Who's afraid of a negative lapse?

We rederive the Arnowitt-Deser-Misner equations in a framework in which the zeros of the lapse are innocuous, whether with or without changes of sign. We further develop and analyse a covariantized version of the Anderson-York equations, which provide a well posed system of tensorial evolution equations with freely prescribable shift vector and densitised lapse. The causality properties of the resulting equations are explored. We show how to relate solutions of the Anderson-York equations to the maximal globally hyperbolic development of the initial data.