Hyperbolicity Constraints in Extended Gravity Theories

TL;DR

This paper investigates how quadratic curvature modifications to Einstein's gravity affect the mathematical well-posedness of the theory, revealing that significant contributions lead to ill-posed initial value problems and non-hyperbolic behavior.

Contribution

It provides a general expression for the effective metric in field space and analyzes its properties in cosmological and spherically symmetric spacetimes, highlighting issues with non-Lorentzian signatures.

Findings

Quadratic terms cause the effective metric to become non-Lorentzian.

The initial value problem becomes ill-posed with significant quadratic contributions.

Such theories are only valid as perturbative extensions, not as alternatives to GR.

Abstract

We study the characteristic structure of the Einstein-Hilbert (EH) action when modifications of the form of $R^2,~ R_{\mu\nu}^2$, $R_{\mu\nu\rho\sigma}^2$ and $C_{\mu\nu\rho\sigma}^2$ are included. We show that when these quadratic terms are significant, the initial value problem is generically ill-posed. We do so by demanding the hyperbolicity of the effective metric for propagation of perturbations. Here, we find a general expression for the effective metric in field space and calculate it explicitly about the cosmological Friedman-Robertson-Walker (FRW) spacetime, and the spherically symmetric Schwarzschild solution. We find that when these quadratic contributions are non-negligible, the signature of the effective metric becomes non-Lorentzian and hence non-hyperbolic. As a consequence, we conclude that theories suggesting the inclusion of these terms can only be considered as a perturbative extension of the EH action and therefore cannot constitute a true alternative to general relativity (GR).