A family of solutions to the inverse problem in gravitation: building a theory around a metric

TL;DR

This paper introduces a method to construct scalar-tensor theories tailored to specific metrics, enabling the design of gravitational models that exactly match pre-selected geometries, including non-Kerr black hole spacetimes.

Contribution

It presents a novel approach to build explicit gravitational theories around a given metric, diverging from traditional action-first methods.

Findings

Constructed scalar-tensor theories matching specified metrics.

Demonstrated the method with non-Kerr black hole spacetimes.

Provided a framework for inverse problem in gravitation.

Abstract

A method is presented to construct a particular, non-minimally coupled scalar-tensor theory such that a given metric is an exact vacuum solution in that theory. In contrast to the standard approach in studies of gravitational dynamics, where one begins with an action and then solves the equations of motion, this approach allows for an explicit theory to be built around some pre-specified geometry. Starting from a parameterized black hole spacetime with generic, non-Kerr hairs, it is shown how an overarching family of theories can be designed to fit the metric exactly.