Classical Gravitation as free Membrane Dynamics

TL;DR

This paper explores a novel formulation of General Relativity by modeling 4D space-time as a deformable membrane embedded in higher-dimensional flat space, introducing new embeddings and variational principles that reproduce Einstein's equations.

Contribution

It introduces new classes of free embeddings inspired by Nash's theory, enabling physically realistic deformations and gravitational wave descriptions within a membrane framework.

Findings

Explicit embeddings for Schwarzschild and other space-times.

Demonstration of gravitational waves as membrane deformations.

New variational principles recovering Einstein's equations.

Abstract

The formulation of General Relativity in which the 4-dimensional space-time is embedded in a flat host space of higher dimension is reconsidered. New classes of embeddings (modeled after Nash's classical free embeddings) are introduced. They present the important advantage of being deformable and therefore physically realistic. Explicit examples of embeddings whose deformations DO describe gravitational waves around their respective backgrounds are given for several space-times, including the Schwarzschild black hole. New variational principles which give back Einstein's General Relativity are proposed. In this framework, the 4-D space-time is a membrane moving in a flat host space.