# Initial conditions and degrees of freedom of non-local gravity

## TL;DR

This paper reformulates non-local gravity with exponential operators into a local higher-dimensional system, enabling analysis of initial conditions and degrees of freedom, and providing a framework for finding analytic solutions.

## Contribution

It establishes an equivalence between non-local gravity and a local higher-dimensional system, facilitating the study of initial conditions and degrees of freedom.

## Key findings

- Non-local scalar theories have two initial conditions in any dimension.
- Non-local gravitational theories have four initial conditions in any dimension.
- The number of degrees of freedom in four-dimensional non-local gravity is eight.

## Abstract

We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor. Thanks to this reformulation, we use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates. This method, first illustrated with a scalar field theory and then applied to gravity, allows one to solve the Cauchy problem and count the number of initial conditions and of non-perturbative degrees of freedom, which is finite. In particular, the non-local scalar and gravitational theories with exponential operators are characterized by, respectively, two and four initial conditions in any dimension and, respectively, by one and eight degrees of freedom in four dimensions. The fully covariant equations of motion are written in a form convenient to find analytic non-perturbative solutions.