Backreaction for Einstein-Rosen waves coupled to a massless scalar field

Abstract

We present a one-parameter family of exact solutions to Einstein equations that may be used to study the nature of the Green-Wald backreaction framework. Our explicit example is a family of Einstein-Rosen waves coupled to a massless scalar field. This solution may be reinterpreted as a generalized three-torus polarized Gowdy cosmology with scalar and gravitational waves. We use it to illustrate essential properties of the Green-Wald approach. Among other things we show that within our model the Green-Wald framework uniquely determines backreaction for finite size inhomogeneities on a predefined background. The results agree with those calculated in the Charach-Malin approach. In the vacuum limit, the Green-Wald, the Charach-Malin and the Isaacson method imply identical backreaction as expected.Comment: 26 pages; minor changes to match published versio

Standing waves in general relativity

“…The effective energy-momentum tensor corresponds to the superposition of ingoing and outgoing gravitational radiation (and to the superposition of ingoing and outgoing waves of the scalar field). This remains true if the solution is rewritten as the three-torus Gowdy cosmology [15]. The properties of the background metric confirm our interpretation of solutions studied in [15].…”

“…This remains true if the solution is rewritten as the three-torus Gowdy cosmology [15]. The properties of the background metric confirm our interpretation of solutions studied in [15]. Although the background metric has a remarkably simple form, it has not been studied extensively in literature before (except the Morgan's article [16] where it appears indirectly and the Kramer's article [17] where spherically symmetric equivalent of this metric is derived)…”

“…The parameter β controls an amplitude of the scalar field. If β = 0, then the one-parameter family of solutions studied in [15] satisfies vacuum Einstein's equations. Therefore, the parameter α is related to the amplitude of the gravitational radiation.…”

“…For the one-parameter family of solutions studied in the article [15], the Cenergy (cylindrical energy as defined by Thorne [19]) is constant 'on average' for a fixed ρ (at nodes it is strictly constant). Therefore, there is no energy transfer.…”

An hyperbolic system of P.D.E. relevant in general relativity

“…This case for its simplicity has been, and still is, the object, in various contexts, of many papers we quote, among others [26], [27], [28], [5] and [17]. In this paper we consider the case ω = 0.…”