Nongeneric null vectors

We consider the generic condition for null directions at a fixed point. A nongeneric vector is one which violates the generic condition: a vectorX is nongeneric ifX ^cX^d X[_aR_b]cd[_e X _f] = 0. The presence of null nongeneric vectors at a point can force the curvature tensor to be uniform, i.e., be that of a constant-curvature space; in particular, if there are 11 null nongeneric directions generically situated, then the curvature is uniform. For non-uniform curvature, the locus of nongeneric null directions must obey a cubic relation; examples show that it need not obey a linear relation.     

Existence and stability of circular orbits in static and axisymmetric spacetimes

The existence and stability of timelike and null circular orbits (COs) in the equatorial plane of general static and axisymmetric (SAS) spacetime are investigated in this work. Using the fixed point approach, we first obtained a necessary and sufficient condition for the non-existence of timelike COs. It is then proven that there will always exist timelike COs at large \(\rho \) in an asymptotically flat SAS spacetime with a positive ADM mass and moreover, these timelike COs are stable. Some other sufficient conditions on the stability of timelike COs are also solved. We then found the necessary and sufficient condition on the existence of null COs. It is generally shown that the existence of timelike COs in SAS spacetime does not imply the existence of null COs, and vice-versa, regardless whether the spacetime is asymptotically flat or the ADM mass is positive or not. These results are then used to show the existence of timelike COs and their stability in an SAS Einstein-Yang-Mills-Dilaton spacetimes whose metric is not completely known. We also used the theorems to deduce the existence of timelike and null COs in some known SAS spacetimes.     

Global hyperbolicity and completeness

We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric. We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature K are integrable. This last condition is required only for the tracefree part of K if the universe is expanding.     

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.     

Null Geodesics in Five-Dimensional Manifolds

We analyze a class of 5D non-compact warped-product spaces characterized by metrics that depend on the extra coordinate via a conformal factor. Our model is closely related to the so-called canonical coordinate gauge of Mashhoon et al. We confirm that if the 5D manifold in our model is Ricci-flat, then there is an induced cosmological constant in the 4D sub-manifold. We derive the general form of the 5D Killing vectors and relate them to the 4D Killing vectors of the embedded spacetime. We then study the 5D null geodesic paths and show that the 4D part of the motion can be timelike—that is, massless particles in 5D can be massive in 4D. We find that if the null trajectories are affinely parameterized in 5D, then the particle is subject to an anomalous acceleration or fifth force. However, this force may be removed by reparameterization, which brings the correct definition of the proper time into question. Physical properties of the geodesics—such as rest mass variations induced by a variable cosmological "constant," constants of the motion and 5D time-dilation effects—are discussed and are shown to be open to experimental or observational investigation.     

Existence and stability of circular orbits in general static and spherically symmetric spacetimes

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large r will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that three SSS spacetimes whose metric function is not completely known will allow the existence of timelike and/or null COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics.     

Concircular Curvature Tensor and Fluid Spacetimes

In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.     

Exact solutions of Einstein and Einstein-scalar equations in 2 + 1 dimensions

A nonstatic and circularly symmetric exact solution of the Einstein equations (with a cosmological constant Λ and null fluid) in 2 + 1 dimensions is given. This is a nonstatic generalization of the uncharged spinless BTZ metric. For Λ = 0, the spacetime is though not flat, the Kretschmann invariant vanishes. The energy, momentum, and power output for this metric are obtained. Further a static and circularly symmetric exact solution of the Einsteinmassless scalar equations is given, which has a curvature singularity atr = 0 and the scalar field diverges atr = 0 as well as at infinity.     

Killing vectors in empty space algebraically special metrics. I

Empty space algebraically special metrics possessing an expanding degenerate principal null vector and a Killing vector are investigated. It is shown that the Killing vector falls into one of two classes. The class containing all asymptotically timelike Killing vectors is investigated in detail and the associated metrics are identified. Several theorems concerning these metrics are given, among which is a proof that if the metric is regular and possesses an asymptotically timelike Killing vector, then it must be typeD. In addition some relations between Killing vectors in general spaces are developed along with a set of tetrad symmetry equations stronger than those of Killing.     

Stability of generic thin shells in conformally flat spacetimes

Some important spacetimes are conformally flat; examples are the Robertson–Walker cosmological metric, the Einstein–de Sitter spacetime, and the Levi-Civita–Bertotti–Robinson and Mannheim metrics. In this paper we construct generic thin shells in conformally flat spacetime supported by a perfect fluid with a linear equation of state, i.e., \(p=\omega \sigma .\) It is shown that, for the physical domain of \(\omega \), i.e., \(0<\omega \le 1\), such thin shells are not dynamically stable. The stability of the timelike thin shells with the Mannheim spacetime as the outer region is also investigated.     

