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Title: More on the covariant retarded Green's function for the electromagnetic field in de Sitter spacetime

Abstract

In a recent paper 2 it was shown in examples that the covariant retarded Green's functions in certain gauges for electromagnetism and linearized gravity can be used to reproduce field configurations correctly in spite of the spacelike nature of past infinity in de Sitter spacetime. In this paper we extend the work of Ref. 2 concerning the electromagnetic field and show that the covariant retarded Green's function with an arbitrary value of the gauge parameter reproduces the electromagnetic field from two opposite charges at antipodal points of de Sitter spacetime.

Authors:
; ;  [1]
  1. Department of Mathematics, University of York, Heslington, York YO10 5DD (United Kingdom)
Publication Date:
OSTI Identifier:
21313421
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 80; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.80.107502; (c) 2009 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; CONFIGURATION; DE SITTER SPACE; ELECTROMAGNETIC FIELDS; ELECTROMAGNETISM; GRAVITATION; GREEN FUNCTION; SPACE-TIME

Citation Formats

Higuchi, Atsushi, Lee, Yen Cheong, and Nicholas, Jack R. More on the covariant retarded Green's function for the electromagnetic field in de Sitter spacetime. United States: N. p., 2009. Web. doi:10.1103/PHYSREVD.80.107502.
Higuchi, Atsushi, Lee, Yen Cheong, & Nicholas, Jack R. More on the covariant retarded Green's function for the electromagnetic field in de Sitter spacetime. United States. doi:10.1103/PHYSREVD.80.107502.
Higuchi, Atsushi, Lee, Yen Cheong, and Nicholas, Jack R. 2009. "More on the covariant retarded Green's function for the electromagnetic field in de Sitter spacetime". United States. doi:10.1103/PHYSREVD.80.107502.
@article{osti_21313421,
title = {More on the covariant retarded Green's function for the electromagnetic field in de Sitter spacetime},
author = {Higuchi, Atsushi and Lee, Yen Cheong and Nicholas, Jack R.},
abstractNote = {In a recent paper 2 it was shown in examples that the covariant retarded Green's functions in certain gauges for electromagnetism and linearized gravity can be used to reproduce field configurations correctly in spite of the spacelike nature of past infinity in de Sitter spacetime. In this paper we extend the work of Ref. 2 concerning the electromagnetic field and show that the covariant retarded Green's function with an arbitrary value of the gauge parameter reproduces the electromagnetic field from two opposite charges at antipodal points of de Sitter spacetime.},
doi = {10.1103/PHYSREVD.80.107502},
journal = {Physical Review. D, Particles Fields},
number = 10,
volume = 80,
place = {United States},
year = 2009,
month =
}
  • We demonstrate in examples that the covariant retarded Green's functions in electromagnetism and linearized gravity work as expected in de Sitter spacetime. We first clarify how retarded Green's functions should be used in spacetimes with spacelike past infinity such as de Sitter spacetime. In particular, we remind the reader of a general formula which gives the field for given initial data on a Cauchy surface and a given source (a charge or stress-energy tensor distribution) in its future. We then apply this formula to three examples: (i) electromagnetism in the future of a Cauchy surface in Minkowski spacetime, (ii) electromagnetismmore » in de Sitter spacetime, and (iii) linearized gravity in de Sitter spacetime. In each example the field is reproduced correctly as predicted by the general argument. In the third example we construct a linearized gravitational field from two equal point masses located at the 'North and South Poles' which is nonsingular on the cosmological horizon and satisfies a covariant gauge condition and show that this field is reproduced by the retarded Green's function with corresponding gauge parameters.« less
  • 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 ismore » 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.« less
  • The gravitational field generated by a mass term and the initial surface through covariant retarded Green's function for linearized gravity in de Sitter spacetime was studied recently [4, 5] with the covariant gauges set to β = 2/3 and α = 5/3. In this paper we extend the work to restore the gauge parameter α in the field coming from the initial data using the method of shifting the parameter. The α terms in the initial field cancels exactly with the one coming from the source term. Consequently, the correct field configuration, with two equal mass points moving in itsmore » geodesic, one located at the North pole and another one located at the South pole, is reproduced in the whole manifold of de Sitter spacetime.« less