# Exactly Solvable Connections in Metric-Affine Gravity

## TL;DR

This paper develops a systematic method to exactly solve for the affine connection in various Metric-Affine gravity theories, including extensions of Einstein-Hilbert and f(R) models, using a set of three theorems.

## Contribution

It introduces a general framework with three theorems for solving the affine connection in Metric-Affine gravity, extending previous methods to more complex actions.

## Key findings

- Derived explicit solutions for the affine connection in multiple gravity models.
- Extended the solution method to actions with arbitrary dependence on the connection.
- Illustrated the approach with simple examples and discussed dynamical versus non-dynamical cases.

## Abstract

This article presents a systematic way to solve for the Affine Connection in Metric-Affine Geometry. We start by adding to the Einstein-Hilbert action, a general action that is linear in the connection and its partial derivatives and respects projective invariance. We then generalize the result for Metric-Affine f(R) Theories. Finally, we generalize even further and add an action (to the Einstein-Hilbert) that has an arbitrary dependence on the connection and its partial derivatives. We wrap up our results as three consecutive Theorems. We then apply our Theorems to some simple examples in order to illustrate how the procedure works and also discuss the cases of dynamical/non-dynamical connections.