Carnot-Carathéodory metrics and spacetime

The Lorentzian metric of general relativity is extraordinarily successful: it governs causal accessibility, encodes the equivalence principle, and provides the standard analytic setting for relativistic field equations. But there are mathematical reasons to ask whether Lorentzian spacetime is the right primitive geometric object. Its tangent space is an algebraic first-order surrogate, not a metric blow-up of the local geometry. In Riemannian geometry, tangent spaces arise by canonical dilation and approximation of the underlying metric space. No analogous positive metric dilation exists for a Lorentzian manifold, no positive metric notion by which nearby events are made close and distant events far away. This is particularly evident in the null sector: a null geodesic traverses the entire spacetime in no proper time at all. In what sense, then, is the Lorentzian tangent space an "approximation"? Indeed, the natural blow-up constructions associated with Lorentzian geometry lead instead toward null-geodesic structures, such as Penrose limits and plane waves, rather than ordinary Lorentzian tangent spaces attached algebraically to events. This talk proposes a different foundation for spacetime geometry, based on contact–Carnot–Carathéodory state spaces. The primitive local object is a normalized space of physical states, because a laboratory must do more than identify nearby events: it must prepare and compare states of motion, including phase, action, momentum, frequency, and transverse variations. The infinitesimal geometry of such data is naturally contact-Heisenberg, with a positive horizontal metric measuring the cost of transverse distinguishability and whose Reeb flow gives the causal evolution. The analytic mechanism for fields is the canonical horizontal sub-Laplacian on the massive contact–CC shell. For positive mass this operator is hypoelliptic upstairs, not hyperbolic downstairs. The wave operator, insofar as it appears downstairs, appears only through a stationary-phase or degeneration limit, while causal structure is recovered from the associated Reeb dynamics.