Nonexistence of conformally flat slices in Kerr and other stationary spacetimes

It is proved that stationary solutions to the vacuum Einstein field equations with a nonvanishing angular momentum have no Cauchy slice that is maximal, conformally flat, and nonboosted. The proof is based on results coming from a certain type of asymptotic expansion near null and spatial infinity--which also show that the development of Bowen-York-type data cannot have a development admitting a smooth null infinity--and from the fact that stationary solutions do admit a smooth null infinity.     

Axiomatic renormalization of the stress tensor of a conformally invariant field in conformally flat spacetimes

Using only the general properties which the renormalized stress-energy tensor T_μν should satisfy—and not relying on any assumptions associated with specific renormalization techniques—we derive the expression for T_μν for conformally invariant fields in conformally flat spacetimes of two and four dimensions. In two dimensions, these arguments rederive the Davies-Fulling-Unruh expression for the stress tensor of a scalar field; in four dimensions the results agree with those of Brown and Cassidy, except that we exclude the local curvature term depending on fourth-order derivatives of the metric. The dynamics of a k = 0 Robertson-Walker universe filled with radiation of the conformally invariant field is investigated and it is found that the equations cease to admit a solution when the Planck density is reached.     

Dissipative fluid in conformally flat space-time

We consider a conformally flat, inhomogeneous solution of the Einstein equations for a dissipative fluid. The production of entropy is found to depend on some arbitrary functions of time. By some subsidiary conditions, such a model is shown to evolve into a homogeneous Friedmann-type universe.     

Kinemaitcs in spatially flat FLRW spacetimes

The kinematics on spatially flat FLRW spacetimes is presented for the first time in local charts with physical coordinates, i.e., the cosmic time and proper Cartesian space coordinates of Painlevé-type. It is shown that there exists a conserved momentum that determines the form of the covariant four-momentum on geodesics in terms of physical coordinates. Moreover, with the help of this conserved momentum, the peculiar momentum can be defined, thus separating the peculiar and recessional motions without ambiguity. It is shown that the energy and peculiar momentum satisfy the mass-shell condition of special relativity while the recessional momentum does not produce energy. In this framework, the measurements of the kinetic quantities along geodesics performed by different observers are analyzed, pointing out an energy loss of the massive particles similar to that producing the photon redshift. The examples of the kinematics on the de Sitter expanding universe and a new Milne-type spacetime are extensively analyzed.     

On classification of conformally flat spaces

Classification of conformally flat n-dimensional pseudo-Riemannian spaces via Plebanski's method is discussed. It is based on embedding these spaces into a flat (n + 2)-dimensional space and on finding their minimal isometry groups which are subgroups of the conformal group. In particular, the case n = 4 is given in full detail and compared with incomplete results known in the literature. The found conformally flat spacetimes are identified with the associated solutions of the Einstein equations and with the spacetimes used in various cosmological considerations.     

Long-range scalar field in conformally flat space-time

Consequences of a massless scalar field in conformally flat space-time are studied. Then a wide class of solutions of the scalar field is obtained.     

The backreaction for conformally invariant fields on a conformally flat background

We study conformally invariant fields within the context of semi-classical gravity. We claim that, generically, conformal flatness implies Friedmann-Robertson-Walker behaviour. A proof is presented here for the case in which the Ricci tensor is of the perfect fluid type. We also rewrite the field equations as a quadratic three dimensional autonomous system of ordinary differential equations, the critical points of which are Minkowski space and de Sitter space. Both these critical points are unstable in the linear as well as in the non-linear theory.This essay received an honorable mention from the Gravity Research Foundation, 1990 —Ed.     

Maximal hypersurfaces in stationary asymptotically flat spacetimes

Existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike near infinity.. The first class consists of strongly causal asymptotically flat spacetimes which contain no black hole or white hole (but may contain ergoregions where the Killing orbits fail to be timelike). The second class of spacetimes possess a black hole and a white hole, with the black and white hole horizons intersecting in a compact 2-surfaceS.Supported in part by KBN grant #2 1047 9101Supported in part by NSF grant PHY-8918388.     

Meron and elliptic solutions in conformally flat space-time

The idea recently advanced by the author that particles arise as distortions of a riemannian background is pursued further. Such distortions represent conformally flat solutions of Einstein's “cosmological” equations extremely large “cosmological” constant. It is shown in particular that merons can be generated by perfect fluid or neutral superfluid distributions of energy and momentum. Perfect fluids can also generate elliptic plane waves of the type discussed by Petiau.     

Existence of maximal surfaces in asymptotically flat spacetimes

We prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition. This uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the Dirichlet problem.     

The existence of maximal slicings in asymptotically flat spacetimes

We consider Cauchy data (g, ) on IR³ that are asymptotically Euclidean and that satisfy the vacuum constraint equations of general relativity. Only those (g, ) are treated that can be joined by a curve of sufficiently bounded initial data to the trivial data (, 0). It is shown that in the Cauchy developments of such data, the maximal slicing condition tr =0 can always be satisfied. The proof uses the recently introduced weighted Sobolev spaces of Nirenberg, Walker, and Cantor.Research partially supported by National Science Foundation Grants GP-39060 and GP-15735Research partially supported by National Science Foundation Grant GP-43909 to the University of North Carolina     

Spherically symmetric lightlike radiation and conformally flat space-times

The problem of the coexistence of spherical symmetry of a 3-space, tracelessness of the energy-momentum tensor, conformally flat 4-metrics, and the validity of the Einstein equations is investigated. The assertion is proved that when spherical symmetry is present nonequilibrium lightlike radiation with the energymomentum tensor T_µv = l _µ l _v (l v = 0) cannot serve as the source of the gravitational field corresponding to a conformally flat space-time (type 0 according to the algebraic classification). An exact spherically symmetric solution with a conformally flat metric is obtained which describes dust and equilibrium isotropic radiation without energy exchange between them. This solution is rewritten for a synchronous reference frame in which it is evident that it describes a homogeneous and isotropic universe. In the limit of the absence of radiation the solution changes into the well-known Friedmann solution for an open universe.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 32–35, July, 1984.In conclusion the authors express their gratitude to all the participants in the seminar of the gravitation section of the Scientific and Technical Council of Minvuz of the USSR (the physics faculty of Moscow State University) for a useful discussion of the results of this paper.     

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

Quantum fluctuations in the conformally flat and the schwarzschild space-times

A general technique is described for dealing with the quantum fluctuations between conformally flat space-times. The second part of the paper deals with the Schwarzschild spacetime. It is shown there that this space-time is stable against fluctuations of mass, but transitions between two space-times of different masses can be obtained via conformai fluctuations. Purely conformal fluctuations of the Schwarzschild metric are, however, damped at the event horizon. Similar conclusions are drawn about the Reissner-Nordstrom space-time.     

Trace anomaly for gauge theories in a conformally flat space

We study the trace anomaly of the energy-momentum tensor for gauge theories coupled to massless fermions in the context of a self-consistent cooperative phenomenon for the creation of the universe.     

On conformally flat and type N pure radiation metrics

We study pure radiation spacetimes of algebraic types O and N with a possible cosmological constant. In particular, we present explicit transformations which put these metrics, that were recently re-derived by Edgar, Vickers and Machado Ramos, into a general Ozsváth–Robinson–Rózga form. By putting all such metrics into the unified coordinate system we confirm that their derivation based on the GIF formalism is correct. We identify only few trivial differences.