The Spiralling Binary System of Black Holes

A novel approach to the numerical analysis of the dynamic system of two spiraling black holes is developed using the initial value formalism of general relativity. As the two black holes spiral in on each other they will lose energy and angular momentum in the form of gravitational radiation. When the amplitude of this gravitational radiation is small enough, the space-time for the two orbiting black holes can be approximated by a geometry which is unchanging as seen by an observer in a frame of reference co-rotating with the two black holes. Then a time-like Killing vector field is assumed to exist over a finite region of the space-time geometry. A variational principle is found for the total mass of the binary system based on the Hamiltonian of general relativity and is used to study the dynamics and stability of the close orbits. The emission of gravitational waves within the context of the quadrupole moment approximation is used to determine the secular evolution of the system. For black holes of equal mass, approximately 3% of the their initial mass, as deter- mined when the two black holes are at rest at large separations, is emitted as gravitational radiation with frequencies less than the quadrupole normal mode frequency of the final coalesced Kerr black hole.

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