Geodesically Complete Research Articles

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Anisotropic generalized polytropic spheres: Regular 3D black holes

We model gravitating relativistic 3D spheres composed of an anisotropic fluid in which the radial and transverse components of the pressure correspond to the vacuum energy and a generalized polytropic equation-of-state, respectively. By using the generalized Tolman–Oppenheimer–Volkoff (TOV) equation, and solving the complete system of equations for these anisotropic generalized polytropic spheres, for a given range of model parameters, we find three novel classes of asymptotically AdS black hole solutions with regular core. We show the regularity of the solutions using curvature scalars and the formalism of geodesic completeness. Then, using the eigenvalues of the Riemann curvature tensor, we consider the effects of repulsive gravity in the three static 3D regular black holes, concluding that their regular behavior can be explained as due to the presence of repulsive gravity near the center of the objects. Finally, we study the stability of the regular black holes under the flow of the energy through the Cauchy horizon.

Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics

In this work, we consider holomorphically projective mappings of (pseudo-) Kähler spaces. We determine the conditions for finite complete geodesics that must be satisfied for the mappings to be trivial; i.e., these spaces are rigid.

Geodesic completeness of effective null geodesics in regular space-times with non-linear electrodynamics

We study the completeness of light trajectories in certain spherically symmetric regular geometries found in Palatini theories of gravity threaded by non-linear (electromagnetic) fields, which makes their propagation to happen along geodesics of an effective metric. Two types of geodesic restoration mechanisms are employed: by pushing the focal point to infinite affine distance, thus unreachable in finite time by any sets of geodesics, or by the presence of a defocusing surface associated to the development of a wormhole throat. We discuss several examples of such geometries to conclude the completeness of all such effective paths. Our results are of interest both for the finding of singularity-free solutions and for the analysis of their optical appearances e.g. in shadow observations.

Geodesic completeness of a ring wormhole

In this work the possible geodesic completeness of an electromagnetic dipole wormhole is studied in detail. The space-time contains a curvature singularity and belongs to a class of solutions to the Einstein-Maxwell equations with a coupled scalar field. Specifically, a numerical analysis is performed to examine congruences of null geodesics that are directed toward the singularity. The results found here show that, depending on the strength of the coupling between the scalar and electromagnetic fields, the wormhole can be either geodesically incomplete or complete. We then focus on those wormholes that are geodesically complete and study the geometry of the neighborhood of their singularity using Kaluza-Klein theory in a five-dimensional space-time. This process allows us to provide a possible explanation of the completeness of geodesics despite unbounded curvature.

Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups

This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2,R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2,ℂ).

Past-completeness of inflationary spacetimes

We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past. Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation. It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate $H_{avg}>0$) must be incomplete in null and timelike past directions. This well-known conjecture relies on specific bounds on the integral of the Hubble parameter over a past-directed timelike or null geodesic. As stated, we show this claim is an open issue. We show that the calculation of $H_{avg}$ yields a continuum of results for a given spacetime predicated upon the underlying topological assumptions. We present an improved definition for $H_{avg}$ and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having $H_{avg}>0$. We discuss a standardized definition for inflationary spacetimes as well as quantum (semi-classical) cosmological concerns over physically reasonable scale factors.

Trans-Planckian Censorship and Spacetime Singularities

We study the effects of trans-Planckian censorship conjecture (TCC) bounds on geodesic completeness of spacetime and the associated existence for an infinite proper time. Using Gronwall’s lemma, TCC bounds can be derived directly, leading to a result about the absence of blowup solutions. We show that the TCC provides part of the required criteria for geodesic completeness, and we then provide the remaining ones, the norm of the extrinsic curvature being bounded away from zero. We also discuss the importance of these results for the classical evolution of Friedmann universes under the assumptions of global and regular hyperbolicity.

