# Tidal coupling of a Schwarzschild black hole and circularly orbiting moon

## Abstract

We describe the possibility of using the laser interferometer space antenna (LISA) 's gravitational-wave observations to study, with high precision, the response of a massive central body (e.g. a black hole or a soliton star) to the tidal gravitational pull of an orbiting, compact, small-mass object (a white dwarf, neutron star, or small-mass black hole). Motivated by this LISA application, we use first-order perturbation theory to study tidal coupling for a special, idealized case: a Schwarzschild black hole of mass M, tidally perturbed by a 'moon' with mass {mu}<<M in a circular orbit at a radius b>>M with orbital angular velocity {omega}. We investigate the details of how the tidal deformation of the hole gives rise to an induced quadrupole moment I{sub ij} in the hole's external gravitational field at large radii, including the vicinity of the moon. In the limit that the moon is static, we find, in Schwarzschild coordinates and Regge-Wheeler gauge, the surprising result that there is no induced quadrupole moment. We show that this conclusion is gauge dependent and that the static, induced quadrupole moment for a black hole is inherently ambiguous, and we contrast this with an earlier result of Suen, which gave, in amore »

- Authors:

- Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125 (United States)

- Publication Date:

- OSTI Identifier:
- 20774537

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 72; Journal Issue: 12; Other Information: DOI: 10.1103/PhysRevD.72.124016; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANGULAR MOMENTUM; ANGULAR VELOCITY; ANTENNAS; BLACK HOLES; COUPLING; DEFORMATION; GRAVITATIONAL FIELDS; GRAVITATIONAL WAVE DETECTORS; GRAVITATIONAL WAVES; INTERFEROMETRY; MASS; NEUTRON STARS; PERTURBATION THEORY; PHASE SHIFT; POLARIZABILITY; QUADRUPOLE MOMENTS; SCHWARZSCHILD METRIC; SOLITONS; WHITE DWARF STARS

### Citation Formats

```
Fang Hua, and Lovelace, Geoffrey.
```*Tidal coupling of a Schwarzschild black hole and circularly orbiting moon*. United States: N. p., 2005.
Web. doi:10.1103/PhysRevD.72.124016.

```
Fang Hua, & Lovelace, Geoffrey.
```*Tidal coupling of a Schwarzschild black hole and circularly orbiting moon*. United States. doi:10.1103/PhysRevD.72.124016.

```
Fang Hua, and Lovelace, Geoffrey. Thu .
"Tidal coupling of a Schwarzschild black hole and circularly orbiting moon". United States.
doi:10.1103/PhysRevD.72.124016.
```

```
@article{osti_20774537,
```

title = {Tidal coupling of a Schwarzschild black hole and circularly orbiting moon},

author = {Fang Hua and Lovelace, Geoffrey},

abstractNote = {We describe the possibility of using the laser interferometer space antenna (LISA) 's gravitational-wave observations to study, with high precision, the response of a massive central body (e.g. a black hole or a soliton star) to the tidal gravitational pull of an orbiting, compact, small-mass object (a white dwarf, neutron star, or small-mass black hole). Motivated by this LISA application, we use first-order perturbation theory to study tidal coupling for a special, idealized case: a Schwarzschild black hole of mass M, tidally perturbed by a 'moon' with mass {mu}<<M in a circular orbit at a radius b>>M with orbital angular velocity {omega}. We investigate the details of how the tidal deformation of the hole gives rise to an induced quadrupole moment I{sub ij} in the hole's external gravitational field at large radii, including the vicinity of the moon. In the limit that the moon is static, we find, in Schwarzschild coordinates and Regge-Wheeler gauge, the surprising result that there is no induced quadrupole moment. We show that this conclusion is gauge dependent and that the static, induced quadrupole moment for a black hole is inherently ambiguous, and we contrast this with an earlier result of Suen, which gave, in a very different gauge, a nonzero static induced quadrupole moment with a sign opposite to what one would get for a fluid central body. For the orbiting moon and the central Schwarzschild hole, we find (in agreement with a recent result of Poisson) a time-varying induced quadrupole moment that is proportional to the time derivative of the moon's tidal field, I{sub ij}=(32/45)M{sup 6}E{sub ij} and that therefore is out of phase with the tidal field by a spatial angle {pi}/4 and by a temporal phase shift {pi}/2. This induced quadrupole moment produces a gravitational force on the moon that reduces its orbital energy and angular momentum at the same rate as the moon's tidal field sends energy and angular momentum into the hole's horizon. As a partial analog of a result derived long ago by Hartle for a spinning hole and a static distant companion, we show that the orbiting moon's tidal field induces a tidal bulge on the hole's horizon, and that the rate of change of the horizon shape (i.e. the horizon shear) leads the perturbing tidal field at the horizon by an angle 4M{omega}. We prefer to avoid introducing an ingoing null geodesic, as Hartle did in his definition of the phase shift, because the moon is in the central body's near zone (b<<1/{omega}) and thus should interact with the horizon instantaneously, not causally. We discuss the implications of these results for LISA's future observations of tidal coupling, including the inappropriateness of using the concepts of tidal polarizability and tidal lag or lead angle, for the massive central body, when discussing LISA's observations.},

doi = {10.1103/PhysRevD.72.124016},

journal = {Physical Review. D, Particles Fields},

number = 12,

volume = 72,

place = {United States},

year = {Thu Dec 15 00:00:00 EST 2005},

month = {Thu Dec 15 00:00:00 EST 2005}

}