Geodesically Complete Research Articles

Complétude et flots nul-géodésibles en géométrie lorentzienne

We study geodesic completeness of null-pregeodesic flows on compact Lorentz manifold, obtaining an obstruction to be null-geodesic. We show that when the orthogonal distribution to the vectorfield generating the considered flow integrates into a foliation ℱ, the completeness of the flow can be read on the holonomie of ℱ. We obtain this way that there are no smooth null-geodesic flows on S 3 . We also prove that a Lorentzian 2-torus is null-complete if and only if its lightlike foliations are both 𝒞 0 linearisable.

The intersection of complete geodesics on S 2 with positive curvature

Let M be a Riemannian manifold. A complete geodesic γ on M means that γ : (−∞, +∞) → M is a normalized geodesic. In this paper, we prove that on (S 2 , g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times.

On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity, causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of $H(x,u)$ with $x$ at infinity determines many properties of geodesics. Essentially, a subquadratic growth of $H$ ensures geodesic completeness and connectedness, while the critical situation appears when $H(x,u)$ behaves in some direction as $|x|^2$, as in the classical model of exact gravitational waves

On future asymptotics of polarized Gowdy \U0001d54b3-models

Gowdy's model of cosmological spacetimes is a much investigated subject in classical and quantum gravity. Depending on spatial topology, re-collapsing as well as expanding models are known. Several analytical tools were used in order to clarify singular behaviour in this class of spacetimes. Here we investigate the structure of a certain subclass, the polarized Gowdy models with spatial \U0001d54b3-topology, in general. The asymptotics for general solutions of the dynamical equation for one of the gravitational degrees of freedom play a key role while the asymptotic behaviour of the remaining metric function is a result of solving the Hamiltonian constraint equation. Using both, we prove future geodesic completeness in all spacetimes of this type.

Geodesically complete nondiagonal inhomogeneous cosmological solutions in dilatonic gravity with a stiff perfect fluid

New nondiagonal $G_{2}$ inhomogeneous cosmological solutions are presented in a wide range of scalar-tensor theories with a stiff perfect fluid as a matter source. The solutions have no big-bang singularity or any other curvature singularities. The dilaton field and the fluid energy density are everywhere regular, too. The geodesic completeness of the solutions is investigated.

A wide family of singularity-free cosmological models

In this paper a family of non-singular cylindrical perfect fluid cosmologies is derived. The equation of state corresponds to a stiff fluid. The family depends on two independent functions under very simple conditions. A sufficient condition for geodesic completeness is provided.

Doubly warped products

In this paper we study geodesic completeness of Riemannian doubly warped products and Lorentzian doubly warped products. We give necessary conditions for generalized Robertson–Walker space-times with doubly warped product spacial parts to be globally hyperbolic. We also state some results about Killing and conformal vector fields of doubly warped products.

Multidimensional Toroidal Compactification with Random Gaussian Modulus: Derivation and Potential Applications of a Stochastic General Relativity

A stochastic general relativity is derived via compactification of (n + v)-dimensional Kaluza–Klein gravity, or effective superstring theory, on an internal space (a v-torus Tv = S1 ⊗ S1 ⊗ ... ⊗ S1) whose volume is parametrised by a random Gaussian modulus (scalar) field. The dimensional reduction, Mn + v → Mn ⊗ Tv, leads to stochastic vacuum Einstein equations on Mn, with a source term arising from random fluctuations or 'turbulence' in geometry on the order of the compactification scale. The source plays the role of a stochastic cosmological constant and the new Einstein vacuum equations still obey the Bianchi identities. The methodology attempts to extend the smooth manifold paradigm of general relativity—in lieu of any existing theory of quantum gravity—to accommodate short distance stochastic structure near the Planck scale. An equivalent description, and interpretation, is possible in terms of random conformal metric fluctuations in n-dimensions: given the usual vacuum Einstein equations, for (Mn, g), there exists conformal metric fluctuations Ω2g, such that the stochastic vacuum equations—derived previously via the dimensional reduction—are reproduced on (M, Ω2g). The stochastic extension of general relativity is tentatively applied to a number of key issues and scenarios of interest. These include geodesic focusing, completeness and conjugate points; energy conditions, singularities and global structure; and gravitational collapse and cosmology.

Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h ij gives Langevin stochastic differential equations for the Cauchy evolution of h ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h′ij away from h ij is considered to be related to h ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the “geometro-hydrodynamics” of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence θ′>0 at s′ the singularity θ=−∞ at future proper time s=3/|θ′|$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s→∞) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that θ→−∞ for s→∞. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.

