Spherically symmetric space-times transparent to scalar multipole waves

Using nonscattering potentials of Chang and Janis, a large class of spherically symmetric space-times is constructed on which all multipole solutions to the minimally coupled scalar wave equation are expressible in terms of characteristic data functions in essentially as simple a fashion as for flat space-time. The space-times are transparent to multipole waves in the same sense that flat space-time is. Both conformally flat and not conformally flat space-times are obtained. Some examples are discussed which show that the variety of transparent space-times is large even within the class of Robertson-Walker spaces.     

The uniqueness of the Schwarzschild interior metric

It is shown that every conformally flat axisymmetric stationary space-time is necessarily static, and that if the source is a perfect fluid then the space-time metric is the usual Schwarzschild interior metric.     

Generalized Darboux maps and massless spin-s fields

We consider conformally invariant massless spin-s field equations on a spherically symmetrical space-time. Precisely when these equations are consistent appropriately defined field components are shown to satisfy wave equations related by a generalization of the classical Darboux map.     

The apparent size of distant objects

A formula in terms of the exponential map is derived, which reveals for any observer in an arbitrary pseudo-Riemannian space-time the apparent diameter (i.e., the angular diameter) of any distant object. The formula shows immediately the existence of the effect of apparent distortion of the objects. It is proved that this distortion is absent if and only if the space-time is conformally flat.     

Conformal groups and conformally equivalent isometry groups

It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper conformal gradient every (local) conformal group is conformally homothetic. In the Riemannian case there is always a conformal scalar unless the metric is conformally Euclidean. In the case of a Lorentzian 4-manifold it is proved that the only metrics with no conformal scalars (and hence the only ones admitting a (local) conformal group not conformally isometric) are either conformal to the plane wave metric with parallel rays or conformally Minkowskian.     

M-theory on eight-manifolds

We show that in certain compactifications of M-theory on eight-manifolds to three-dimensional Minkowski space-time the four-form field strength can have a non-vanishing expectation value, while an N = 2 supersymmetry is preserved. For these compactifications a warp factor for the metric has to be taken into account. This warp factor is non-trivial in three space-time dimensions due to Chern-Simons corrections to the fivebrane Bianchi identity. While the original metric on the internal space is not Kahler, it can be conformally transformed to a metric that is Kähler and Ricci flat, so that the internal manifold has SU(4) holonomy.     

Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.     

The Hadamard Construction of Green's Functions on a Curved Space-Time with Symmetries

The Hadamard constituents of Green's functions for a ζ-parametrized generalization of the massless scalar d'Alembert equation to a curved space-time including the conformally invariant wave equation: the world function of space-time, the transport scalar, and the tail-term coefficients, being simultaneously coefficients in the Schwinger-DeWitt expansion of the Feynman propagator for the corresponding invariant Klein-Gordon equation, are considered on a general static spherically symmetric and (2,2)-decomposable metric. The construction equations determining the Hadamard building elements are cast into a symmetry-adapted form and used to obtain, on a specific model metric, exact explicit solutions.     

Conformally plane solutions of the maxwell-einstein equations

A special case of the Rainich problem is considered. The class of conformally plane solutions of the Einstein equations containing the energy-momentum tensor of Isotropic electromagnetic radiation is determined. The relationship between the conformal function which defines the metric of the conformally plane space-time and the electromagnetic field intensities which produce this space-time is established.Translated from Izvestiya VUZ. Fizika, Vol. 11, No. 8, pp. 76–80, August, 1968.     

Solitary waves on a curved space-time

A nonlinear partial differential equation is derived which admits plane solitary waves on a conformally flat Riemannian space-time. The metric is determined by the amplitude of these waves. By interpreting these solitary waves as particles we arrive at the following picture: these particles are confined to regions exhibiting singular (very large) amplitudes in an otherwise continuous wavetrain. There is, thus, no distinction between the notion of a particle and that of a wave.     

Three-surface twistors and conformal embedding

The three-surface twistor equation is defined for an arbitrary three-surface in an arbitrary curved spaceM. It is proved that three-surface twistors exist on 2 if and only if can be embedded in a conformally flat space-time with the same first and second fundamental forms.     

On the rescaling problem of space-times admitting groups of conformal motions

The necessary and sufficient conditions are given that a space-time admitting a group of conformal motions can be mapped conformally on a space-time admitting the same group but of Killing symmetries.     

Quantum conformal fluctuations near the classical space-time singularity

This paper investigates the behavior of conformal fluctuations of space-time geometry that are admissible under the quantized version of Einstein's general relativity. The approach to quantum gravity is via path integrals. It is shown that considerable simplification results when only the conformal degrees of freedom are considered under this scheme, so much so that it is possible to write down a formal kernel in the most general case where the space-time contains arbitrary distributions of particles with no other interaction except gravity. The behavior of this kernel near the classical space-time singularity then shows that quantum fluctuations inevitably diverge near the singularity. It is shown further that the root cause of this divergence lies in the fact that the Green's function for the conformally invariant scalar wave equation diverges at the singularity. The limitations on the validity of classical general relativity near the space-time singularity are discussed and it is argued that the notion of singularity itself needs to be radically modified once the quantum effects are taken into account.On leave of absence from the Tata Institute of Fundamental Research, Bombay, India     

New Lagrangians for Bosonic m-branes with vanishing cosmological constant

A class of conformally invariant σ model actions in 2n dimensions is shown to be classically equivalent to the Nambu-Goto action for an extended object, an m-brane (m+1=2n), embedded in a higher dimensional space-time (d⪖m+1). when m is even, a (2n + 1)-dimensional σ model action is also constructed, which is classically equivalent to the Nambu-Goto action, but in this case there is no conformal invariance. In both cases the cosmological constant can be set to zero.     

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.     

Cosmological Particle Creation and Dynamical Casimir Effect

We compute particle creation for a real massive scalar field conformally coupled to a spatially closed Robertson–Walker space-time background, with time-dependent scale factor. This is a dynamical Casimir effect with moving boundaries.     

pp waves of conformal gravity with self-interacting source

Recently, Deser, Jackiw and Pi have shown that three-dimensional conformal gravity with a source given by a conformally coupled scalar field admits pp wave solutions. In this paper, we consider this model with a self-interacting potential preserving the conformal structure. A pp wave geometry is also supported by this system and, we show that this model is equivalent to topologically massive gravity with a cosmological constant whose value is given in terms of the potential strength.     

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.     

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.     

An introduction to the theory of local twistors

This paper deals with the formalism of local twistors, which has developed from the twistor algebra, and extends some of the basic twistor concepts to curved space-time. Essentially, the central ideas are to define a twistor space at each point of the spacetime, and to define a covariant derivative so that an operation of local twistor transport is possible; this leads to the definition of a conformally invariant curvature twistor. In an appendix, some conformally invariant spinors are discussed.