When two black holes constitute a binary system, what are the laws
which describe their motion? The famous three laws stated by Kepler
in the XVII Century are not valid in the case of black holes, for
these objects require general relativity for their correct description.
Researchers at Paris Observatory and CNRS have just computed numerically
the orbital motion of two black holes close to each other. Their
results, presented in two articles in press in Physical Review D, have
put a term to a debate between the theoricians who were trying to
compute these orbits and whose results did not agree among each other.
The new results are important for the detection of the gravitational
waves emitted by binary black holes by the detectors VIRGO and LIGO
currently under construction.

In Newtonian theory of gravitation, the determination of the orbits
in a system of two point-like bodies is an elementary problem. The
classical result, known as the Kepler’s Third Law, stipulates that for
two bodies in circular orbit around their common centre of mass, the
square of the orbital frequency is proportional to the sum of their
masses and inversely proportional to the cube of their distance.

When the sizes of the two bodies are not negligible compared to the
distance between them, they cannot any more be considered as point-
like. The deformation of each body by the tidal forces of its
companion implies that the Kepler’s Third Law is no longer satisfied.
It is then necessary to carry out numerical calculations to solve
the problem. In the case of two stars, the system is called a tight
binary system.

The theoretical astrophysicists have been interested for a few years
in somewhat peculiar tight binary systems, namely those made up of
two black holes. These systems indeed constitute one of the principal
sources of gravitational waves expected by the interferometric
detectors VIRGO and LIGO, currently in construction. Gravitational
waves are deformations of space-time which propagate at the speed
of light. They were predicted by Einstein in 1918, as direct
consequences of his theory of General Relativity.

A system of two very relativistic objects, such as black holes,
emits gravitational waves and thus loses energy and angular momentum.
Consequently, the orbits described by each body are not exactly
circular but form a slow spiral which leads to the coalescence of
the two black holes. It is this cataclysmic event which is hoped to
be observed using VIRGO and LIGO.

A first step in the study of this phenomenon is the calculation of
the orbits of the two black holes, neglecting the reaction to the
gravitational radiation, i.e. by approximating the orbits with exact
circles. Not only it is a tight binary system, not obeying the
Kepler’s Third Law, but also an additional complication comes from
the need to consider General Relativity. This results from the very
nature of black holes, that the Newtonian theory of the gravitation
cannot describe.

Researchers from the Department of Relativistic Astrophysics and
Cosmology of the Observatory of Paris have just carried out such a
computation. The structure of space-time containing two black holes
had already been computed by American groups, but for arbitrary
motions, not expected in nature. Realistic motions must be circular
orbits, the emission of gravitational waves transforming initially
elliptic orbits into circular ones.

The researchers of Paris Observatory obtained configurations which
describe a system of two identical black holes on circular orbits.
They calculated a sequence of such configurations by gradually
decreasing the distance between the two black holes, in order to
simulate the evolution of the system under the effect of energy
loss through gravitational radiation.

The orbital frequency thus obtained strongly deviates from that given
by Kepler’s Third Law (by more than 100% when the black holes are
close to each other). These configurations will be used as initial
conditions to compute the coalescence of the two black holes, by the
researchers of the European Network of Gravitational Wave Astrophysics.

Notes for editor


* E. Gourgoulhon, P. Grandclément and S. Bonazzola: Binary black holes
in circular orbits. I. A spacetime approach, submitted to Physical
Review D [preprint: http://arXiv.org/abs/gr-qc/0106015]

* P. Grandclément, E. Gourgoulhon and S. Bonazzola: Binary black holes
in circular orbits. II. Numerical methods and first results,
submitted to Physical Review D,
[preprint: http://arXiv.org/abs/gr-qc/0106016]


Eric Gourgoulhon

E-mail: Eric.Gourgoulhon@obspm.fr

Tél: 33 1 45 07 74 33

Fax: 33 1 45 07 79 71

Philippe Grandclément


Observatoire de Paris – Section de Meudon

E-mail: PGrandclement@northwestern.edu

Silvano Bonazzola


Observatoire de Paris – Section de Meudon

E-mail: Silvano.Bonazzola@obspm.fr