Static spherically symmetric solutions in general projective relativity

Static spherically symmetric solutions have been obtained for general projective relativity withn=0 andn0 both in isotropic and curvature coordinates. In curvature coordinates, only a restricted exact solution is possible. However, an approximate solution can always be obtained following a method similar to Vanden Bergh. In these spacetimes there is no horizon, but only a naked singularity atr=0. Thus there are no black holes. It is shown that there is no solution in static, spherically symmetric, conformally flat spacetime.     

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.     

Gradient conformal killing vectors and exact solutions

We establish a connection between conformally related Einstein spaces and conformai killing vectors (CKV). We begin with the conformal map and prove that (a) under the conformal mapping¯g _ik=^–2g_ik, the necessary and sufficient condition for the tracefree part of the Ricci tensor (S _ik=R_ik–(R/4)g _ik) to remain invariant is that _i is a CKV ofg _ik, and (b) the most general form for for conformally flat Einstein space, which is the de Sitter space, is composed of three terms each of which alone represents a flat space. The existence of gradient CKV (GCKV) is examined in relation to vacuum and perfect fluid spacetimes.     

Jacobi,Ellipsoidal Coordinates and Superintegrable Systems

Abstract

We describe Jacobi’s method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n ? 1 (classically) functionally independent second-order symmetry operators. (The 2n ? 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.     

Separation of variables and symmetry operators for the conformally invariant Klein-Gordon equation on curved spacetime

The separability of the conformally invariant Klein-Gordon equation and the Laplace-Beltrami equation are contrasted on two classes of Petrov type D curved spacetimes, showing that neither implies the other. The second-order symmetry operators corresponding to the separation of variables of the conformally invariant Klein-Gordon equation are constructed in both classes and the most general second-order symmetry operator for the conformally invariant Klein-Gordon operator on a general curved background is characterized tensorially in terms of a valence two-symmetric tensor satisfying the conformal Killing tensor equation and further constraints.     

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.     

Conformal flatness, non-Abelian Kaluza-Klein reduction and quaternions

The non-Abelian Kaluza-Klein reduction of conformally flat spaces is considered for arbitrary dimensions and signatures. The corresponding equations are particularly elegant when the internal space supports a global Killing parallelization. Assuming this imposes the generalized ‘spacetime’ to be maximally symmetric with holonomy in the unitary quaternionic group Sp(d/4). Recalling an analogous result for the complex case, we conclude that all special manifolds with constant properly ‘holonomy-related’ sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza-Klein spaces.     

Conformally flat tilted Bianchi Type-V cosmological models in general relativity

We have investigated two conformally flat tilted Bianchi Type-V cosmological models in general relativity. To get a determinate solution, we have assumed a supplementary conditionA =B ⁿ between metric potentials wheren is a constant. The behaviour of the model forn = 2 is discussed in detail. Various physical and geometrical aspects of the models are also discussed.     

Conformally Flat Spacetimes and Weyl Frames

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordstr?m scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.     

Type O pure radiation metrics with a cosmological constant

In this paper we complete the integration of the conformally flat pure radiation spacetimes with a non-zero cosmological constant Λ, and , by considering the case . This is a further demonstration of the power and suitability of the generalised invariant formalism (GIF) for spacetimes where only one null direction is picked out by the Riemann tensor. For these spacetimes, the GIF picks out a second null direction (from the second derivative of the Riemann tensor) and once this spinor has been identified the calculations are transferred to the simpler GHP formalism, where the tetrad and metric are determined. The whole class of conformally flat pure radiation spacetimes with a non-zero cosmological constant (those found in this paper, together with those found earlier for the case ) have a rich variety of subclasses with zero, one, two, three, four or five Killing vectors.     

Fields of a viscous fluiid without shear and rotation

It is shown that in Einstein's theory of gravitation conformally flat gravitational fields of a viscous fluid without shear and rotation are analogous to the fields of an ideal fluid with a conformally flat space-time. Thus, the fluid particles in such fields move along geodesics, and the three-dimensional hypersurfaces determined by the sections x⁰=const are spaces of constant curvature.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 36–40, January, 1981.     

Topological mass generation of scalar particles in the static flat Clifford-Klein-Robertson-Walker spacetimes

In spacetimes with topologies Tⁿ X R^4?n and flat metrics, n = 2, 3, where T = S¹ is a circle and in the massless ø⁴ theory, the masses of scalar multiplets produced both by self-interaction and by nontrivial spacetime topology are evaluated.     

Comments on a paper by Collinson

The demonstration of the uniqueness of the Schwarzschild interior metric within conformally flat axisymmetric stationary spacetimes is revised. A complete proof containing the three possible branches of interior fluid solutions is given.     

Gauge invariance versus masslessness in de Sitter spaces

The connection between gauge invariance, masslessness and null cone propagation is a flat space property which does not persist even in constant curvature geometries. In particular, we show that both the gauge invariant spin $\text{3}\text{2}$ and 2 fields in anti-de Sitter space have support inside the cone, whereas where are conformally invariant, but gauge variant, models which do propagate on the light cone. The Maxwell field in constant curvature spaces of dimension other than four also does not have null cone propagation; again there is a conformally invariant model which does.     

On CP 1 and CP 2 Maps and Weierstrass Representations for Surfaces Immersed into Multi-Dimensional Euclidean Spaces

Abstract

An extension of the classic Enneper–Weierstrass representation for conformally parametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP ¹ and CP ² sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3- and 8-dimensional spaces, respectively. Relations of Weierstrass type systems to the equations of these sigma models are established. In particular, it is demonstrated that the generalised Weierstrass representation can admit different CMC-surfaces in ?³ which have globally the same Gauss map. A new procedure for constructing CMC-surfaces in ?ⁿ is presented and illustrated in some explicit examples.     

Obtaining a class of type O pure radiation metrics with a negative cosmological constant,using invariant operators

Using the generalised invariant formalism we derive a special subclass of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N, respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors.     

Static conformally flat solutions in a general scalar-tensor theory of gravitation

Explicit field equations in the general scalar-tensor theory of gravitation proposed by Nordtvedt are obtained with the aid of a static spherically symmetric conformally flat metric. Exact static solutions of Nordtvedt-Barker field equations both in vacuum and in the presence of a source-free electromagnetic field are presented and studied. It is shown that there are no spherically symmetric static conformally flat solutions of Nordtvedt-Barker field equations representing perfect fluid distribution with disordered radiation obeying the equation of state=3p, except for the trivial empty flat space-time of Einstein's theory.     

On fluid spheres of uniform density in general relativity

It is shown that for spherically symmetric perfect fluid solutions, with spatial isotropy and uniform density, the free gravitational fieldproduces a singularity at the centre of the sphere and when this singularity is removed the space-time is conformally flat.It is pointed out that the interior geometry of the fluid spheres with uniform density given by Thompson and Whitrow, and others, is conformally flat, and hence the spacetime is of class one.     

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.     

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.