Moving Manifolds and General Relativity (2406.08382v1)

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.