Conformal transformations of metric spaces and Lorentzian pre-length spaces

Abstract

We introduce conformal transformations in the synthetic setting of metric spaces and Lorentzian (pre-)length spaces. Our main focus lies on the Lorentzian case, where, motivated by the need to extend classical notions to spaces of low regularity, we provide the first consistent notion of conformal length, and analyse its fundamental properties. We prove that the conformal time separation function $\tau_{\Omega}$ (and the causal structure it induces) yields a Lorentzian pre-length structure if the original space is intrinsic and strongly causal. This allows us to construct a notion of conformal transformation between spaces within this class, yielding an equivalence relation. As applications, we show that the conformal length functional agrees with the standard conformal length of (strongly causal) spacetimes. We also prove conformal invariance of angles and causality conditions, give a characterisation of global hyperbolicity via finiteness of $\tau_{\Omega}$ for all conformal factors, and establish the behaviour of the Lorentzian Hausdorff measure defined in [MS22a] under conformal changes. Moreover, we apply the same methods to the metric case, which is of interest in its own right. This is exemplified by proving an analog of the Nomizu--Ozeki theorem for metric length spaces, which has the advantage that the resulting complete space is conformally related to the original space.