Threshold for strong hyperbolicity of modified-harmonic formulations with higher-derivative corrections

Determine rigorous quantitative conditions or bounds on the size of the higher-derivative contributions in modified-harmonic-gauge formulations of gravitational effective field theories—specifically, in the Einstein–Maxwell effective field theory—under which the equations remain strongly hyperbolic and the initial value problem stays well-posed, and identify when strong hyperbolicity fails as these contributions grow.

Background

The thesis proves that the modified harmonic gauge yields a strongly hyperbolic, and hence well-posed, initial value formulation of the Einstein–Maxwell effective field theory when the higher-derivative terms are small (weak coupling/regime of validity).

However, the analysis does not provide a quantitative boundary for how large the higher-derivative terms can be before strong hyperbolicity and well-posedness break down. Establishing such a threshold is important both for mathematical control and for assessing the robustness of simulations and theoretical predictions beyond the strict weak-coupling regime.

References

However, it is an open question as to how large the higher derivative terms can be before the formulation loses its strong hyperbolicity and is no longer well-posed.

— Gravitational Effective Field Theories and Black Hole Mechanics  (2411.14023 - Davies, 2024) in Chapter Concluding Remarks

It remains an open question how far these flat-space scalar field results extend beyond flat space and to more general EFTs, in particular, those involving a dynamical metric.

— Nonlinear evolution in Galileon EFTs: Regularization and screening  (2609.05407 - Thaalba et al., 4 Sep 2026) in Section Conclusions