Multiply warped products

In this paper, we first state covariant derivative formulas for multiply warped products and consider the geodesic equations for these spaces. Then we state some basic facts about causality of Lorentzian multiply products and study Cauchy surfaces and global hyperbolicity. Finally, we consider null, time-like and space-like geodesic completeness of Lorentzian multiply products and geodesic completeness of Riemannian multiply warped products.

Geodesic connectedness in Gödel type space-times

In this paper we use a variational approach in order to prove the geodesic connectedness of some Gödel type space-times; moreover direct methods allow to prove the geodesic connectedness of the Gödel Universe. At last a result of geodesic completeness is given.

The unboundedness of certain minimal submanifolds of positively curved Riemannian spaces

We prove that a minimal immersion of a complete Riemannian manifold M into another complete noncompact Riemannian manifold N of positive curvature must have an unbounded image provided that M has scalar curvature bounded away from −∞. This extends the unboundedness theorems of Gromoll and Meyer for complete geodesics and of Galloway and Rodriguez for parabolic minimal surfaces. Furthermore, we prove that in case M is of codimension 1, only the Ricci curvature and not necessarily the full sectional curvature of the ambient space N need be positive in order for the same conclusion to hold.

Geodesic completeness of orthogonally transitive cylindrical space–times

In this paper a theorem is derived in order to provide a wide sufficient condition for an orthogonally transitive cylindrical space–time to be singularity free. The applicability of the theorem is tested on examples provided by the literature that are known to have regular curvature invariants.

Strings of opposite magnetic charges in a gauge field theory

We show that there are soliton–like solutions representing arbitrarily located M vortices and N antivortices with opposite magnetic charges in an Abelian gauge field theory. We establish an existence and uniqueness theorem and prove that the total magnetic flux is proportional to M − N but the total energy is proportional to M + N . In the presence of Einstein's gravity, we establish an existence theorem under a necessary and sufficient condition that ensures the geodesic completeness of the corresponding gravitational metric. The coexisting vortices and antivortices are now cosmic strings and antistrings and the total magnetic flux and matter–gauge energy are given by similar expressions as those for vortices. Besides the energy, the total curvature also provides an exact, quantized, measurement of the number of the strings of the two types.

Exterior differential system for cosmological perfect fluids and geodesic completeness

In this paper a new formalism based on exterior differential systems is derived for perfect-fluid spacetimes endowed with an Abelian orthogonally transitive group of motions acting on spacelike surfaces. This formulation allows simplifications of Einstein equations and it can be applied for different purposes. As an example a singularity-free metric is rederived in this framework. A sufficient condition for a diagonal metric to be geodesically complete is also provided.

On the Geometry of Generalized Robertson-Walker Spacetimes: Geodesics

The geometry and, especially, the geodesics of a class of spacetimes generalizing Robertson-Walker ones (without any assumption on the fiber) is studied, under a global point of view. Our study covers geodesic connectedness, geodesic completeness and stability of completeness.

Geodesic and metric completeness in infinite dimensions

Geodesic and metric completeness in infinite dimensions

MATHEMATICAL THEORY OF SELF-DUAL COSMIC STRINGS: EXISTENCE AND OBSTRUCTIONS

This is a survey article concerning some recent mathematical results for the static selfdual cosmic-string solutions in the Abelian Higgs model and in the Weinberg-Salam standard model unifying electromagnetic and weak interactions, both coupled with gravity through the Einstein equations. For the Abelian Higgs strings there is a nearly complete picture. If the Riemann surface M on which the strings reside is compact, it can be shown that M must be S2 up to topological equivalence and there are only countably many values of the Higgs vacuum states for strings to exist. When M is noncompact and conformally a plane there are exact obstructions to the finiteness of energies and geodesic completeness of solutions. For the Weinberg-Salam strings, much is to be achieved. It can be shown in this case that self-dual strings generated from W and Higgs condensation lead to an explicit formula for a positive cosmological constant and the gravitational metric is always noncomplete. This feature leads to the properties that the metric decays sufficiently rapidly at infinity and there exist non-Abelian electroweak strings of finite energies. It is established that for any integer N there are always suitable ranges of the electroweak parameters to allow the existence of W- and Higgs-condensed N-vortex solutions of finite energies.

Denjoy–Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces

We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in $\R{n}$ has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are also discussed.

On singularity-free spacetimes

We consider here the metric for the singularity-free family of fluid models. The metric is unique for cylindrically symmetric space-time with metric potentials being separable functions of radial and time coordinates in the comoving coordinates. It turns out that fluid models separate out into two classes, withρ ≠µp in general butρ = 3p in particular andp =ρ. It is shown that in both the cases radial heat flow can be incorporated without disturbing the singularity-free character of the spacetime. The geodesics of the singularity-free metric are studied and the geodesic completeness is established. Several previously known solutions are derived as particular cases.