Incompleteness of timelike submanifolds with nonvanishing second fundamental form

LetM be a properly immersed timelike hypersurface of Minkowski space and assume thatM has a strictly positive second fundamental form. If each point ofM is of diagonal type and dimM 3, then the Ricci curvature ofM is strictly positive on all (nonzero) nonspacelike vectors. ThusM satisfies both the generic and strong energy conditions and a singularity theorem forM may be established.Supported in part by a grant from the Weldon Spring fund of the University of Missouri.     

A class of algebraically general solutions of the Einstein-Maxwell equations for non-null electromagnetic fields

The Einstein-Maxwell field equations for non-null electromagnetic fields are studied under the conditions that the null tetrad is parallelly propagated along both principal null congruences. It is shown that the resulting spacetime solutions are necessarily algebraically general. The twist-free solution found in a previous article is shown to be the most general twist-free solution. An expansionfree solution with twist and shear is also found.     

Spacelike metric foliations

In this paper, we investigate spacelike metric foliations in lightlike complete spacetimes. When such a foliation satisfies the strong energy condition Ric^V (e) ≥ 0 for timelike vectors e, it must be totally geodesic, and the metric is of higher rank, in the sense that each point of the spacetime is contained inside a flat, totally geodesic, timelike rectangle. If in addition Ric^V(e) = 0, then the metric is (at least locally) a product metric, with the leaves of the foliation tangent to one of the factors.     

Detecting the classical harmonic vibrations of micro amplitudes and low frequencies with an atomic Mach–Zehnder interferometer

Here we present an example of an axially symmetric spacetime, representing pure radiation, and admitting circular closed timelike curves (CTCs) on the $z= \hbox {constant plane}$ . The spacetime is regular everywhere, having no curvature singularities and is locally isometric to (non-vacuum) pp wave spacetimes. The stability of the CTCs under linear perturbations is studied and they are found to be stable from a calculation of the Lyapunov exponent for the deviation vector. We also demonstrate that the spacetime also admits non-circular CTCs which do not lie in this plane. A modification of the metric is also studied and we find that a region of this spacetime behaves like a time-machine where CTCs appear after a certain instant of time.     

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 such that (M, ² g) 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.     

Horizons and singularities in static,spherically symmetric spacetimes

We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (static) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show the completeness of all timelike and spacelike curves despite the superficial failure of the metric.     

Inflationary spacetimes are incomplete in past directions

Many inflating spacetimes are likely to violate the weak energy condition, a key assumption of singularity theorems. Here we offer a simple kinematical argument, requiring no energy condition, that a cosmological model which is inflating--or just expanding sufficiently fast--must be incomplete in null and timelike past directions. Specifically, we obtain a bound on the integral of the Hubble parameter over a past-directed timelike or null geodesic. Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime.     

Cosmological black holes: A black hole in the Einstein-de Sitter universe

As an example of a dynamical cosmological black hole, a spacetime that describes an expanding black hole in the asymptotic background of the Einstein-de Sitter universe is constructed. The black hole is primordial in the sense that it forms ab initio with the big bang singularity and its expanding event horizon is represented by a conformal Killing horizon. The metric representing the black hole spacetime is obtained by applying a time dependent conformal transformation on the Schwarzschild metric, such that the result is an exact solution with a matter content described by a two-fluid source. Physical quantities such as the surface gravity and other effects like perihelion precession, light bending and circular orbits are studied in this spacetime and compared to their counterparts in the gravitational field of the isolated Schwarzschild black hole. No changes in the structure of null geodesics are recorded, but significant differences are obtained for timelike geodesics, particularly an increase in the perihelion precession and the non-existence of circular timelike orbits. The solution is expressed in the Newman-Penrose formalism.     

Late-time Evolution of the Yang-Mills Field in the Spherically Symmetric Gravitational Collapse

We investigate the late-time evolution of theYang-Mills field in the self-gravitating backgrounds:Schwarzschild and Reissner-Nordstrom spacetimes. Thelate-time power-law tails develop in the threeasymptotic regions: the future timelike infinity, thefuture null infinity and the black hole horizon. Inthese two backgrounds, however, the late-time evolutionhas quantitative and qualitative differences. In the Schwarzschild black hole background, thelate-time tails of the Yang-Mills field are the same asthose of the neutral massless scalar field withmultipole moment l = 1. The late-time evolutionis dominated by the spacetime curvature. When the backgroundis the Reissner-Nordstrom black hole, the late-timetails have not only a smaller power-law exponent, butalso an oscillatory factor. The late-time evolution is dominated by the self-interacting term ofthe Yang-Mills field. The cause responsible for thedifferences is revealed.     

Fermat principle in Finsler spacetimes

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point q (emission event) to a timelike curve γ (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here “arrival time” refers to a parametrization of the timelike curve γ. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.