# Schwarzschild tests of the Wahlquist-Estabrook-Buchman-Bardeen tetrad formulation for numerical relativity

## Abstract

A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift, which specify the evolution of the coordinates relative to the tetrad congruence, are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges ('Nester' and 'Lorentz') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.

- Authors:

- Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20774535

- 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.72.124014; (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; BLACK HOLES; BOUNDARY-VALUE PROBLEMS; COORDINATES; EINSTEIN FIELD EQUATIONS; FINITE DIFFERENCE METHOD; GEOMETRY; SCHWARZSCHILD METRIC; SPHERICAL CONFIGURATION; STABILITY; SYMMETRY

### Citation Formats

```
Buchman, L.T., Bardeen, J.M., and Physics Department, University of Washington, Seattle, Washington.
```*Schwarzschild tests of the Wahlquist-Estabrook-Buchman-Bardeen tetrad formulation for numerical relativity*. United States: N. p., 2005.
Web. doi:10.1103/PhysRevD.72.124014.

```
Buchman, L.T., Bardeen, J.M., & Physics Department, University of Washington, Seattle, Washington.
```*Schwarzschild tests of the Wahlquist-Estabrook-Buchman-Bardeen tetrad formulation for numerical relativity*. United States. doi:10.1103/PhysRevD.72.124014.

```
Buchman, L.T., Bardeen, J.M., and Physics Department, University of Washington, Seattle, Washington. Thu .
"Schwarzschild tests of the Wahlquist-Estabrook-Buchman-Bardeen tetrad formulation for numerical relativity". United States.
doi:10.1103/PhysRevD.72.124014.
```

```
@article{osti_20774535,
```

title = {Schwarzschild tests of the Wahlquist-Estabrook-Buchman-Bardeen tetrad formulation for numerical relativity},

author = {Buchman, L.T. and Bardeen, J.M. and Physics Department, University of Washington, Seattle, Washington},

abstractNote = {A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift, which specify the evolution of the coordinates relative to the tetrad congruence, are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges ('Nester' and 'Lorentz') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.},

doi = {10.1103/PhysRevD.72.124014},

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}

}