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**

References:

* 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]

**Contact:**

Eric Gourgoulhon

E-mail: Eric.Gourgoulhon@obspm.fr

Tél: 33 1 45 07 74 33

Fax: 33 1 45 07 79 71

Philippe Grandclément

DARC/LUTH, CNRS

Observatoire de Paris – Section de Meudon

E-mail: PGrandclement@northwestern.edu

Silvano Bonazzola

DARC/LUTH, CNRS

Observatoire de Paris – Section de Meudon

E-mail: Silvano.Bonazzola@obspm.fr