# The Algebraic-Hyperbolic Approach to the Linearized Gravitational Constraints on a Minkowski Background

## TL;DR

This paper investigates an algebraic-hyperbolic method for solving Einstein's gravitational constraints, revealing limitations in its application to linearized Minkowski space despite its success with nonlinear perturbations of black holes.

## Contribution

It introduces a novel algebraic-hyperbolic approach to Einstein's constraints and analyzes its applicability to linearized Minkowski space, highlighting fundamental limitations.

## Key findings

- No suitable Cauchy hypersurfaces exist for the well-posedness of the linearized algebraic-hyperbolic problem in Minkowski space.
- The method is well posed for nonlinear perturbations of Schwarzschild black holes.
- The approach offers potential for binary black hole data construction, despite limitations in linearized Minkowski space.

## Abstract

An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to solving the constraints of Einstein's equations which does not involve elliptic equations and has potential importance for the construction of binary black hole data. In order to shed light on the underpinnings of this approach, we consider its application to obtain solutions of the constraints for linearized perturbations of Minkowski space. In that case, we find the surprising result that there are no suitable Cauchy hypersurfaces in Minkowski space for which the linearized algebraic-hyperbolic constraint problem is well posed.