# An hyperbolic system of P.D.E. relevant in general relativity

## TL;DR

This paper investigates a hyperbolic PDE system derived from Einstein's field equations, proving existence and uniqueness of small amplitude gravitational waves and constructing solutions for larger cases, relevant in general relativity.

## Contribution

It applies the implicit function theorem to establish existence and uniqueness results for gravitational waves in a hyperbolic PDE system related to Einstein's equations.

## Key findings

- Proved existence and uniqueness of small amplitude gravitational waves.
- Constructed solutions for larger amplitude cases.
- Established a theorem for stationary problem solutions.

## Abstract

Assuming as starting point the validity of the Einstein-Rosen metric, we study the hyperbolic system of P.D.E. to which the Einstein's field's equations can be reduced. We prove using the implicit function theorem in Banach spaces, the existence and uniqueness of gravitational waves of small amplitude. A class of solutions, not necessarily small is also constructed. In the last Section a theorem of existence and uniqueness is given for the corresponding stationary problem.