# The future is not always open

## TL;DR

This paper reveals that in low regularity Lorentzian spacetimes, fundamental causality concepts such as chronological futures can behave unexpectedly, challenging traditional assumptions and impacting synthetic causality approaches.

## Contribution

It characterizes spacetimes with non-open chronological futures and relates these phenomena to causal bubble structures and curve deformation properties.

## Key findings

- Chronological futures may be non-open in low regularity spacetimes.
- Differences between causality sets defined by Lipschitz and $C^1$-curves are demonstrated.
- Conditions for the occurrence of causal bubbles are characterized.

## Abstract

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise $C^1$-curves. By refining the notion of a causal bubble from [CG:12],we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.