# Well-posed constrained evolution of 3+1 formulations of general relativity

## Abstract

We present an analysis of well-posedness of constrained evolution of 3+1 formulations of general relativity. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of Arnowitt-Deser-Misner (ADM), Baumgarte-Shapiro-Shibata-Nakamura and other 3+1 formulations derived from the ADM formulation by adding combinations of constraints to the right-hand side of the ADM formulation and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical ADM and many other 3+1 formulations of general relativity.

- Authors:

- Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 (United States)
- (United States)
- Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 (United States)
- (Denmark)
- (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21010896

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.024026; (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; COSMOLOGY; DISTURBANCES; EINSTEIN FIELD EQUATIONS; FOUR-DIMENSIONAL CALCULATIONS; GAUGE INVARIANCE; GENERAL RELATIVITY THEORY; MATHEMATICAL SOLUTIONS; TRANSFORMATIONS

### Citation Formats

```
Paschalidis, Vasileios, Khokhlov, Alexei, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, Novikov, Igor, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, and Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810.
```*Well-posed constrained evolution of 3+1 formulations of general relativity*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.024026.

```
Paschalidis, Vasileios, Khokhlov, Alexei, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, Novikov, Igor, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, & Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810.
```*Well-posed constrained evolution of 3+1 formulations of general relativity*. United States. doi:10.1103/PHYSREVD.75.024026.

```
Paschalidis, Vasileios, Khokhlov, Alexei, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, Novikov, Igor, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, and Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810. Mon .
"Well-posed constrained evolution of 3+1 formulations of general relativity". United States.
doi:10.1103/PHYSREVD.75.024026.
```

```
@article{osti_21010896,
```

title = {Well-posed constrained evolution of 3+1 formulations of general relativity},

author = {Paschalidis, Vasileios and Khokhlov, Alexei and Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 and Novikov, Igor and Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen and Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810},

abstractNote = {We present an analysis of well-posedness of constrained evolution of 3+1 formulations of general relativity. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of Arnowitt-Deser-Misner (ADM), Baumgarte-Shapiro-Shibata-Nakamura and other 3+1 formulations derived from the ADM formulation by adding combinations of constraints to the right-hand side of the ADM formulation and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical ADM and many other 3+1 formulations of general relativity.},

doi = {10.1103/PHYSREVD.75.024026},

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

number = 2,

volume = 75,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}