PN, EFT and geodesics

Post Newtonian formalism

Analytic or semi analytic treatments of the relativistic two body problem have been studied within the PN approximation [23], perturbatively or in the self-force context [24]. These methods are complementary to numerical approaches, that are still computationally expensive. Here we will focus on the PN approach. 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 [25], 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 [26]. The PN framework provides a modeling 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.

Effective field theory

EFT [19] is a formalism commonly used in quantum field theory. It consists in giving an effective description of some parts of a full – fundamental – theory, by essentially truncating the theory in a mathematically consistent way. More precisely, the idea is typically to integrate out the ultra violet regime, i.e. the small scales out of the theory, leaving a low energy effective model, that can be studied on its own right. In some cases where the fundamental theory is too complex for this procedure to be carried out explicitly, an effective model is built from the symmetries of the theory, in this case, there are unknown couplings that need to be matched to the full model. In the astrophysical context, this is exactly the same principle: the compact objects, including their fine details like internal structure are effectively modeled by point particles [20]. Finite size effects are incorporated in multipolar degrees of freedom propagating along the objects’ world-lines [20]. There has been considerable amount of work relying on the EFT approach; for example the energy momentum up to the quadrupolar approximation is given in [21], inclusion of dissipative effects in the black hole dynamics has been considered in [22]. As all EFTs, the approach has a limit of validity and breaks down once the system probes smaller scales, for example during the merger in a binary system.

Binary systems: a dance-floor for gravity – In a binary system, the main channel of interaction is gravity, through tidal interactions. Tidal forces have been well understood in Newton’s gravity for more than a century [27]. The so-called Love numbers – named after Augustus Edward Hough Love – encode how much the body responds to a tidal external perturbation. In the relativistic case, they are better understood in the adiabatic limit, i.e. when the deformation is aligned with the perturbing source. Relativistic tidal interactions of compact objects were studied by Damour and Nagar [28], and Binnington and Poisson [29] in the effective action (resp. EFT) context and in the adiabatic limit. The perturbing field was chosen to be static, which means that the time scale of the response of the compact object to the perturbation is much shorter than the time scale of the perturbation itself. Among the results, it turns out that tidal coefficients (Love numbers) of stationary black holes vanish in 4 dimensions. This is in fact an accident, in arbitrary number of dimensions and depending on the angular momentum, the Love numbers can be divergent [30]. They can be renormalized using standard regularization schemes, since all usual tools of statistical field theory apply to effective field theory. Note that relativistic tidal coefficients including rotation are presently unknown. Observations of binary systems involving compact objects provides invaluable informations, see the case of the historical Hulse-Taylor binary pulsar.

The geodesic equation

Gravitating bodies are freely falling in a cruved spacetime. They follow geodesics, i.e. the shortest path in curved spacetime. Said differently, they are not accelerated in this spacetime: Let X(t) be a curve in a Riemanian manifold, with t a parameter (such as proper time). The velocity is given by the derivative of X w.r.t. X: u = dX/dt and the acceleration by a=du/dt = u.D u, where D is the covariant derivative. A geodesic curve satisfies a ~ u, if t is the proper time, then the geodesic satisfies a = 0.

Geodesics describe the trajectories of test bodies, i.e. bodies that do no cause further bending to the spacetime. This is valid as a first approximation for ‘small’ or light bodies. Corrections can be computed within perturbation theory, using the following principle: the lagrangian of the freely falling body (particle following a geodesic) is L = – m ds. This leads to a stress tensor, that is considered as sourcing a metric perturbation around flat space. A formal solution can be constructed, and energy or angular momentum loss can be computed (see Wald’s book Gravitation for a derivation). This procedure is well suited for describing extreme mass ration, for instance a celestial body orbiting around of falling into a supermassive black hole.

In some sense, the perturbation theory gives the first correction. Taking into account the backreaction of the emitted radiation on the trajectory of the perturber leads to the so-called self-force problem.