A Curvature Inequality Characterizing Totally Geodesic Null Hypersurfaces

A well-known application of the Raychaudhuri equation shows that, under geodesic completeness, totally geodesic null hypersurfaces are unique which satisfy that the Ricci curvature is nonnegative in the null direction. The proof of this fact is based on a direct analysis of a differential inequality. In this paper, we show, without assuming the geodesic completeness, that an inequality involving the squared null mean curvature and the Ricci curvature in a compact three-dimensional null hypersurface also implies that it is totally geodesic. The proof is completely different from the above, since Riemannanian tools are used in the null hypersurface thanks to the rigging technique.

Geodesic (in) Completeness in General Metric Frames

The geometric concept of geodesic completeness depends on the choice of the metric field or “metric frame”. We develop a frame-invariant concept of “generalised geodesic completeness” or “time completeness”. It is based on the notion of physical time defined by counting oscillations for some physically allowed process. Oscillating solutions of wave functions for particles with varying mass permit the derivation of generalised geodesics and the associated notion of completeness. Time completeness involves aspects of particle physics and is no longer a purely geometric concept.

The $L^{\infty }$-positivity Preserving Property and Stochastic Completeness

We say that a Riemannian manifold satisfies the $L^p$-positivity preserving property if $(-\Delta + 1)u\ge 0$ in a distributional sense implies $u \ge 0$ for all $ u \in L^p$.While geodesic completeness of the manifold at hand ensures the $L^p$-positivity preserving property for all $p \in (1, +\infty)$, when $p = + \infty$ some assumptions are needed. In this paper we show that the $L^\infty$-positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of $-\Delta + 1 \ge 0$, which are of independent interest.

Cyclic cosmology and geodesic completeness

We consider recently proposed bouncing cosmological models for which the Hubble parameter is periodic in time, but the scale factor grows from one cycle to the next as a mechanism for shedding entropy. Since the scale factor for a flat universe is equivalent to an overall conformal factor, it has been argued that this growth corresponds to a physically irrelevant rescaling, and such bouncing universes can be made perfectly cyclic, extending infinitely into the past and future. We show that any bouncing universe which uses growth of the scale factor to dissipate entropy must necessarily be geodesically past-incomplete, and therefore cannot be truly cyclic in time.

Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form

Let \(\mathscr{G}\) be the manifold of all (unparametrized) oriented lines of \(\mathbb{R}^{3}\). We study the controllability of the control system in \(\mathscr{G}\) given by the condition that a curve in \(\mathscr{G}\) describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed \(\alpha \). Actually, we pose the analogous more general problem by means of a control system on the manifold \(\mathscr{G}_{\kappa }\) of all the oriented complete geodesics of the three dimensional space form of curvature \(\kappa \): \(\mathbb{R}^{3}\) for \(\kappa =0\), \(S^{3}\) for \(\kappa =1\) and hyperbolic 3-space for \(\kappa =-1\). We obtain that the system is controllable if and only if \(\alpha ^{2}\neq \kappa \). In the spherical case with \(\alpha =\pm 1\), an admissible curve remains in the set of fibers of a fixed Hopf fibration of \(S^{3}\).We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.

Charged BTZ-type solutions in Eddington-inspired Born-Infeld gravity

We construct an axially symmetric solution of Eddington-inspired Born-Infeld gravity coupled to an electromagnetic field in 2+1 dimensions including a (negative) cosmological constant term. This is achieved by using a recently developed mapping procedure that allows to generate solutions in certain families of metric-affine gravity theories starting from a known seed solution of General Relativity, which in the present case corresponds to the electrically charged Banados-Teitelboim-Zanelli (BTZ) solution. We discuss the main features of the new configurations, including the modifications to the ergospheres and horizons, the emergence of wormhole structures, and the consequences for the regularity (or not) of these space-times via geodesic completeness.

On the geometry of stationary Galilean spacetimes

In this work we introduce a new family of non-relativistic spacetimes: standard stationary Galilean spacetimes, which constitute the local geometric model of stationary Galilean spacetimes. We also study the geodesic completeness of stationary Galilean spacetimes as well as the the geometric conditions for these spacetimes that guarantee the existence of a global splitting as a standard stationary Galilean spacetime.

