The Mechanics of Spacetime - A Solid Mechanics Perspective on the Theory of General Relativity

TL;DR

This paper models spacetime as an elastic fabric, using solid mechanics principles to derive gravitational phenomena and connect elasticity with general relativity, especially in weak-field conditions.

Contribution

It introduces a novel elastic hyperplate model of spacetime, linking solid mechanics with general relativity through a Lagrangian approach and weak-field approximation.

Findings

Fabric's Lagrangian matches Einstein-Hilbert Lagrangian in free space.

Elastic properties relate to gravitational phenomena like waves and potential.

Solid mechanics tools can be applied to cosmological problems.

Abstract

We present an elastic constitutive model of gravity where we identify physical space with the mid-hypersurface of an elastic hyperplate called the "cosmic fabric" and spacetime with the fabric's world volume. Using a Lagrangian formulation, we show that the fabric's behavior as derived from Hooke's Law is analogous to that of spacetime per the Field Equations of General Relativity. The study is conducted in the limit of small strains, or analogously, in the limit of weak and nearly static gravitational fields. The Fabric's Lagrangian outside of inclusions is shown to have the same form as the Einstein-Hilbert Lagrangian for free space. Properties of the fabric such as strain, stress, vibrations, and elastic moduli are related to properties of gravity and space, such as the gravitational potential, gravitational acceleration, gravitational waves, and the energy density of free space. By introducing a mechanical analogy of General Relativity, we enable the application of Solid Mechanics tools to address problems in Cosmology.