### 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 noncompact discrete subgroup of SL(2,R) whose structure is related to the Pell equation. A discrete version of the Bañados-Teitelboim-Zanelli (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.

Original language | English (US) |
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Article number | 106007 |

Journal | Physical Review D |

Volume | 94 |

Issue number | 10 |

DOIs | |

State | Published - Jan 1 2016 |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

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## Cite this

*Physical Review D*,

*94*(10), [106007]. https://doi.org/10.1103/PhysRevD.94.106007