Transformations of units and world's geometry

TL;DR

This paper advocates for Weyl-integrable geometry as a consistent framework for gravity, emphasizing its invariance under point-dependent unit transformations and its implications for singularities and quantum effects.

Contribution

It demonstrates that Weyl-integrable geometry, unlike Riemann geometry, provides a unit-invariant formulation of gravitational laws, offering new insights into singularities and quantum effects.

Findings

Weyl-integrable geometry is invariant under point-dependent unit transformations.

Riemann geometry does not support invariance under units transformations.

Singularities are linked to incorrect geometric formulations of gravity.

Abstract

The issue of the transformations of units is treated, mainly, in a geometrical context. It is shown that Weyl-integrable geometry is a consistent framework for the formulation of the gravitational laws since the basic law on which this geometry rests is invariant under point-dependent transformations of units. Riemann geometry does not fulfill this requirement. Spacetime singularities are then shown to be a consequence of a wrong choice of the geometrical formulation of the laws of gravitation. This result is discussed, in particular, for the Schwrazschild black hole and for Friedmann-Robertson-Walker cosmology. Arguments are given that point at Weyl-integrable geometry as a geometry implicitly containing the quantum effects of matter. The notion of geometrical relativity is presented. This notion may represent a natural extension of general relativity to include invariance under the group of units transformations.