John K. Beem

Conformal changes and geodesic completeness

Let (M, g) be a causal spacetime. ConditionN will be satisfied if for each compact subsetK ofM there is no future inextendible nonspacelike curve which is totally future imprisoned inK. IfM satisfies conditionN, then wheneverE is an open and relatively compact subset ofM the spacetimeE with the metricg restricted toE is stably causal. Furthermore, there is a conformal factor Ώ such that (M, Ώ2g) is both null and timelike geodesically complete. IfM is an open subset of two dimensional Minkowskian space, thenM is conformal to a geodesically complete spacetime.

Pseudoconvexity and geodesic connectedness

SummaryLet M be a smooth manifold with a linear connection. If M is pseudoconvex, disprisoning, and has no conjugate points, then each pair of points of M is joined by at least one geodesic.

Cauchy horizon end points and differentiability

Cauchy horizons are shown to be differentiable at end points where only a single null generator leaves the horizon. A Cauchy horizon fails to have any null generator end points on a given open subset iff it is differentiable on the open subset and also iff the horizon is (at least) of class C1 on the open subset. Given the null convergence condition, a compact horizon which is of class C2 almost everywhere has no end points and is (at least) of class C1 at all points.

Some Examples of Incomplete Space-Times

An example is given of a space-time which is timelike and spacelike complete but null incomplete. An example is also given of a space-time which is geodesically complete but contains an inextendible timelike curve of bounded acceleration and finite length. These two examples may be modified so that in each case they become globally hyperbolic and retain the stated properties. All of the examples are conformally equivalent to open subsets of the two-dimensional Minkowski space.

Proper homothetic maps and fixed points

Let f be a proper homothetic map of the pseudo-Riemannian manifold M and assume f has a fixed point p. If all of the eigenvalues of either f*p or f-1*p have absolute values less than unity, then M is topologically Rn and M has a flat metric. This yields three characterizations of Minkowski spacetime. In general, a homothetic map of a complete pseudo-Riemannian manifold need not have fixed points. Furthermore, an example shows the existence of a proper homothetic map with a fixed point does not imply M is flat. The scalar curvature vanishes at a fixed point, but some of the sectional curvatures may be nonzero.

The space of geodesics

Let M be a manifold with linear connection ▽. The space G(M) of all geodesics of M may be given a topological structure and may be realized as a quotient space of the reduced tangent bundle of M. The space G(M) is a T1 space iff the image of each geodesic is a closed subset of M. It is Hausdorff iff each tangentially convergent sequence of geodesics converges in the Hausdorff limit sense to the limit geodesic. If M has no conjugate points and G(M) is Hausdorff, then M is geodesically connected.

Sectional curvature and tidal accelerations

Sectional curvature is related to tidal accelerations for small objects of nonzero rest mass. Generically, the magnification of tidal accelerations due to high speed goes as the square of the magnification of energy. However, some space‐times have directions with bounded increases in tidal accelerations for relativistic speeds. These investigations also yield a characterization of null directions that fail to satisfy the generic condition used in singularity theorems. For Ricci flat four‐dimensional space‐times, tidally nondestructive directions are characterized as repeated principal null directions.

The generic condition is generic

We consider the generic condition for vectors—both null and non-null—at a fixed pointp of a spacetime, and ask just how generic this condition is. In a general spacetime, if the curvature is not zero at the pointp, then the generic condition is found to be generic in the mathematical sense that it holds on an open dense set of vectors atp; more specifically, if there are as many as five non-null vectors in general position atp which fail to satisfy the generic condition, then the curvature vanishes atp. If the Riemann tensor is restricted to special forms, then stronger statements hold: An Einstein spacetime with three linearly independent nongeneric timelike vectors atp is flat atp. A Petrov type D spacetime may not have any nongeneric timelike vectors except possibly those lying in the plane of the two principal null directions; if any of the non-null vectors in such a plane are nongeneric, then so are all the vectors of that plane, as well as the plane orthogonal to it.

Stability of geodesic incompleteness for Robertson-Walker space-times

Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = −dt2 ⊕fh, where −∞⩽a<b⩽+∞, (H, h) is a Riemannian manifold andf: (a, b)→(0, ∞) is a smooth function. We show that ifa>−∞ and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC0 perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC1 perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.

The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg′ conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.