# Further applications of harmonic mappings of Riemannian manifolds to gravitational fields

## Abstract

We consider static, spherically-symmetric and stationary, axially-symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton--Jacobi techniques, we obtain the geodesics and construct the resulting space--time geometries. For static space--times with spherical symmetry we obtain the well-known solutions of Schwarzschild, Reissner--Nordstroem, and Janis--Newman--Winicour, where in the latter two cases the source for the geometry is the static, source-free Maxwell field and the massless scalar field respectively. When we consider the source to be a triad of massless scalar fields with the internal symmetry SU(2) x SU(2), we find that the space--time geometry is once again that of Janis--Newman--Winicour. Finally we formulate the stationary, axisymmetric gravitational field problem in terms of composite mappings and obtain the Weyl, Papapetrou, and the generalized Lewis and van Stockum classes of axisymmetric solutions.

- Authors:

- Publication Date:

- Research Org.:
- Department of Physics, Middle East Technical University, Ankara, Turkey

- OSTI Identifier:
- 7119998

- Resource Type:
- Journal Article

- Journal Name:
- J. Math. Phys. (N.Y.); (United States)

- Additional Journal Information:
- Journal Volume: 18:4

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GRAVITATIONAL FIELDS; RIEMANN SPACE; TOPOLOGICAL MAPPING; EINSTEIN FIELD EQUATIONS; GENERAL RELATIVITY THEORY; GEODESICS; SU-2 GROUPS; EQUATIONS; FIELD EQUATIONS; FIELD THEORIES; LIE GROUPS; MATHEMATICAL SPACE; SPACE; SU GROUPS; SYMMETRY GROUPS; TRANSFORMATIONS; 657003* - Theoretical & Mathematical Physics- Relativity & Gravitation

### Citation Formats

```
Eris, A.
```*Further applications of harmonic mappings of Riemannian manifolds to gravitational fields*. United States: N. p., 1977.
Web. doi:10.1063/1.523311.

```
Eris, A.
```*Further applications of harmonic mappings of Riemannian manifolds to gravitational fields*. United States. https://doi.org/10.1063/1.523311

```
Eris, A. Fri .
"Further applications of harmonic mappings of Riemannian manifolds to gravitational fields". United States. https://doi.org/10.1063/1.523311.
```

```
@article{osti_7119998,
```

title = {Further applications of harmonic mappings of Riemannian manifolds to gravitational fields},

author = {Eris, A},

abstractNote = {We consider static, spherically-symmetric and stationary, axially-symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton--Jacobi techniques, we obtain the geodesics and construct the resulting space--time geometries. For static space--times with spherical symmetry we obtain the well-known solutions of Schwarzschild, Reissner--Nordstroem, and Janis--Newman--Winicour, where in the latter two cases the source for the geometry is the static, source-free Maxwell field and the massless scalar field respectively. When we consider the source to be a triad of massless scalar fields with the internal symmetry SU(2) x SU(2), we find that the space--time geometry is once again that of Janis--Newman--Winicour. Finally we formulate the stationary, axisymmetric gravitational field problem in terms of composite mappings and obtain the Weyl, Papapetrou, and the generalized Lewis and van Stockum classes of axisymmetric solutions.},

doi = {10.1063/1.523311},

url = {https://www.osti.gov/biblio/7119998},
journal = {J. Math. Phys. (N.Y.); (United States)},

number = ,

volume = 18:4,

place = {United States},

year = {1977},

month = {4}

}