A locally finite model for gravity

TL;DR

This paper proposes a classical, UV-finite toy model for gravity in 3+1 dimensions by representing matter as straight string defects embedded in a tessellated locally flat spacetime. String segments interact only at intersections or vanish into shorter pieces, producing new segments governed by a two-parameter moduli space at each junction, and yielding globally fractal growth. The framework leverages $SL(2,\mathbb{C})$ holonomies to encode string geometry and enforces causality via constraints on junctions and holonomies, while deliberately avoiding standard quantization paths in favor of a pre-quantization approach. The model offers a tractable arena for analytical GR questions and as a stepping stone toward quantum gravity, though it leaves open many issues, including positivity of deficit angles, completeness of evolution rules, and robust quantization schemes.

Abstract

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate matter of a form that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Globally, however, the model is not finite, because solutions tend to generate infinite fractals. The model is not (yet) quantized, but could serve as an interesting setting for analytical approaches to classical general relativity, as well as a possible stepping stone for quantum models. Details of its properties are explained, but some problems remain unsolved, such as a complete description of the most violent interactions, which can become quite complex.

Figures (9)

• Figure 1: $a$) 2 dimensional space surrounding a point particle. Points $A$ and $A'$ are identified. $b$) artist's impression of 2 dimensional space embedded in higher dimensions.
• Figure 2: $a$) Cross section of 3-space surrounding a positive string, with deficit angle $A\,A'$; $b$) Cross section of 3-space surrounding a negative string, showing a surplus angle $A\,A'$.
• Figure 3: junction connecting three strings. The holonomy $C_A$ of string $A$ equals the product $C_B\,C_C$ of the holonomies of strings $B$ and $C$
• Figure 4: Two types of string junctions. In case $a$, there is always an obtuse angle present. This is the case when all three strings are positive or all three are negative. In case $b$, one of the strings, in fact string #$B$, has a sign opposite to the sign that the other two have in common.
• Figure 5: Six string segments connected into a triangle, forming a 'localized' particle.