Evolution of segmented strings

TL;DR

This paper introduces a purely algebraic, discrete method for evolving segmented strings in de Sitter and anti-de Sitter spaces, applicable to rational points and related to black hole models, avoiding continuum spacetime.

Contribution

It presents a novel algebraic evolution rule for segmented strings in curved spacetimes, extending to rational points and connecting to discrete black hole models.

Findings

Discrete evolution rules are purely algebraic and applicable to rational points.

A simplified evolution rule for 3D anti-de Sitter space derived from Wess-Zumino-Witten equations.

A discrete BTZ black hole model that avoids the firewall paradox.

Abstract

I explain how to evolve segmented strings in de Sitter and anti-de Sitter spaces of any dimension in terms of forward-directed null displacements. The evolution is described entirely in terms of discrete hops which do not require a continuum spacetime. Moreover, the evolution rule is purely algebraic, so it can be defined not only on ordinary real de Sitter and anti-de Sitter, but also on the rational points of the quadratic equations that define these spaces. For three-dimensional anti-de Sitter space, a simpler evolution rule is possible that descends from the Wess-Zumino-Witten equations of motion. In this case, one may replace three-dimensional anti-de Sitter space by a non-compact discrete subgroup of SL(2,R) whose structure is related to the Pell equation. A discrete version of the BTZ black hole can be constructed as a quotient of this subgroup. This discrete black hole avoids the firewall paradox by a curious mechanism: even for large black holes, there are no points inside the horizon until one reaches the singularity.