On the projectivity of conformally flat spaces created by a hydrodynamical energy-momentum tensor

It is shown that under the condition _ju _k = _k u_j imposed on the mapping function the geodesics in conformai gravitational fields are the same. The following fact is also established: all conformally flat spaces satisfying this condition correspond to the gravitational fields of an ideal fluid.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 16–19, 1972.     

Flat twistor spaces,conformally flat manifolds and four-dimensional field theory

A definition is proposed of four-dimensional conformal field theory in which the Riemann surfaces of two-dimensional CFT are replaced by (Riemannian) conformally flat four-manifolds and the holomorphic functions are replaced by solutions of the Dirac equation. The definition is investigated from the point of view of twistor theory, allowing methods from complex analysis to be employed. The paper fills in the main mathematical details omitted from the preliminary announcement [15].     

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.     

Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics

Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras–McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras–McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we obtain complete agreement with the results obtained in intrinsic coordinates. We have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.     

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.     

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.     

On the curvature of piecewise flat spaces

We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.Supported in part by NSF MCS-810-2758-A-02Supported in part by Deutsche Forschungsgemeinschaft and NSF PHY-81-09110-A-01On leave of absence from Freie Universität Berlin     

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.     

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.     

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.     

Yano vector fields and conformally related spaces

We discuss some geometrical properties of Killing forms of order three and exhibit a procedure for constructing spaces admitting such tensors. As an example we prove that all conformally flat spaces admitting such a tensor are related to spaces of constant curvature.On leave April 1979 from the Service de Mécanique et Relativité Générales, Université de l'Etat à Mons, Belgium.     

Nontrivial conformally steckel metrics in einstein spaces

All isotropic conformally Steckel metrics are derived that satisfy the Einstein vacuum equations containing a Λ term and do not reduce to the purely Steckel type. The metrics allow the integration of the Hamilton-Jacobi equation for a massless particle by complete variable separation. Tomsk State Pedagogic University. Tomsk State University. High-Current Electronics Institute, Siberian Division, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 74–78, October, 1997.     

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.     

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.     

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.