A Rigidity Property of Asymptotically Simple Spacetimes Arising from Conformally Flat Data

Given a time symmetric initial data set for the vacuum Einstein field equations which is conformally flat, it is shown that the solutions to the regular finite initial value problem at spatial infinity extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data coincides with Schwarzschild data.     

Transformation Laws for Decomposable World-Spin Affinities in a Class of Conformally Flat Spacetimes

A class of conformally flat spacetimes that admit certain decomposable world-spin affine patterns is considered. New coordinate-derivative relations are particularly utilized to demonstrate that the procedures involved in a well-known spinor translation of the corresponding Riemann and Ricci tensors bear world invariance. The establishment of this invariance property will presumably shed some light upon the overall spacetime situation taken up by the underlying works.     

Schwarzschild Interior in Conformally Flat Form

A unified conformally flat form of the static Schwarzschild interior space–times is provided. A new parameter that allows us to analyze the symmetry (spherical, plane or hyperbolic) of the three well known classes of metrics is introduced. In the spherically symmetric case, this parameter is related to the historical limit value of the mass to radius ratio found by Schwarzschild for a sphere of incompressible fluid.     

Perfect Fluid Spacetimes with a Purely Magnetic Weyl Tensor

We present a solution of the gravitational fieldequations which is similar in form to that given byWainwright. Several cases are considered, in particularwe find a general algebraic perfect fluid solution with equation of state p = whose Weyl tensor is of the purely magnetic type within a finiteregion of the spacetime. It is shown, for an observerwith four-velocity, u_mag say, that themetric's Weyl tensor is purely magnetic within the finiteregion while it is purely electric, as read by anotherobserver with four-velocity u_ele, elsewhere.Another observer, independent of the observers whomeasure the Weyl tensor to be purely electric ormagnetic, interprets the perfect fluid to have anequation of state p = . The Petrov type of themetric, in this case, is I(M) by theArianrhod-McIntosh classification and therefore there exists noconformally related metric which is vacuum. The vacuumseed metrics are derived for the perfect fluidsolutions.     

Equatorial Circular Orbits of Neutral Test Particlesin Weyl Spacetimes

A general stability study of equatorial circular orbits in static axially symmetric gravitating systems is presented. We investigate the motion of neutral test particles in circular geodesics such as the marginally stable orbit, the marginally bounded orbit, and the photon orbit are analyzed. We find general expressions for the radius, specific energy, specific angular momentum, and the radius of the marginally stable orbit, both for null and timelike circular geodesics. Different solutions were expressed in different coordinates systems: cylindrical coordinates, oblate spheroidal coordinates, and prolate spheroidal coordinates are considered. We show that all null circular trajectories are unstable, and that there aren’t marginally stable null geodesics, whereas for timelike geodesics the motion can be unbounded, bounded, or circular.     

Lie Symmetries for a Conformally Flat Radiating Star

We consider a relativistic radiating spherical star in conformally flat spacetimes. In particular we study the junction condition relating the radial pressure to the heat flux at the boundary of the star which is a nonlinear partial differential equation. The Lie symmetry generators that leave the equation invariant are identified and we generate an optimal system. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analysed. We identify new categories of exact solutions to the boundary conditions. Two classes of solutions are of interest. The first class depends on a self similar variable. The second class is separable in the spacetime variables.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when is a generalized Robertson-Walker spacetime.     

Creation of Scalar and Dirac Particles in Asymptotically Flat Robertson-Walker Spacetimes

In the present article we obtain the exact solutions of the Klein-Gordon and Dirac equations for two models of Robertson-Walker spaces with asymptotically Minkowskian regions. Using the obtained exact solutions we calculate the density of scalar and Dirac particles created through Bogolubov transformations technique. For Dirac field it is shown that the creation rate of particles and anti particles are equal.     

Transverse Frames for Petrov Type I Spacetimes: A General Algebraic Procedure

We develop an algebraic procedure to rotate a general Newman-Penrose tetrad in a Petrov type I spacetime into a frame with Weyl scalars ₁ and ₃ equal to zero, assuming that initially all the Weyl scalars are non vanishing. The new frame highlights the physical properties of the spacetime. In particular, in a Petrov type I spacetime, setting ₁ and ₃ to zero makes apparent the superposition of a Coulomb-type effect ₂ with transverse degrees of freedom ₀ and ₄.     

Static Conformally Flat Spherically Symmetric Space-Time in Einstein-Cartan Theory of Gravitation

The exact solution of Einstein-Cartan field equations for static, conformally flat spherically symmetric space-time is derived and it is proved to be Petrov-type D.     

The Invariant-Relat ed Unitary Transformation Method and the Evolution of a Dirac Field in Friedmann-Robertson-Walker Flat Spacetimes

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of a quantum Dirac field in Friedmann-Robertson-Walker spatially flat spacetime. We first solve the functional Schrodinger equation for a free Dirac field and obtain the exact solutions. We then investigate the way of extending the method to treat the case in which there is an interaction between the Dirac field and a scalar field.     

