Perturbative Analysis of the Two-body Problem in (2+1)-AdS gravity

TL;DR

This paper develops a perturbative approach to analyze the interaction between point sources and (2+1)-dimensional Anti-de Sitter gravity, linking the problem to polydromic mappings and geodesic motion on a constrained hypersurface.

Contribution

It introduces a novel perturbative scheme for point source interactions in AdS gravity using polydromic mappings and explores solutions constrained by monodromy groups.

Findings

Derived a perturbative scheme for point source interactions in AdS gravity.

Established a connection between the interaction problem and polydromic mappings with O(2,2) monodromies.

Obtained solutions perturbatively in the cosmological constant.

Abstract

We derive a perturbative scheme to treat the interaction between point sources and AdS-gravity. The interaction problem is equivalent to the search of a polydromic mapping $X^A= X^A(x^\mu)$, endowed with 0(2,2) monodromies, between the physical coordinate system and a Minkowskian 4-dimensional coordinate system, which is however constrained to live on a hypersurface. The physical motion of point sources is therefore mapped to a geodesic motion on this hypersuface. We impose an instantaneous gauge which induces a set of equations defining such a polydromic mapping. Their consistency leads naturally to the Einstein equations in the same gauge. We explore the restriction of the monodromy group to O(2,1), and we obtain the solution of the fields perturbatively in the cosmological constant.