Time-reversal of Einstein-frame geometry: The curious case of phantom Chaplygin gas

The phantom Dirac--Born--Infeld (DBI) model exhibits superluminal propagation. In addition, the model can dynamically develop sudden singularities, where the pressure diverges, and evolution seemingly grinds to a halt. We demonstrate that the former property cures the latter, making all equations of motion nonsingular throughout. This is accomplished by promoting the induced metric defined by the DBI scalar field to the fundamental gravitational degree of freedom, relegating the Einstein-frame metric to the status of emergent. By choosing the induced (or causal-frame) metric's lapse function to be everywhere positive, we show that the Einstein-frame lapse function emerges as a dynamical field that smoothly changes sign. This has the effect of spontaneously reversing the flow of time for general relativity, as well as for any degrees of freedom minimally coupled to the Einstein-frame metric. The DBI scalar field couples to the induced metric, which we denote by the causal-frame metric, since its time is monotonically increasing. This realizes a transient mismatch in thermodynamic arrows of time. All degrees of freedom, including the Einstein-frame lapse function, have non-singular equations of motion in the causal frame. In a cosmological setting, the scale factor can smoothly bounce in causal frame, despite the Hubble expansion rate remaining positive definite in Einstein frame. Big Bang and Big Crunch singularities are thereby avoided for generic initial conditions.