The applications of spherically symmetric solutions of the massless scalar Einstein equations to cosmic censorship are discussed. A new nonstatic solution to these equations is given. The Vaidya form of Wyman's solution is constructed and is shown to obey reasonable energy conditions.

Scalar Field Counterexamples to the Cosmic Censorship Hypothesis

It is known that Lorentzian wormholes must be threaded by matter that violates the null energy condition. We phenomenologically characterize such exotic matter by a general class of microscopic scalar field Lagrangians and formulate the necessary conditions that the existence of Lorentzian wormholes imposes on them. Under rather general assumptions, these conditions turn out to be strongly restrictive. The most simple Lagrangian that satisfies all of them describes a minimally coupled massless scalar field with a reversed sign kinetic term. Exact, nonsingular, spherically symmetric solutions of Einstein's equations sourced by such a field indeed describe traversable wormhole geometries. These wormholes are characterized by two parameters: their mass and charge. Among them, the zero mass ones are particularly simple, allowing us to analytically prove their stability under arbitrary space-time dependent perturbations. We extend our arguments to nonzero mass solutions and conclude that at least a nonzero measure set of these solutions is stable.

On a class of stable, traversable Lorentzian wormholes in classical general relativity

We describe the different possibilities that a simple and apparently quite harmless classical scalar field theory provides to violate the energy conditions. We demonstrate that a non-minimally coupled scalar field with a positive curvature coupling ξ>0 can easily violate all the standard energy conditions, up to and including the averaged null energy condition (ANEC). Indeed, this violation of the ANEC suggests the possible existence of traversable wormholes supported by non-minimally coupled scalars. To investigate this possibility we derive the classical solutions for gravity plus a general (arbitrary ξ) massless non-minimally coupled scalar field, restricting attention to the static and spherically symmetric configurations. Among these classical solutions we find an entire branch of traversable wormholes for every ξ>0. (This includes and generalizes the case of conformal coupling ξ = (1/6) we considered in 1999 Phys. Lett. B 466 127-34.) For these traversable wormholes to exist we demonstrate that the scalar field must reach trans-Planckian values somewhere in the geometry. We discuss how this can be accommodated within the current state of the art regarding scalar fields in modern theoretical physics. We emphasize that these scalar field theories, and the traversable wormhole solutions we derive, are compatible with all known experimental constraints from both particle physics and gravity.

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Scalar fields, energy conditions and traversable wormholes

http://homepages.ecs.vuw.ac.nz/~visser/Articles/Journals/plb-wormhole.pdf

Traversable wormholes from massless conformally coupled scalar fields

We show that the well-known most general static and spherically symmetric exact solution to the Einstein-massless scalar equations given by Wyman is the same as one found by Janis, Newman and Winicour several years ago. We obtain the energy associated with this space–time and find that the total energy for the case of the purely scalar field is zero.

http://cds.cern.ch/record/318047/files/9701021.pdf

Janis-Newman-Winicour and Wyman solutions are the same

The Schwarzschild, Reissner-Nordstroem, and Kerr exterior solutions in general relativity are reconsidered adding to the vacuum a massless scalar field. The event horizons in the modified solutions all reduce to a point, thus preventing the formation of black holes.

Gravitation without black holes.

In recent years much work has been devoted to the development of the (i singularity theorems ~ of Hawking and Penrose in order to have a bet ter understanding of the properties of space-times in Einstein 's general theory of re la t iv i ty (1). In spite, however, of the significant advances made in the subject, many impor tant problems of the classical theory of singularities remain still unsolved. The results obtained t i l l now, support the feeling tha t physically realistic solutions to the Einstein 's equations that are singulari ty free can only be obtained either by modifying the field equations or by looking for some missing physics in the applied general relat ivi ty. To be more concrete, let us consider the singulari ty which occurs in deriving the metric external to a static spherically symmetr ic body of mass qn, tha t is the Schwarzschild curvature singularity. This is a part icularly simple situation so, assuming the val id i ty of Einstein 's equations of general relativity, we can t ry to look for a minimal physical ansatz which can lead to singularity-free solutions. The most obvious suggestion might be to explore which physical proper ty one could at t r ibute to the body, besides its mass m, in order to get a well-behaved line element. Otherwise stated, between the many parameter familes of asymptot ical ly flat solutions which generalize che Schwarzschild case, one should succeed in choosing a suitable set of parameter to which correspond regular solutions. In stat ionary nonspherieal situations, however, i t is known tha t the Kerr-Newman solutions, where the parameters are the mass m, the electric charge q, the magneticmonopole charge p and a rotation parameter a, still display horizons, naked singularities and other unwanted phenomena. The simplest generalization of the Schwarzsehild problem, in which to the mass m is associated a scalar charge a, does not seem to have been adequately t reated in the li terature.

It is well known [1] that the three-dimensional space 

$$ {d^{(3)}}{8^2} = {{d{r^2}} \over {1 - {{{r^2}} \over {{R^2}}}}} + {r^2}(d{\vartheta ^2} + {\sin ^2}\vartheta d{\varphi ^2}) $$

(1)

of some models of closed homogeneous and isotropic universes has an especially simple geometry which can be seen best introducing a angular coordinate 0 ≤ ϰ ≤ π via r = R sin ϰ and transforming the line element (1) into the form 

$$ {d^{(3)}}{8^2} = {R^2}(d{\chi ^2} + {\sin ^2}\chi d{\Omega ^2}) $$

(2)

where 

$$ d{\Omega ^2} = d{\vartheta ^2} + {\sin ^2}\vartheta d{\varphi ^2} $$

(3)

The metric (2) is that of a three-dimensional hypersurface of radius R which can be represented in a flat, four-dimensional Euclidean embedding space.

Schwarzschild Metrics, Quasi-Universes and Wormholes

The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows us to obtain some results that otherwise are less apparent or even hidden in other coordinate systems.

Schwarzschild Metrics and Quasi-Universes

A one-parameter family of static and spherically symmetric solutions to Einstein equations with a traceless energy-momentum tensor is found. When the nonzero parameter $\beta$ lies in the open interval $(0,1)$ one obtains traversable Lorentzian wormholes. One also obtains naked singularities when either $\beta 1$ and the Schwarzschild black hole for $\beta = 1$.

A. G. Agnese

Papers

Gravitation without black holes.

Gravitation without black holes.

Schwarzschild Metrics, Quasi-Universes and Wormholes

Schwarzschild Metrics and Quasi-Universes

Traceless stress-energy and traversable wormholes