Embeddings in Spacetimes Sourced by Scalar Fields

TL;DR

This paper extends the Campbell-Magaard embedding theorem to general relativity with scalar fields, providing new embeddings and exploring theoretical relationships with initial value problems.

Contribution

It formulates and proves an extension of the Campbell-Magaard theorem for scalar fields in general relativity, including applications to self-interacting fields and Brans-Dicke theory.

Findings

New embeddings for scalar field configurations

Extension of the theorem to Brans-Dicke theory

Insights into the relationship with Cauchy and initial value problems

Abstract

The extension of the Campbell-Magaard embedding theorem to general relativity with minimally-coupled scalar fields is formulated and proven. The result is applied to the case of a self-interacting scalar field for which new embeddings are found, and to Brans-Dicke theory. The relationship between Campbell-Magaard theorem and the general relativity, Cauchy and initial value problems is outlined.