a review.

The mode spectrum of NS also consists in QNMs, since gravity is taking away energy in the form of gravitational wave. It must be stressed that the spectrum is somehow richer since the system now consists in a fluid, gravity and coupling between both, see for a review. QNMs are similar to harmonic oscillators with friction, and are characterized by a frequency and a damping time. Gravitating system typically dissipate energy from emission of gravitational waves. Normal modes instead do not dissipate.

The Post-Newtonian framework

The two body problem in general relativity has always been challenging. Unlike in the Newtonian framework, there is no analytic solution available. This is what motivated the introduction of the so-called Post-Newtonian formalism (see for a review).

Note that analytic or semi analytic treatments of the relativistic two body problem have been studied within the PN approximation, perturbatively or in the self-force context ( These methods are complementary to numerical approaches, that are still computationally expensive.

The first work using the principle of the PN approximation was performed by Einstein in order to predict the perihelion advance of Mercury and formalized by Drost in 1918, where the equations of motion were derived up to 1 PN order (first order in 1/c²). In 1938 Einstein, Infeld and Hoffmann justified the point particle approximation and Chandrasekhar incorporated fluids in the sources in 1960.

The rebirth of the PN formalism is due to Damour and collaborators in the early 80’s (, and led to the 2.5 PN equations of motion (1/c5). Blanchet (from IAP) extended and systematized the construction of higher PN equations of motion in harmonic gauge and developed the formalism for gravitational waveform extraction. The PN framework provides a modelling of the evolution of binary systems, but also allows to reconstruct the metric, in turn allowing to extract the waveform and all geometrical quantities. This is very important for testing the theories against astrophysical observations.