# Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk

## Abstract

Will, in 1974, treated the perturbation of a Schwarzschild black hole due to a slowly rotating, light, concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to perturbation by a thin disk (which is more relevant astrophysically), but, due to rather bad convergence properties, the resulting expansions are not suitable for specific (numerical) computations. However, we show that Green’s functions, represented by Will’s result, can be expressed in closed form (without multipole expansion), which is more useful. In particular, they can be integrated out over the source (a thin disk in our case) to yield good converging series both for the gravitational potential and for the dragging angular velocity. The procedure is demonstrated, in the first perturbation order, on the simplest case of a constant-density disk, including the physical interpretation of the results in terms of a one-component perfect fluid or a two-component dust in a circular orbit about the central black hole. Free parameters are chosen in such a way that the resulting black hole has zero angular momentum but non-zero angular velocity, as it is just carried along bymore »

- Authors:

- Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, Prague (Czech Republic)

- Publication Date:

- OSTI Identifier:
- 22661092

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal, Supplement Series; Journal Volume: 232; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCRETION DISKS; ANGULAR MOMENTUM; ANGULAR VELOCITY; ASTROPHYSICS; BLACK HOLES; CALCULATION METHODS; CONVERGENCE; COSMIC DUST; DISTURBANCES; GRAVITATION; IDEAL FLOW; MASS; MULTIPOLES; ORBITS; ROTATION; SCHWARZSCHILD METRIC; VISIBLE RADIATION

### Citation Formats

```
Čížek, P., and Semerák, O., E-mail: oldrich.semerak@mff.cuni.cz.
```*Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk*. United States: N. p., 2017.
Web. doi:10.3847/1538-4365/AA876B.

```
Čížek, P., & Semerák, O., E-mail: oldrich.semerak@mff.cuni.cz.
```*Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk*. United States. doi:10.3847/1538-4365/AA876B.

```
Čížek, P., and Semerák, O., E-mail: oldrich.semerak@mff.cuni.cz. Fri .
"Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk". United States.
doi:10.3847/1538-4365/AA876B.
```

```
@article{osti_22661092,
```

title = {Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk},

author = {Čížek, P. and Semerák, O., E-mail: oldrich.semerak@mff.cuni.cz},

abstractNote = {Will, in 1974, treated the perturbation of a Schwarzschild black hole due to a slowly rotating, light, concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to perturbation by a thin disk (which is more relevant astrophysically), but, due to rather bad convergence properties, the resulting expansions are not suitable for specific (numerical) computations. However, we show that Green’s functions, represented by Will’s result, can be expressed in closed form (without multipole expansion), which is more useful. In particular, they can be integrated out over the source (a thin disk in our case) to yield good converging series both for the gravitational potential and for the dragging angular velocity. The procedure is demonstrated, in the first perturbation order, on the simplest case of a constant-density disk, including the physical interpretation of the results in terms of a one-component perfect fluid or a two-component dust in a circular orbit about the central black hole. Free parameters are chosen in such a way that the resulting black hole has zero angular momentum but non-zero angular velocity, as it is just carried along by the dragging effect of the disk.},

doi = {10.3847/1538-4365/AA876B},

journal = {Astrophysical Journal, Supplement Series},

number = 1,

volume = 232,

place = {United States},

year = {Fri Sep 01 00:00:00 EDT 2017},

month = {Fri Sep 01 00:00:00 EDT 2017}

}