Gravitational radiation from a particle in circular orbit around a black hole. I. Analytical results for the nonrotating case

Among the most promising and interesting sources of gravitational waves for interferometric detectors, such as the ground-based Laser Interferometer Gravitational-wave Observatory (LIGO)/VIRGO system and the proposed space-based Laser Gravitational-Wave Observatory in Space (LAGOS), is the last several minutes of inspiral of a compact binary (one made of neutron stars and/or black holes). This paper is the first in a series that will carry out detailed calculations relevant to such binaries, in the case where one body is a small-mass black hole or neutron star and the other is a much more massive black hole, and the orbit is circular (aside from its gradual inspiral). These papers will focus primarily on the emitted waveforms and especially their phasing—which is crucial for extraction of information from the detectors' measurements. This first paper is restricted to the case where the massive black hole is nonrotating. The paper begins by bringing the already well-developed formalisms for computing the waveforms (the "Regge-Wheeler" and "Teukolsky" formalisms) into a combined form that is particularly well suited both for high accuracy numerical calculations (to be carried out in paper II), and for analytic calculations (this paper). Then analytic solutions to the formalism's equations are found in the limiting case of orbits with large radii $r_0$, and correspondingly small values of $v={\left(\frac{M}{r_0}\right)}^{\frac{1}{2}}={\left(M\Omega \right)}^{\frac{1}{3}}$. Here $M$ is the mass of the large black hole; $v$ and $\Omega$ are, respectively, the orbital linear and angular velocities as measured far from the hole. In particular, (i) the leading-order (in $v$) contribution of each spherical harmonic to the waveforms and to the energy loss is computed analytically, and (ii) the full wave-forms and full energy loss are computed analytically up through fractional corrections of order $v^3$ beyond Newtonian, i.e. up through ${\mathrm{post}}^{\frac{3}{2}}$-Newtonian order. It is shown that propagation of the waves through the intermediate zone (which connects the near zone to the wave zone) distorts the waveforms and changes their power (and hence phasing), at ${\mathrm{post}}^{\frac{3}{2}}$-Newtonian order, in ways that have not previously been computed—except abstractly and nonconcretely as formal "tail terms" in the waves. It is demonstrated that these ${\mathrm{post}}^{\frac{3}{2}}$-Newtonian corrections will be of considerable importancee for the extraction of information from the waveforms that LIGO/VIRGO expects to measure.