[Paper Review] Transformations of units and world's geometry

Israel Quirós|ArXiv.org|Apr 5, 2000

Mathematics and Applications1 references3 citations

TL;DR

This paper argues that Weyl-integrable geometry (WIG) provides a consistent geometric framework for gravity because it remains invariant under point-dependent unit transformations—unlike Riemannian geometry, which breaks under such transformations. The key result is that spacetime singularities (e.g., in Schwarzschild black holes or Friedmann-Robertson-Walker cosmology) are artifacts of choosing an inconsistent geometric formulation, and when transformed via conformal rescaling, they become singularity-free wormhole-like geometries in WIG, suggesting WIG may naturally incorporate quantum effects.

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.

Motivation & Objective

• To address the inconsistency of general relativity under point-dependent unit transformations, which affect curvature scalars and thus physical laws.
• To argue that Riemannian geometry is physically inadequate due to its lack of invariance under unit transformations, leading to unphysical singularities.
• To propose Weyl-integrable geometry (WIG) as a more fundamental geometric framework that preserves physical laws under unit transformations.
• To show that singularities in Schwarzschild and FRW spacetimes are artifacts of the Riemannian formulation, not physical realities.
• To introduce 'geometrical relativity' as a natural extension of general relativity, incorporating invariance under unit transformations and potentially encoding quantum effects.

Proposed method

• Formalizing the invariance of physical laws under point-dependent unit transformations, particularly for length, time, and mass.
• Applying conformal rescaling (ĝ_ab = Ω²(x)g_ab) to map Riemannian spacetimes into Weyl-integrable ones, preserving spacetime coincidences.
• Using the Brans-Dicke action in both original and conformal frames to demonstrate how unit transformations affect the geometric structure of gravity.
• Deriving the Weyl-integrable formulation from the conformal transformation of the Brans-Dicke action, showing invariance of the underlying geometric postulate (Eq. 2.6).
• Analyzing the Schwarzschild and Friedmann-Robertson-Walker solutions under conformal rescaling to show that singularities disappear in the WIG framework.
• Proposing a new postulate of equivalence among Weyl-integrable spacetimes with different conformal factors, forming an equivalence class of physically indistinguishable geometries.

Experimental results

Research questions

• RQ1 Why do spacetime singularities appear in general relativity, and could they be a consequence of the geometric formulation rather than physical reality?
• RQ2 Is general relativity invariant under point-dependent transformations of units, and if not, what are the implications for physical laws?
• RQ3 Can Weyl-integrable geometry serve as a consistent geometric framework for gravity that remains invariant under unit transformations?
• RQ4 How does the conformal rescaling of the metric transform singularities in Schwarzschild and FRW spacetimes, and what does this imply for astrophysical observations?
• RQ5 Can the invariance under unit transformations in WIG naturally incorporate quantum effects, suggesting a deeper unification with quantum mechanics?

Key findings

• Spacetime singularities in the Schwarzschild solution are not physical but result from the use of Riemannian geometry, which is not invariant under point-dependent unit transformations.
• Under conformal rescaling (ĝ_ab = Ω²(x)g_ab), the Schwarzschild black hole is mapped into a singularity-free wormhole spacetime in Weyl-integrable geometry, with no change in spacetime coincidences.
• The Friedmann-Robertson-Walker cosmological model also transforms into a singularity-free configuration under the same conformal mapping, suggesting that big bang singularities may be artifacts of geometric choice.
• Weyl-integrable geometry preserves the fundamental geometric postulate (Eq. 2.6) under unit transformations, making it a consistent framework for physical laws, unlike Riemannian geometry.
• The paper introduces 'geometrical relativity' as a new principle: an infinite equivalence class of Weyl-integrable spacetimes with different conformal factors are observationally and physically equivalent, extending general covariance to include unit transformations.
• The results suggest that Weyl-integrable geometry may implicitly contain quantum effects of matter, offering a natural extension of general relativity that could resolve long-standing issues with singularities.