# Null geodesics and embedding diagrams of the interior Schwarzschild--de Sitter spacetimes with uniform density

## Abstract

Null geodesics and embedding diagrams of central planes in the ordinary space geometry and the optical reference geometry of the interior Schwarzschild--de Sitter spacetimes with uniform density are studied. For completeness, both positive and negative values of the cosmological constant are considered. The null geodesics are restricted to the central planes of these spacetimes, and their properties can be reflected by an 'effective potential.' If the interior spacetime is extremely compact, the effective potential has a local maximum corresponding to a stable circular null geodesic around which bound null geodesics are concentrated. The upper limit on the size of the interior spacetimes containing bound null geodesics is R=3M, independently of the value of the cosmological constant. The embedding diagrams of the central planes of the ordinary geometry into three-dimensional Euclidean space are well defined for the complete interior of all spacetimes with a repulsive cosmological constant, but the planes cannot be embedded into the Euclidean space in the case of spacetimes with subcritical values of an attractive cosmological constant. On the other hand, the embedding diagrams of the optical geometry are well defined for all of the spacetimes, and the turning points of these diagrams correspond to the radii ofmore »

- Authors:

- Publication Date:

- Sponsoring Org.:
- (US)

- OSTI Identifier:
- 40230591

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 64; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.64.044004; Othernumber: PRVDAQ000064000004044004000001; 054114PRD; PBD: 15 Aug 2001; Journal ID: ISSN 0556-2821

- Publisher:
- The American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COSMOLOGICAL CONSTANT; EUCLIDEAN SPACE; GEODESICS; GEOMETRY

### Citation Formats

```
Stuchlik, Zdenek, Hledik, Stanislav, Soltes, Jiri, and Ostgaard, Erlend.
```*Null geodesics and embedding diagrams of the interior Schwarzschild--de Sitter spacetimes with uniform density*. United States: N. p., 2001.
Web. doi:10.1103/PhysRevD.64.044004.

```
Stuchlik, Zdenek, Hledik, Stanislav, Soltes, Jiri, & Ostgaard, Erlend.
```*Null geodesics and embedding diagrams of the interior Schwarzschild--de Sitter spacetimes with uniform density*. United States. doi:10.1103/PhysRevD.64.044004.

```
Stuchlik, Zdenek, Hledik, Stanislav, Soltes, Jiri, and Ostgaard, Erlend. Wed .
"Null geodesics and embedding diagrams of the interior Schwarzschild--de Sitter spacetimes with uniform density". United States. doi:10.1103/PhysRevD.64.044004.
```

```
@article{osti_40230591,
```

title = {Null geodesics and embedding diagrams of the interior Schwarzschild--de Sitter spacetimes with uniform density},

author = {Stuchlik, Zdenek and Hledik, Stanislav and Soltes, Jiri and Ostgaard, Erlend},

abstractNote = {Null geodesics and embedding diagrams of central planes in the ordinary space geometry and the optical reference geometry of the interior Schwarzschild--de Sitter spacetimes with uniform density are studied. For completeness, both positive and negative values of the cosmological constant are considered. The null geodesics are restricted to the central planes of these spacetimes, and their properties can be reflected by an 'effective potential.' If the interior spacetime is extremely compact, the effective potential has a local maximum corresponding to a stable circular null geodesic around which bound null geodesics are concentrated. The upper limit on the size of the interior spacetimes containing bound null geodesics is R=3M, independently of the value of the cosmological constant. The embedding diagrams of the central planes of the ordinary geometry into three-dimensional Euclidean space are well defined for the complete interior of all spacetimes with a repulsive cosmological constant, but the planes cannot be embedded into the Euclidean space in the case of spacetimes with subcritical values of an attractive cosmological constant. On the other hand, the embedding diagrams of the optical geometry are well defined for all of the spacetimes, and the turning points of these diagrams correspond to the radii of the circular null geodesics. All the embedding diagrams, for both the ordinary and optical geometry, are smoothly matched to the corresponding embedding diagrams of the external vacuum Schwarzschild--de Sitter spacetimes.},

doi = {10.1103/PhysRevD.64.044004},

journal = {Physical Review D},

issn = {0556-2821},

number = 4,

volume = 64,

place = {United States},

year = {2001},

month = {8}

}