Spacelike Orbits Research Articles

Spatially-homogeneous cosmologies

AbstractThe necessary and sufficient conditions for a perfect fluid solution to define a spatially-homogeneous cosmology are achieved. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic, and they offer an IDEAL labeling of these geometries. When a three-dimensional group acts on the three-dimensional space-like orbits, the Bianchi type of the model is also obtained.

Domain wall space-times with a cosmological constant

We solve vacuum Einstein's field equations with the cosmological constant in space-times admitting 3-parameter group of isometries with 2-dimensional space-like orbits. The general exact solutions, which are represented in the advanced and retarded null coordinates, have two arbitrary functions due to the freedom of choosing null coordinates. In the thin-wall approximation, the Israel's junction conditions yield one constraint equation on these two functions in spherical, planar, and hyperbolic domain wall space-times with reflection symmetry. The remain freedom of choosing coordinates are completely fixed by requiring that when surface energy density $\sigma_0$ of domain walls vanishes, the metric solutions will return to some well-known solutions. It leads us to find a planar domain wall solution, which is conformally flat, in the de Sitter universe.

Observations on arithmetic invariants and U-duality orbits in $ \mathcal {N} $ = 8 supergravity

We establish a relation between time-like, light-like and space-like orbits of the non-compact E{7(7)}(R)symmetry and discrete E{7(7)}(Z) invariants. We discuss the U-duality invariant formula for the degeneracy of states d(Q) which in the approximation of large occupation numbers reproduces the Bekenstein-Hawking entropy formula for regular black holes with AdS_2 horizon. We explain why the states belonging to light-like orbits, corresponding to classical solutions of "light black holes with null singularity", decouple from the corresponding index. We also present a separate U-duality invariant formula for the class of light-like orbits specified by discrete E{7(7)}(Z) invariants. We conclude that the present study of the non-perturbative sector of the theory does not reveal any contradiction with the conjectured all-loop perturbative finiteness of D=4 N=8 supergravity.

Chain propagator, mass, and universality in polymer solutions from Brownian relativity

A Lagrangian theory for single chains in polymer solutions is addressed via a recent Brownian relativity. By employing generalized diffusive coordinates, statements of covariance and diffusivity invariance result into free particle Lagrangians, where mass turns out to rise as a universal spacetime property. It descends from lowering diffusivity (or curving spacetime), so identifying a mechanism which conceptually resemble those ruling macromolecular scaling laws. An extended chain propagator recovers the Gaussian end-to-end distribution and, in the limits of time-like and space-like orbits, the dualism for diffusive paths and polymer random-walks.

Brane cosmological evolution with a general bulk matter configuration

Using a fully covariant treatment for the description of the bulk geometry, we study the brane cosmological evolution in the presence of a smooth bulk matter distribution. We focus on the case of a Friedmann-Robertson-Walker (FRW) brane, invariantly characterized by the existence of a six-dimensional group of isometries acting on 3D spacelike orbits. With a FRW brane, the bulk geometry can be regarded as the 5D generalization of the \emph{inhomogeneous orthogonal family of Locally Rotationally Symmetric (LRS)} spacetimes. We show that, for \emph{any bulk matter configuration}, the expansion rate on the brane depends only on the covariantly defined \emph{comoving mass} $\mathcal{M}$ of the bulk fluid within a radius equal to the average length scale of the 3D spacelike hypersurfaces of constant curvature. This unique contribution incorporates the effects of the 5D Weyl tensor and the projected tensor related to the bulk matter, and gives a transparent physical picture that includes an effective conservation equation between the brane and the bulk matter.

Nonsingular G2 stiff fluid cosmologies

In this paper we analyze Abelian diagonal orthogonally transitive space–times with spacelike orbits for which the matter content is a stiff perfect fluid. The Einstein equations are cast in a suitable form for determining their geodesic completeness. A sufficient condition on the metric of these space–times is obtained, that is fairly easy to check and to implement in exact solutions. These results confirm that nonsingular space–times are abundant among stiff fluid cosmologies.

Fuchsian analysis of S2 × S1 and S3 Gowdy spacetimes

The Gowdy spacetimes are vacuum solutions of the Einstein equations with two commuting Killing vectors having compact spacelike orbits with T3, S2 × S1 or S3 topology. In the case of T3 topology, Kichenassamy and Rendall have found a family of singular solutions which are asymptotically velocity dominated by construction. In the case when the velocity is between 0 and 1, the solutions depend on the maximal number of free functions. We consider the similar case with S2 × S1 or S3 topology, where the main complication is the presence of symmetry axes. The results for T3 may be applied locally except at the axes, where one of the Killing vectors degenerates. We use Fuchsian techniques to show the existence of singular solutions similar to the T3 case. We first solve the analytic case and then generalize to the smooth case by approximating smooth data with a sequence of analytic data. However, for the metric to be smooth at the axes, the velocity must be −1 or 3 there, which is outside the range where the constructed solutions depend on the full number of free functions. A plausible explanation is that in general a spiky feature may develop at the axis, a situation which is unsuitable for a direct treatment by Fuchsian methods.

