*by Luc Blanchet and Alexandre Le Tiec*

**The first law of binary black hole mechanics can be extended to include non local tail effects. **

Ever since Kepler’s discovery of the laws of planetary motion, the “two-body problem” has always played a central role in gravitational physics. In Einstein’s general theory of relativity, the simplest and most “universal,” purely gravitational, two-body problem is that of a binary system of black holes. The inspiral and merger of two compact objects (i.e. bodies whose radius is comparable to their mass, in “geometric” units where *G* = *c* = 1) produces copious amounts of gravitational radiation, as was recently discovered by LIGO’s multiple detections of gravitational waves from black hole binary systems.

In general relativity, the inspiral and onset of the merger of two compact objects is indeed universal, as it does not depend on the nature of the bodies, be they black holes or neutron stars, or possibly more exotic objects like boson stars or even naked singularities. However, the gravitational waves generated during the post-merger phase depend on the internal structure of the compact objects, and in the case of neutron stars should reveal many details about the scenario for the formation of the final black hole after merger, and the equation of state of nuclear matter deep inside the neutron stars

The theoretical and numerical works on the two-body problem in general relativity, say, in the case of two black holes, play a very important role while deciphering and interpreting the gravitational wave signals, and to perform tests of general relativity theory. In order to model accurately the gravitational-wave emission, it is crucial to properly model the binary’s orbital motion as well.

The equations of orbital motion of two black holes have been developed since the early days of general relativity by means of the post-Newtonian (PN) approximation scheme. After one century of progress on the problem of motion, the state of the art is the 4PN approximation, i.e. the order (*v/c*)^{8} beyond the Newtonian force. At the 4PN order, an interesting effect appears. Indeed, at that order it becomes impossible to disentangle the problem of motion to that of wave generation and propagation. In particular, the orbital dynamics becomes *non-local* in time, because of the effect of gravitational-wave *tails*: gravitational radiation that gets scattered off the background spacetime curvature backreacts on the orbital motion at later times, such that the binary’s dynamics at a given moment in time depends on its entire past history.

On the other hand, the dynamics of two compact objects enjoys a nice property which has been baptized the *first law of binary mechanics*. This is a variational relationship that generalizes to compact binary systems the celebrated first law of black hole mechanics. Remarkably, as a consequence of the first law of binary mechanics, and despite the nonlinearity of the Einstein equations, *global* quantities that characterize the binary system turn out to be related in a very simple manner to *local* quantities that characterize each individual compact object. This implies that a piece of information about one of the two bodies can be used to improve our knowledge of the binary system, such as its binding energy and orbital angular momentum.

Taking full advantage of this property, the first law of binary mechanics has been extensively used for comparing predictions from PN theory, which concern global quantities such as the total energy of the binary system, to predictions of a completely different approximation scheme called the gravitational self-force (GSF) framework, which is based on black hole perturbations and deals with properties of individual compact objects.

In our recent paper, motivated by current 4PN equations of motion and PN/GSF comparisons, we proved that the first law of compact binary mechanics still holds at the 4PN order, by consistently taking into account the non locality in time due to the occurrence of the gravitational-wave tail effect.

About the authors:

Luc Blanchet is a senior researcher (Directeur de Recherche au CNRS) at l’Institut d’Astrophysique de Paris.

Alexandre Le Tiec is a researcher (Chargé de Recherche au CNRS) at l’Observatoire de Paris and Guest Editor of *CQG* focus issue Approaches to the two-body problem.

*Read the full article in Classical and Quantum Gravity:*

First law of compact binary mechanics with gravitational-wave tails

* Blanchet and Le Tiec 2017 Class. Quantum Grav. 34 164001*

This article is also part of a focus issue on gravitational waves.

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