Operational Spacetimes

Time, positions and spacetimes are modelled by groups and subgroup classes. Their Hilbert space representations define particles and interactions. Electroweak spacetime, proposed as nature-realized manifold, represents the homogeneous groups of the electroweak standard model. The leptonic and hadronic (quark) sectors, characteristic for particles and interactions, are related to the embedded time and position operations, respectively.     

Invariant Solutions of Einstein’s Field Equations for Conformally Flat Fluid Spheres of Embedding Class One

In the present paper, conformal spherical symmetric perfect fluid solutions to Einstein’s field equations are obtained by using the invariance of the equations under the Lie group of transformations. All the solutions thus obtained are analyzed physically.     

Spacetimes with killing tensors

For Einstein-Maxwell fields for which the Weyl spinor is of type {2, 2}, and the electromagnetic field spinor is of type {1, 1} with its principal null directions coaligned with those of the Weyl spinor, the integrability conditions for the existence of a certain valence two Killing tensor are shown to reduce to a simple criterion involving the ratio of the amplitude of the Weyl spinor to the amplitude of a certain test solution of the spin two zero restmass field equations. The charged Kerr solution provides an example of a spacetime for which the criterion is satisfied; the chargedC-metric provides an example for which it is not.This piece of work was completed, in part, during the authors' summer 1972 stays at The University of Texas at Dallas, Division of Mathematics and Mathematical Physics, the Max-Planck-Institut für Physik und Astrophysik in München, and the Black Hole session of the Ecole d'été de Physique Théorique in Les Houches; supported, in part, by the National Science Foundation, Grants GP-8868, GP-3463 9X, GP-20023, and GU-1598; the Air Force Office of Scientific Research, Grant 903-67; the National Aeronautics and Space Administration, Grant 44-004-001; the Westinghouse Corporation; the Clark Foundation; and the Rhodes Trust.     

Uniqueness Theorem for BMS-Invariant States of Scalar QFT on the Null Boundary of Asymptotically Flat Spacetimes and Bulk-Boundary Observable Algebra Correspondence

This work concerns some features of scalar QFT defined on the causal boundary of an asymptotically flat at null infinity spacetime and based on the BMS-invariant Weyl algebra .(a) (i) It is noticed that the natural BMS invariant pure quasifree state λ on , recently introduced by Dappiaggi, Moretti and Pinamonti, enjoys positivity of the self-adjoint generator of u-translations with respect to every Bondi coordinate frame on , ( being the affine parameter of the complete null geodesics forming and complex coordinates on the transverse 2-sphere). This fact may be interpreted as a remnant of the spectral condition inherited from QFT in Minkowski spacetime (and it is the spectral condition for free fields when the bulk is the very Minkowski space). (ii) It is also proved that the cluster property under u-displacements is valid for every (not necessarily quasifree) pure state on which is invariant under u displacements. (iii) It is established that there is exactly one algebraic pure quasifree state which is invariant under u-displacements (of a fixed Bondi frame) and has positive self-adjoint generator of u-displacements. It coincides with the GNS-invariant state λ. (iv) Finally it is shown that in the folium of a pure u-displacement invariant state ω (like λ but not necessarily quasifree) on is the only state invariant under u-displacement.(b) It is proved that the theory can be formulated for spacetimes asymptotically flat at null infinity which also admit future time completion i ⁺ (and fulfill other requirements related with global hyperbolicity). In this case a -isomorphism ı exists - with a natural geometric meaning - which identifies the (Weyl) algebra of observables of a linear field propagating in the bulk spacetime with a sub algebra of . Using ı a preferred state on the field algebra in the bulk spacetime is induced by the BMS-invariant state λ on .     

Projectively and Conformally Invariant Star-Products

We consider the Poisson algebra S(M) of smooth functions on T ^* M which are fiberwise polynomial. In the case where M is locally projectively (resp. conformally) flat, we seek the star-products on S(M) which are SL(n+1,) (resp. SO(p+1,q+1))-invariant. We prove the existence of such star-products using the projectively (resp. conformally) equivariant quantization, then prove their uniqueness, and study their main properties. We finally give an explicit formula for the canonical projectively invariant star-product.     

Conformally connected universes

A well-known difficulty associated with the conformal method for the solution of the general relativistic Hamiltonian constraint is the appearance of an aphysical “bag of gold” singularity at the nodal surfaces of the conformal factor. This happens whenever the background Ricci scalar is too large. Using a simple model we demonstrate that some of these singular solutions do have a physical meaning, and that these can be considered as initial data for Universes containing black holes, which are connected, in a conformally nonsingular way with each other. The relation between the ADM mass and the horizon area in this solution supports the cosmic censorship conjecture.     

Conformally flat spaces and gravitation

This article considers the theory of gravity which is defined by R ² as the free Lagrangian. The resulting equations are conformally invariant, and their equivalence to Einstein's equation is demonstrated (provided the stress tensor is traceless). The possibility of adapting this theory to massive point particles on a conformally flat background is discussed.     

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.