Abstract
The existence of Closed Timelike Curves (CTCs) in a generic spacetime is often associated with a non-physical choice of coordinates and can be cured by limiting the admissibility of such coordinates. Lichnerowicz conditions, for instance, represent a criterion for admissibility. The result, however, is a very restrictive limitation which may imply “removal” of important regions (with respect to the peculiarity of phenomena which may happen there) of the spacetime manifold. We consider here the point of view of a family of observers (Fundamental Slicing Observers, FSO) having their world lines orthogonal to the surfaces of constant coordinate time. We say that the time coordinate has not a global character if the associated FSO change their causality condition in the domain of validity of the coordinates themselves. Furthermore, in those regions where FSO have no more timelike world lines, CTCs are present and one may think of special devices or investigation tools apt to operationally detect them. We will discuss in detail theoretical approaches involving (scalar) waves or photons.
Highlights
Consider a generic spacetime with metric g in a coordinate system xα = (t, xa).1 Let the spacetime admit a non-null foliation, i.e. an integrable distribution of hypersurfaces which can be either timelike or spacelike, but not lightlike and assume that such a foliation is parametrized by the coordinate time, i.e. the “slices" are the t =const. hypersurfaces
The existence of Closed Timelike Curves (CTCs) in a generic spacetime is often associated with a non-physical choice of coordinates and can be cured by limiting the admissibility of such coordinates
We say that the time coordinate has not a global character if the associated FSO change their causality condition in the domain of validity of the coordinates themselves. In those regions where FSO have no more timelike world lines, CTCs are present and one may think of special devices or investigation tools apt to operationally detect them
Summary
There exists a congruence of curves which is orthogonal to the foliation itself and is geometrically characterized by the vanishing of the vorticity associated with the unit normal vector field. In those regions where the foliation is timelike, the congruence of orthogonal curves is spacelike and cannot be associated anymore with test observers. Limiting situations such that the foliation is always or never spacelike exist (even for the same spacetime with the metric expressed in different coordinate systems). We may learn much about this topic from black hole or black hole-related stationary axisymmetric spacetimes
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