Timelike Convergence Condition Research Articles

Hawking's singularity theorem under a bounded integral norm of Ricci curvature

In classical general relativity, the timelike convergence condition is Ric(v,v)≥0 for every timelike vector v. Hawking's singularity theorem is that the timelike convergence condition with a positive future convergence entails the future timelike incompleteness.But recent astronomical observations show that the timelike convergence condition does not hold. We prove a singularity theorem using an integral norm of Ricci curvature instead of Ric(v,v)≥0.

Generic warp drives violate the null energy condition

Three recent articles have claimed that it is possible to, at least in theory, either set up positive energy warp drives satisfying the weak energy condition (WEC), or at the very least, to minimize the WEC violations. These claims are, at best, incomplete since the arguments as presented only assert but do not prove the existence of one set of timelike observers, the comoving Eulerian observers, who see relatively ``nice'' physics. While these particular observers might arguably see a positive energy density, the WEC requires all timelike observers to see positive energy density. Therefore, one should carefully revisit this issue. A more careful analysis shows that the situation is actually much grimmer than advertised---within the framework adopted by those three papers all physically reasonable warp drives will certainly violate the WEC, and both the strong and dominant energy conditions. Under plausible subsidiary conditions the null energy condition is also violated. While warp drives are certainly interesting examples of speculative physics, the violation of the energy conditions, at least within the framework of standard general relativity, is unavoidable. Even in modified gravity, physically reasonable warp drives will still violate the purely geometrical null convergence condition and the timelike convergence condition which, in turn, will place very strong constraints on any modified-gravity warp drive.

Cosmological singularities from high matter density without global topological assumptions

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a ‘past null focusing’ condition. Such a condition forces all null geodesics gamma :[0,a)rightarrow M with future endpoint gamma (0) to develop a pair of conjugate points if past complete. By the Einstein field equations, such a condition will be satisfied if the density of matter fields remains sufficiently high towards the past of the spacetime, as may be expected in certain cosmological scenarios. The theorem obtained doesn’t make starting assumptions about the spacetime’s topology, such as the existence of a compact achronal slice, and if in addition to a ‘past null focusing’ condition we assume the timelike convergence condition, then further consequences pertaining to the existence of CMC foliations and the character of the singularity are obtained. With the addition of the timelike convergence condition, we obtain the conclusion that all timelike geodesics are past incomplete, rather than the existence of a single incomplete non-spacelike geodesic.

Stable maximal hypersurfaces in Lorentzian spacetimes

We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target space has constant sectional curvature as well as give sufficient conditions on the geometry of the ambient spacetime (e.g., the validity of TCC) to ensure stability. Some rigidity results and height estimates are also proven in GRW spacetimes. In the last part of the paper we consider k-stability of spacelike hypersurfaces, a concept related to mean curvatures of higher orders.

Bartnik’s splitting conjecture and Lorentzian Busemann function

In 1988 Bartnik posed the splitting conjecture about the cosmological space-time. This conjecture has been proved by several people, with different approaches and by using some additional assumptions such as ‘S-ray condition’ and ‘level set condition’. It is known that the ‘S-ray condition’ yields the ‘level set condition’. We have proved that the two are indeed equivalent, by giving a different proof under the assumption of the ‘level set condition’. In addition, we have shown several properties of the cosmological space-time, under the presence of the ‘level set condition’. Finally we have provided a proof of the conjecture under a different assumption on the cosmological space-time. But we first prove some results without the timelike convergence condition which help us to state our proofs.

On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field

We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hypersurface is totally geodesic. Moreover, we give an extension of the classical Calabi–Bernstein theorem to this class of pp-wave spacetimes.

Can extra dimensional effects allow wormholes without exotic matter?

We explore the existence of Lorentzian wormholes in the context of an effective on-brane, scalar-tensor theory of gravity. In such theories, the timelike convergence condition, which is always violated for wormholes, has contributions, via the field equations, from on-brane matter as well as from an effective geometric stress energy generated by a bulk-induced radion field. It is shown that, for a class of wormholes, the required on-brane matter, as seen by an on-brane observer in the Jordan frame, is not exotic and does not violate the Weak Energy Condition. The presence of the effective geometric stress energy in addition to on-brane matter is largely responsible for creating this intriguing possibility. Thus, if such wormholes are ever found to exist in the Universe, they would clearly provide pointers towards the existence of a warped extra dimension as proposed in the two-brane model of Randall and Sundrum.

A note on inverse mean curvature flow in cosmological spacetimes

In Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface.We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition.

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.