Global Foliations of Vacuum Spacetimes withT2Isometry

We prove a global existence theorem (with respect to a geometrically defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit aT2isometry group with two-dimensional spacelike orbits, acting onT3spacelike surfaces.

SPACE–TIMES WITH A TWO-DIMENSIONAL GROUP OF CONFORMAL MOTIONS

The necessary and sufficient conditions for a space–time to admit a two-dimensional group of conformal motions (and, in particular, of homothetic motions) acting on nonnull orbits are found in the compacted spin-coefficient formalism. Although the discussion is restricted to the case of spacelike orbits, similar results are readily obtained for timelike orbits via the (modified) Sachs star operation. A number of theorems are obtained dealing with such topics as the Gaussian curvature of the group orbits, orthogonal transitivity, and hypersurface orthogonality of the conformal Killing vectors. A simple proof is presented of a generalization of a theorem due to Papapetrou.

Lorentz manifolds with homogeneous spacelike cuts

Lorentz manifolds Vn whose fundamental tensor satisfies the Einstein equations with hydrodynamical energymomentum tensor and admits a group of motions with spacelike orbits of codimension one are studied. Problems of geodesic completeness and the possibility of imbedding Vn into a Lorentz V′n satisfying certain conditions are considered.

Higher symmetries in a class of cosmological models

A particular class of solutions of Einstein's field equations, in which the source is a perfect fluid and the geometry admits a two-parameter Abelian isometry group with spacelike orbits, is examined for the possible admission of higher symmetries. It is shown that in certain cases the geometry is spatially homogeneous, being either of the ‘orthogonal’ Bianchi type I or of the ‘tilted’ Bianchi-Behr type VIUh withh = −4, and that no other cases of higher symmetry (in the sense of isometry groups) are possible. The global behaviour of the Bianchi-Behr type VIUh, models is studied, and conformal diagrams are given. This investigation is extended to the case when additional homothetic vectors are admitted, and a parallel study of the global behaviour, complete with conformal diagrams, is provided. A brief argument also shows that, within the class considered, whenever the fluid flow-lines possess identical thermal histories, the geometry is necessarily spatially homogeneous.

Spacetimes admitting a three-parameter group of isometries and quasilocal gravitational mass.

Spacetimes admitting a three-parameter group of motions multiply transitive on two-dimensional spacelike orbits allow construction of a simple geometrical invariant out of the magnitudes of Killing fields, their gradients, and the metric. We show that this invariant covariantizes and geometrizes certain forms of quasilocal introduced and discussed in the literature, in particular, Misner's in spherical symmetry and Taub's mass in plane symmetry.

Invariant conformal vectors in space-times admitting a group of G3 of motions acting on spacelike orbits S2

The paper deals with four-dimensional space-times admitting locally a three-dimensional group of motions G3 acting on two-dimensional spacelike orbits S2. The local existence problem for conformal vectors invariant under G3 is shown to be equivalent to the local existence problem for Killing vectors of a given two-dimensional pseudo-Riemannian metric g. This problem is explicitly solved in terms of the Gaussian curvature R of g and two of its scalar differential concomitants. The results are applied to the case of dust-filled space-times, where an exhaustive list of metrics has been obtained by using the algebraic computing language smp. The metrics are either homogeneous, self-similar, or Friedmann models.

Exact power law metrics in cosmology

Spatially homogeneous and non-exceptional spatially self-similar spacetime metrics which are 'exact power law metrics' are defined, explicitly parametrised and shown to have fixed conformal 3-geometry in the natural slicing of the spacetime by the orbits of the symmetry group and to admit a homothetic Killing vector field not tangent to that slicing. In fact the exact power metrics are exactly those spatially homogeneous and spatially self-similar metrics which admit a homothetic Killing vector field not tangent to the spacelike orbits of the homogeneity or self-similarity group. Such metrics arise as 'singular point solutions' of gravitational field equations when formulated as a certain system of first order differential equations. These special exact solutions play an important role in the qualitative behaviour of the general solution of a given set of field equations and sources with these symmetries.

A class of solutions of Einstein's equations which admit a 3-parameter group of isometries

The Plebanski and Stachel and Goenner and Stachel lists of metrics which are solutions of Einstein's field equations, have two double eigenvalues and admit 3-parameter groups of isometries with 2-dimensional spacelike orbits are completed by the addition of metrics which result from the use of a more general metric form.