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

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.