Moving Manifolds and General Relativity

TL;DR

This paper revises general relativity using calculus for moving surfaces, revealing it as an approximation within a broader framework that accounts for dynamic, higher-dimensional manifolds, and offers insights into cosmological phenomena and wave behavior.

Contribution

It introduces a new derivation of Einstein's equations based on moving manifold calculus, highlighting GR's limitations and proposing a broader geometric framework for understanding space-time.

Findings

GR is an approximation of moving manifold equations

CMS predicts fluctuations between inflation and collapse

Solutions suggest wave-like behavior of moving points

Abstract

We revise general relativity (GR) from the perspective of calculus for moving surfaces (CMS). While GR is intrinsically constructed in pseudo-Riemannian geometry, a complete understanding of moving manifolds requires embedding in a higher dimension. It can only be defined by extrinsic Gaussian differential geometry and its extension to moving surfaces, known as CMS. Following the recent developments in CMS, we present a new derivation for the Einstein field equation and demonstrate the fundamental limitations of GR. Explicitly, we show that GR is an approximation of moving manifold equations and only stands for dominantly compressible space-time. While GR, with a cosmological constant, predicts an expanding universe, CMS shows fluctuation between inflation and collapse. We also show that the specific solution to GR with cosmological constant is constant mean curvature shapes. In the end, by presenting calculations for incompressible but deforming two-dimensional spheres, we indicate that material points moving with constant spherical velocities move like waves, strongly suggesting a resolution of the wave-corpuscular dualism problem.