On the construction of global models describing rotating bodies; uniqueness of the exterior gravitational field

TL;DR

This paper investigates the construction and uniqueness of exterior gravitational fields for rotating bodies in equilibrium, highlighting overdetermination issues and providing a proof of uniqueness using harmonic map methods.

Contribution

It analyzes the overdetermined nature of constructing global models and proves the uniqueness of exterior vacuum solutions for rotating bodies.

Findings

Construction of global models is overdetermined when boundary data are specified.

The exterior vacuum gravitational field is unique given the interior matter distribution.

Harmonic map formulation is used to prove the uniqueness of solutions.

Abstract

The problem of constructing global models describing isolated axially symmetric rotating bodies in equilibrium is analyzed. It is claimed that, whenever the global spacetime is constructed by giving boundary data on the limiting surface of the body and integrating Einstein's equations both inside and ouside the body, the problem becomes overdetermined. Similarly, when the spacetime describing the interior of the body is explicitly given,the problem of finding the exterior vacuum solution becomes overdetermined. We discuss in detail the procedure to be followed in order to construct the exterior vacuum field created by a given but arbitrary distribution of matter. Finally, the uniqueness of the exterior vacuum gravitational field is proven by exploiting the harmonic map formulation of the vacuum equations and the boundary conditions prescribed from the matching.