# A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case

## Abstract

Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimes – particularly around black holes – we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional Euclidean spaces. The general formula is applied to (d ≥ 2)-dimensional de Sitter spacetime, and the scalar Green's function is demonstrated to be sourced by a line emanating infinitesimally close to the origin of the ambient (d + 1)-dimensional Minkowski spacetime and piercing orthogonally through the de Sitter hyperboloids of all finite sizes. This method does not require solving the de Sitter wave equation directly. Only the zero mode solution to an ordinary differential equation, the “wave equation” perpendicular to the hyperboloid – followed by a one-dimensional integral – needs to be evaluated. A topological obstruction to the general construction is also discussed by utilizing it to derive a generalized Green's function of the Laplacian on the (d ≥ 2)-dimensionalmore »

- Authors:

- Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (United States)

- Publication Date:

- OSTI Identifier:
- 22306039

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BLACK HOLES; DE SITTER GROUP; DE SITTER SPACE; EUCLIDEAN SPACE; GREEN FUNCTION; LAPLACIAN; MINKOWSKI SPACE; SPACE-TIME; TOPOLOGY; WAVE EQUATIONS

### Citation Formats

```
Chu, Yi-Zen.
```*A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case*. United States: N. p., 2014.
Web. doi:10.1063/1.4895506.

```
Chu, Yi-Zen.
```*A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case*. United States. doi:10.1063/1.4895506.

```
Chu, Yi-Zen. Mon .
"A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case". United States. doi:10.1063/1.4895506.
```

```
@article{osti_22306039,
```

title = {A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case},

author = {Chu, Yi-Zen},

abstractNote = {Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimes – particularly around black holes – we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional Euclidean spaces. The general formula is applied to (d ≥ 2)-dimensional de Sitter spacetime, and the scalar Green's function is demonstrated to be sourced by a line emanating infinitesimally close to the origin of the ambient (d + 1)-dimensional Minkowski spacetime and piercing orthogonally through the de Sitter hyperboloids of all finite sizes. This method does not require solving the de Sitter wave equation directly. Only the zero mode solution to an ordinary differential equation, the “wave equation” perpendicular to the hyperboloid – followed by a one-dimensional integral – needs to be evaluated. A topological obstruction to the general construction is also discussed by utilizing it to derive a generalized Green's function of the Laplacian on the (d ≥ 2)-dimensional sphere.},

doi = {10.1063/1.4895506},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 9,

volume = 55,

place = {United States},

year = {2014},

month = {9}

}