Causality of spacetimes admitting a parallel null vector and weak KAM theory

Abstract

The causal spacetimes admitting a covariantly constant null vector provide a connection between relativistic and non-relativistic physics. We explore this relationship in several directions. We start proving a formula which relates the Lorentzian distance in the full spacetime with the least action of a mechanical system living in a quotient classical space time. The timelike eikonal equation satisfied by the Lorentzian distance is proved to be equivalent to the Hamilton-Jacobi equation for the least action. We also prove that the Legendre transform on the classical base corresponds to the musical isomorphism on the light cone, and the Young-Fenchel inequality is nothing but a well known geometric inequality in Lorentzian geometry. A strategy to simplify the dynamics passing to a reference frame moving with the E.-L. flow is explained. It is then proved that the causality properties can be conveniently expressed in terms of the least action. In particular, strong causality coincides with stable causality and is equivalent to the lower semi-continuity of the classical least action on the diagonal, while global hyperbolicity is equivalent to the coercivity condition on the action functional. The classical Tonelli's theorem in the calculus of variations corresponds, then, to the well known result that global hyperbolicity implies causal simplicity. The well known problem of recasting the metric in a global Rosen form is shown to be equivalent to that of finding global solutions to the Hamilton-Jacobi equation having complete characteristics. * Work presented at the conferences "Non-commutative structures and non-relativistic (super)symmetries", Tours (2010) [59], and "Weak KAM Theory in Italy", Cortona (2011). This first version includes results which are mostly on causality theory; the next version will include a chapter with more results related to weak KAM theory. Some relevant references can be missing in this version.

Completeness of first and second order ODE flows and of Euler–Lagrange equations

“…Eisenhart [27] realized that the spacelike geodesics of (M, g) project to E := T × Q into solutions of the above Euler-Lagrange equation and that any such solution can be regarded as such projection. The author showed that the same can be said with spacelike replaced by lightlike [28], a fact particularly useful for its connection with causality theory [28,31,32]. Thus there is a very fruitful oneto-one correspondence between generalized gravitational waves and rheonomic mechanical systems, which allows one to import methods and ideas from one field to the other [32].…”

“…The author showed that the same can be said with spacelike replaced by lightlike [28], a fact particularly useful for its connection with causality theory [28,31,32]. Thus there is a very fruitful oneto-one correspondence between generalized gravitational waves and rheonomic mechanical systems, which allows one to import methods and ideas from one field to the other [32].…”

“…For instance, under the assumption that (Q, a t ) are complete, the completeness of the E.-L. flow is equivalent to the geodesic completeness of (M, g), thus Corollary 3.6 establishes conditions for the geodesic completeness of (M, g). However, we shall leave the details of this result to a different work [32].…”

A novel notion of null infinity for c-boundaries and generalized black holes

“…In [8, Thm. 6.9] (see also [24]) the authors provide mild conditions under which a plane wave is causally continuous. So, according to Corollary 6.4, these conditions joined to the previously cited ones for geodesic completeness ensure that a plane wave has no black holes.…”

Lifting the Eisenhart-Duval Lift to a Minimal Brane

“…In addition to elucidating the symmetries, of and equivalences between, dynamical systems the Eisenhart-Duval construction offers the prospect of applying global techniques in Lorentzian geometry and causal theory to the dynamical systems which are their shadows [12,13].…”