Lorentzian distance functions in contact geometry

Abstract

An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.

Remarks on the oscillation energy of Legendrian isotopies

“…Using the dichotomy established in [RZ20], the paper [DRS21] proves this pseudometric is non-degenerate on all isotopy classes of compact Legendrians, in the class of contact manifolds where the Chekanov-Eliashberg DGA technology 2 is established. The non-degeneracy of the metric was established previously in [Ush21] for hypertight Legendrians, and in [Hed21] for the case of orderable Legendrian isotopy classes. One of the results in this paper is the construction of a Legendrian isotopy class for which the pseudo-metric is degenerate (Theorem 2).…”

On the space of compact diamonds of Lorentzian length spaces

“…As an example of the value of building a causal theory on axiomatic grounds we cite the causet approach to quantum gravity (see [46] and references therein), in which no spacetime metric is considered at all. The existence of a time separation function, as proposed in [28], opens the scope of applications to settings where no regular spacetime metric is given a priori, as is the case of Lorentzian manifolds with timelike boundary [2], causal completions of globally hyperbolic spacetimes [1] or even some class of contact structures [24].…”

On the space of compact diamonds of Lorentzian length spaces

“…As an example of the value of building a causal theory on axiomatic grounds we cite the causet approach to Quantum Gravity (see [50] and references therein), in which no spacetime metric is considered at all. The existence of a time separation function, as proposed in [33], opens the scope of applications to settings where no regular spacetime metric is given a priori, as is the case of Lorentzian manifolds with timelike boundary [2], causal completions of globally hyperbolic spacetimes [1] or even some class of contact structures [27].…”