# Metric Cartesian mechanics of nonlocal energies with tensor internal tensions modifies Navier-Stokes dynamics

## TL;DR

This paper proposes a gauge-invariant tensor-based framework for nonlocal energies in continuum mechanics, modifying Navier-Stokes dynamics to incorporate nonlocal matter extensions and internal tensions, suggesting a shift from Newtonian models.

## Contribution

It introduces a novel tensorial formalism for nonlocal energies and inertial densities, extending classical mechanics and fluid dynamics with gauge invariance and nonlocal effects.

## Key findings

- Tensor inertial feedback relates to non-Newtonian accelerations.
- Modified Navier-Stokes equations incorporate nonlocal matter extensions.
- Proposes replacing Newtonian space with a Cartesian matter-extension model.

## Abstract

We introduce the gauge-invariant vector dynamics of continuous inertial densities through the metric formalism for extended mechanical charges. Ricci scalar density is related to invariant sum of inertial and gravitational mass densities of nonlocal matter-extension. Such a Cartesian continuum of gravitating inertial densities is self-governed by internal tensor tensions toward a static equilibrium state with a Euclidean material 3-space under the equivalence of inertial and gravitational densities of extended masses. External forces and local frictions transform the self-dynamics of an elementary closed continuum into a forced motion of still adaptive energy flows, where high-order space-time derivatives can provide non-Newtonian self-accelerations. If such tensor inertial feedback with the inverse constant of Cavendish 1/G is justified by measurements for the modified Navier-Stokes equation, the Newton empty space model should be replaced by the Cartesian matter-extension for the non-local macroscopic world.