Nonsmooth gravity

General Relativity is built on a contradiction. Spacetime is described by a Lorentzian manifold with at least twice differentiable metric, but the Lorentzian and differentiability conditions cannot be maintained together in general. For example, in cosmology, evolution breaks down once the differentiable and Lorentzian metric reaches the initial singularity.

The basic idea of nonsmooth gravity (inspired by nonsmooth mechancis) is to trade differentiability for better evolvability. The metric is allowed to be nonsmooth. The Lorentzian condition is treated as an unilateral i.e., inequality constraint on the metric. Solutions are found by applying the action principle under unilateral constraints Jia2025a, Jia2025b. This allows evolution pass singularity through nonsmooth bounces.

The new solutions can be viewed both as solutions to nonsmoothly extended classical gravitational theories, and as saddle points to Lorentzian quantum gravitational path integrals. As such they are relevant to both classical and quantum gravity.

In the future, I hope to understand better the phenomenological implications (if any), and develop techniques for equation-solving (e.g., numerical relativity with nonsmooth extensions).