Multicenter solutions in Eddington-inspired Born\u2013Infeld gravity

We find multicenter (Majumdar–Papapetrou type) solutions of Eddington-inspired Born–Infeld gravity coupled to electromagnetic fields governed by a Born–Infeld-like Lagrangian. We construct the general solution for an arbitrary number of centers in equilibrium and then discuss the properties of their one-particle configurations, including the existence of bounces and the regularity (geodesic completeness) of these spacetimes. Our method can be used to construct multicenter solutions in other theories of gravity.

Attractors of the ‘n + 1’ dimensional Einstein-Λ flow

Here we prove a global existence theorem for sufficiently small however fully nonlinear perturbations of a family of background solutions of the ‘n + 1’ dimensional vacuum Einstein equations in the presence of a positive cosmological constant Λ. The future stability of vacuum solutions in the small data and zero cosmological constant limit has been studied previously for both ‘3 + 1’ and higher dimensional spacetimes. However, with the advent of dark energy driven accelerated expansion of the Universe, it is of fundamental importance in mathematical cosmology to include a positive cosmological constant, the simplest form of the dark energy for the vacuum Einstein equations. Such Einsteinian evolution is here designated as the ‘Einstein-Λ’ flow. We study the background solutions of this ‘Einstein-Λ’ flow in ‘n + 1’ dimensional spacetimes in constant mean curvature spatial harmonic gauge, n ⩾ 3 and establish both linear and non-linear stability of such solutions. In the cases of number of spatial dimensions being strictly greater than 3, the finite dimensional Einstein moduli spaces form the centre manifolds of the dynamics. A suitable shadow gauge condition [Andersson L and Moncrief V 2011 J. Differ. Geom. 89 1–47] is implemented in order to treat these cases. In addition, the autonomous character of the suitably re-scaled Einstein flow breaks down as a consequence of including Λ(>0). We construct a Lyapunov functional (controlling a suitable norm of the small data) similar to a wave equation type energy for the non-linear non-autonomous evolution of the data and prove its decay in the direction of cosmological expansion utilizing the structure of the non-autonomous terms and smallness assumption on the data. Our results demonstrate the future stability and geodesic completeness of the perturbed spacetimes, and show that the scale-free geometry converges to an element of the space of constant negative scalar curvature metrics sufficiently close to and containing the Einstein moduli space (a point for n = 3 and a finite dimensional space for n > 3), which has significant consequences for the cosmic topology while restricting to the case of n = 3.

A Birman-Series type result for geodesics with infinitely many self-intersections

Given a hyperbolic surface $\S$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $\S$. We define a self-intersection function for each complete geodesic, which bounds the number of self-intersections in finite length subarcs. We then extend the Birman-Series result to sets of complete geodesics with certain bounds on their self-intersection functions. In fact, we get the same conclusion as the Birman-Series result for sets of complete geodesics whose self-intersection functions are in $o(l^2)$, where $l$ measures arclength.

Singularity Theorems for $$C^1$$-Lorentzian Metrics

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal geodesic completeness is $C^1$-fine stable. This improves a similar older stability result of Beem and Ehrlich where they also used the $C^1$-fine topology to measure closeness but still required smoothness of all metrics. Lastly, we include a brief appendix where we use some of the same techniques in the Riemannian case to give a proof of the classical Myers Theorem for $C^1$-metrics.

Nonlinear σ -models in the Eddington-inspired Born-Infeld gravity

In this paper we consider two different nonlinear $\sigma$-models minimally coupled to Eddington-inspired Born-Infeld gravity. We show that the resultant geometries represent minimal modifications with respect to those found in GR, though with important physical consequences. In particular, wormhole structures always arise, though this does not guarantee by itself the geodesic completeness of those space-times. In one of the models, quadratic in the canonical kinetic term, we identify a subset of solutions which are regular everywhere and are geodesically complete. We discuss characteristic features of these solutions and their dependence on the relationship between mass and global charge.