Embeddings for General Relativity

Abstract

We present a systematic approach to embed $n$-dimensional vacuum general relativity in an $(n+1)$-dimensional pseudo-Riemannian spacetime whose source is either a (non)zero cosmological constant or a scalar field minimally-coupled to Einstein gravity. Our approach allows us to generalize a number of results discussed in the literature. We construct {\it all} the possible (physically distinct) embeddings in Einstein spaces, including the Ricci-flat ones widely discussed in the literature. We examine in detail their generalization, which - in the framework under consideration - are higher-dimensional spacetimes sourced by a scalar field with flat (constant $\ne 0$) potential. We use the Kretschmann curvature scalar to show that many embedding spaces have a physical singularity at some finite value of the extra coordinate. We develop several classes of embeddings that are free of singularities, have distinct non-vanishing self-interacting potentials and are continuously connected (in various limits) to Einstein embeddings. We point out that the induced metric possesses scaling symmetry and, as a consequence, the effective physical parameters (e.g., mass, angular momentum, cosmological constant) can be interpreted as functions of the extra coordinate.

Accepted for publication in Classical and Quantum Gravity

References in corpus (6)

• Decaying Dark Energy in Higher-Dimensional Gravity
• An Embedding for General Relativity and its Implications for New Physics
• In Defense of Campbell's Theorum as a Frame for New Physics
• The Schwarzschild-de Sitter solution in five-dimensional general relativity briefly revisited
• Embeddings for 4D Einstein equations with a cosmological constant
• The Embedding of General Relativity in Five-Dimensional Canonical Space: A Short History and a Review of Recent Physical Progress

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• FLRW embeddings in ${\mathbb{R}}^{n+2}$, differential geometry and conformal photon propagator