A METHOD TO CONSTRUCT 4-DIMENSIONAL SPACETIMES WITH A SPACELIKE CIRCLE ACTION

A general procedure to construct a 4-dimensional spacetime from a 3-dimensional time-oriented Lorentzian manifold and each of its timelike vector fields is exposed. It is based on the construction of the null congruence Lorentzian manifold. As an application, examples of stably causal spacetimes which obey the timelike convergence condition, are semi-symmetric, and admit an isometric spacelike circle action are obtained.

Distance to a focal point and singularity theorems with weak timelike convergence condition

AbstractIn classical general relativity, the timelike convergence condition is Ric(v, v) ≥ 0 for every timelike vector v. But recent astronomical observations show that the timelike convergence condition does not hold. In this paper, we obtain an explicit upper bound of the volume form and the distance to a focal point (such as Hubble distance) along timelike geodesic with integrals of the negative part of Ricci curvature, which entails singularity theorems.

A Study of Spacetimes with vanishing M−projective Curvature Tensor

In this paper we study about the M−projectively flat perfect fluid spacetime. First of all we showed that the Riemannian curvature tensor of an M−projectively flat spacetime is covariantly constant. Then we found the length of the Ricci operator in an M−projectively flat perfect fluid spacetime and proved that the isotropic pressure and entry density of an M−projectively flat perfect fluid spacetime satisfying Einsteins field equation with cosmological constant are constant. Then we showed that an M−projectively flat perfect fluid spacetime satisfying Einsteins field equation with cosmological constant and obeying the timelike convergence condition has positive isotropic pressure. Further we showed that the isotropic pressure and the energy density of an M−projectively flat perfect fluid spacetime satisfying Einsteins field equation with cosmological constant vanishes in a purely electromagnetic distribution. Lastly we showed that an M−projectively flat perfect fluid spacetime with the energy momentum tensor of an electromagnetic field such that the spacetimesatisfies Einsteins field equation without cosmological constant is a Euclidean space

Volume expansion rate of the Lorentzian manifold based on integral Ricci curvature over a timelike geodesic

Recent astronomical observations show that the universe is not only expanding but also undergoing accelerated expansion [A.G. Riess, et al., The farthest known supernova, Astrophys. J. 560 (2001) 49–71; P.K. Townsend, M.N.R. Wohlfarth, Accelerating cosmologies from compactification, Phys. Rev. Lett. 91 (2003) 061302]. Then the timelike convergence condition does not hold every time, i.e. the Ricci curvature Ric ( v , v ) cannot be nonnegative for every timelike vector v . We obtain the volume expansion rate of the universe based on the integral norm of negative part of the Ricci curvature along a timelike geodesic.

On the CMC foliation of future ends of a spacetime

We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corresponding further assumptions are satisfied. In addition we show that the volume of any sequence of spacelike hypersurfaces, which run into the future singularity, decays to zero provided there exists a time function covering a future end, such that the level hypersurfaces have nonnegative mean curvature and decaying volume.

Geodesic connectedness and conjugate points in GRW space–times

Given two points of a generalized Robertson–Walker space–time, the existence, multiplicity and causal character of geodesics connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional space–times and to GRW space–times satisfying the timelike convergence condition are also found.

Existence of maximal hypersurfaces in some spherically symmetric spacetimes

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface and the maximal distance between and . In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume.

Open and closed universes, initial singularities, and inflation.

The existence of initial singularities in expanding universes is proved without assuming the timelike convergence condition. The assumptions made in the proof are ones likely to hold both in open universes and in many closed ones. (It is further argued that at least some of the expanding closed universes that do not obey a key assumption of the theorem will have initial singularities on other grounds.) The result is significant for two reasons: (a)~previous closed-universe singularity theorems have assumed the timelike convergence condition, and (b)~the timelike convergence condition is known to be violated in inflationary spacetimes. An immediate consequence of this theorem is that a recent result on initial singularities in open, future-eternal, inflating spacetimes may now be extended to include many closed universes. Also, as a fringe benefit, the time-reverse of the theorem may be applied to gravitational collapse.

Lines in space-times

We construct a complete timelike maximal geodesic (“line”) in a timelike geodesically complete spacetimeM containing a compact acausal spacelike hypersurfaceS which lies in the past of someS-ray. AnS-ray is a future complete geodesic starting onS which maximizes Lorentzian distance fromS to any of its points. If the timelike convergence condition (strong energy condition) holds, a line exists only ifM is static, i.e. it splits geometrically as space × time. So timelike completeness must fail for a nonstatic spacetime with strong energy condition which contains a “closed universe”S with the above properties.

The topology of asymptotically Euclidean static perfect fluid space-time

It is shown that a geodesically complete, asymptotically Euclidean, static perfect fluid space-time satisfying the time-like convergence condition and having a connected fluid region is diffeomorphic to ℝ3×ℝ.