[Paper Review] Connection by geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field

Rossella Bartolo, Anna María Candela+1 more|arXiv (Cornell University)|May 5, 2014

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes a characterization for when two points in a globally hyperbolic spacetime with a complete lightlike Killing vector field and a complete Cauchy hypersurface can be connected by a geodesic. Using a limit argument on a sequence of perturbed standard stationary spacetimes, the authors prove that such points are geodesically connected if and only if they can be joined by a $C^1$ curve along which the inner product with the Killing field has constant sign or vanishes identically. A key consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with complete Cauchy hypersurfaces.

ABSTRACT

Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauchy hypersurface.

Motivation & Objective

• To determine which pairs of points in a globally hyperbolic spacetime with a complete lightlike Killing vector field and complete Cauchy hypersurface can be connected by a geodesic.
• To extend the known result on geodesic connectedness in stationary spacetimes with timelike Killing fields to the limiting case of lightlike Killing fields.
• To establish a sharp, intrinsic criterion for geodesic connectedness in this class of spacetimes, analogous to the timelike case.
• To demonstrate the applicability of the result to generalized plane waves, confirming their geodesic connectedness under global hyperbolicity and completeness.

Proposed method

• Perturb the original spacetime metric into a sequence of standard stationary metrics that converge to the original one.
• Apply an adapted version of the geodesic connectedness theorem for standard stationary spacetimes to each perturbed metric.
• Use the $C^1$ curve condition involving the inner product with the Killing field to derive uniform estimates on the sequence of connecting geodesics.
• Establish strong convergence of the sequence of geodesics in a suitable Sobolev and uniform topology.
• Use the limit of the sequence of geodesics to prove the existence of a geodesic connecting the original points in the unperturbed spacetime.
• Leverage variational methods and functional convergence to ensure the limit geodesic satisfies the required energy and causality conditions.

Experimental results

Research questions

• RQ1 Under what conditions can two points in a globally hyperbolic spacetime with a complete lightlike Killing vector field be connected by a geodesic?
• RQ2 Does the geodesic connectedness result for spacetimes with timelike Killing fields extend to the case of lightlike Killing fields?
• RQ3 What intrinsic geometric condition on a curve joining two points ensures geodesic connectedness in spacetimes with a lightlike Killing field?
• RQ4 Is the geodesic connectedness of generalized plane waves a consequence of global hyperbolicity and completeness of the Cauchy hypersurface?
• RQ5 Can the limit argument used for timelike Killing fields be adapted to the lightlike case, and what new challenges arise?

Key findings

• Two points in a globally hyperbolic spacetime with a complete lightlike Killing vector field and a complete Cauchy hypersurface are geodesically connected if and only if they can be joined by a $C^1$ curve along which the inner product with the Killing field has constant sign or is identically zero.
• The proof relies on a limit argument: perturbing the metric to a sequence of standard stationary spacetimes, applying the known geodesic connectedness result to each, and taking the limit of the connecting geodesics.
• The limit geodesic is shown to exist and be causal, as the energy functional converges and the limit satisfies the causality condition $f(\gamma) \leq 0$.
• The result confirms that any globally hyperbolic generalized plane wave with a complete Cauchy hypersurface is geodesically connected.
• The condition on the curve is sharp and consistent with previous results on generalized plane waves, providing a unifying criterion.
• The method reveals that geodesic connectedness is not preserved in general under limits of spacetimes, highlighting the subtlety of extending results from timelike to lightlike Killing fields.