# Initial-boundary-value problem of the self-gravitating scalar field in the Bondi-Sachs gauge

## Abstract

It is shown that, in the Bondi-Sachs gauge that fixes the speed of incoming light rays to the value 1, the Einstein equations coupled to a scalar field in spherical symmetry are cast into a symmetric-hyperbolic system of equations for the scalar field, lapse and shift as fundamental variables. In this system of equations, the lapse and shift are incoming characteristic fields, and the scalar field has three components: incoming, outgoing and static. A constraint-preserving boundary condition is prescribed by imposing the projection of the Einstein equation normal to the boundary at the outer value of the radial coordinate. The boundary condition specifies one of the two incoming metric fields. The remaining incoming metric field and the incoming scalar field component need to be specified arbitrarily. Numerical simulations of the scattering of the scalar field by a black hole in the nonlinear regime are presented that illustrate interesting facts about black-hole physics and the behavior of the characteristic variables of the problem.

- Authors:

- Department of Physics, Duquesne University, Pittsburgh, Pennsylvania 15282 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 21011086

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.75.044021; (c) 2007 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; BLACK HOLES; BOUNDARY CONDITIONS; BOUNDARY-VALUE PROBLEMS; COSMOLOGY; EINSTEIN FIELD EQUATIONS; NONLINEAR PROBLEMS; SCALAR FIELDS; SCATTERING; SYMMETRY

### Citation Formats

```
Frittelli, Simonetta, Gomez, Roberto, and Pittsburgh Supercomputing Center, 300 S. Craig Ave, Pittsburgh, Pennsylvania 15213.
```*Initial-boundary-value problem of the self-gravitating scalar field in the Bondi-Sachs gauge*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.044021.

```
Frittelli, Simonetta, Gomez, Roberto, & Pittsburgh Supercomputing Center, 300 S. Craig Ave, Pittsburgh, Pennsylvania 15213.
```*Initial-boundary-value problem of the self-gravitating scalar field in the Bondi-Sachs gauge*. United States. doi:10.1103/PHYSREVD.75.044021.

```
Frittelli, Simonetta, Gomez, Roberto, and Pittsburgh Supercomputing Center, 300 S. Craig Ave, Pittsburgh, Pennsylvania 15213. Thu .
"Initial-boundary-value problem of the self-gravitating scalar field in the Bondi-Sachs gauge". United States.
doi:10.1103/PHYSREVD.75.044021.
```

```
@article{osti_21011086,
```

title = {Initial-boundary-value problem of the self-gravitating scalar field in the Bondi-Sachs gauge},

author = {Frittelli, Simonetta and Gomez, Roberto and Pittsburgh Supercomputing Center, 300 S. Craig Ave, Pittsburgh, Pennsylvania 15213},

abstractNote = {It is shown that, in the Bondi-Sachs gauge that fixes the speed of incoming light rays to the value 1, the Einstein equations coupled to a scalar field in spherical symmetry are cast into a symmetric-hyperbolic system of equations for the scalar field, lapse and shift as fundamental variables. In this system of equations, the lapse and shift are incoming characteristic fields, and the scalar field has three components: incoming, outgoing and static. A constraint-preserving boundary condition is prescribed by imposing the projection of the Einstein equation normal to the boundary at the outer value of the radial coordinate. The boundary condition specifies one of the two incoming metric fields. The remaining incoming metric field and the incoming scalar field component need to be specified arbitrarily. Numerical simulations of the scattering of the scalar field by a black hole in the nonlinear regime are presented that illustrate interesting facts about black-hole physics and the behavior of the characteristic variables of the problem.},

doi = {10.1103/PHYSREVD.75.044021},

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

number = 4,

volume = 75,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}