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Register a new userA new look at the Plebanski-Demianski family of solutions
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The Plebanski-Demianski metric, and those that can be obtained from it by taking coordinate transformations in certain limits, include the complete family of space-times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non-expanding type D solutions are also identified. All special cases are derived in a simple and transparent way.
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We show that the Plebanski-Demianski spacetime persists as a solution of General Relativity when the theory is supplemented with both, a conformally coupled scalar theory and with quadratic curvature corrections. The quadratic terms are of two types and are given by quadratic combinations of the Riemann tensor as well as a higher curvature interaction constructed with a scalar field which is conformally coupled to quadratic terms in the curvature. The later is built in terms of a four-rank tensor $S_{mu u}^{ lambdarho}$ that depends on the Riemann tensor and the scalar field, and that transforms covariantly under local Weyl rescallings. Due to the generality of the Plebanski-Demianski family, several new hairy black hole solutions are obtained in this higher curvature model. We pay particular attention to the C-metric spacetime and the stationary Taub-NUT metric, which in the hyperbolic case can be analytically extended leading to healthy, asymptotically AdS, wormhole configurations. Finally, we present a new general model for higher derivative, conformally coupled scalars, depending on an arbitrary function and that we have dubbed Conformal K-essence. We also construct spherically symmetric hairy black holes for these general models.
The aim of this work is to describe the complete family of non-expanding Plebanski-Demianski type D space-times and to present their possible interpretation. We explicitly express the most general form of such (electro)vacuum solutions with any cosmological constant, and we investigate the geometrical and physical meaning of the seven parameters they contain. We present various metric forms, and by analyzing the corresponding coordinates in the weak-field limit we elucidate the global structure of these space-times, such as the character of possible singularities. We also demonstrate that members of this family can be understood as generalizations of classic B-metrics. In particular, the BI-metric represents an external gravitational field of a tachyonic (superluminal) source, complementary to the AI-metric which is the well-known Schwarzschild solution for exact gravitational field of a static (standing) source.
An infinite family of new exact solutions of the Einstein vacuum equations for static and axially symmetric spacetimes is presented. All the metric functions of the solutions are explicitly computed and the obtained expressions are simply written in terms of oblate spheroidal coordinates. Furthermore, the solutions are asymptotically flat and regular everywhere, as it is shown by computing all the curvature scalars. These solutions describe an infinite family of thin dust disks with a central inner edge, whose energy densities are everywhere positive and well behaved, in such a way that their energy-momentum tensor are in fully agreement with all the energy conditions. Now, although the disks are of infinite extension, all of them have finite mass. The superposition of the first member of this family with a Schwarzschild black hole was presented previously [G. A. Gonzalez and A. C. Gutierrez-Pi~neres, arXiv: 0811.3002v1 (2008)], whereas that in a subsequent paper a detailed analysis of the corresponding superposition for the full family will be presented.
Using the action for the instanton representation of Plebanski gravity (IRPG), we construct minisuperspace solutions restricted to diagonal variables. We have treated the Euclidean signature case with zero cosmological constant, depicting a gravitational analogy to free particle motion. This paper provides a testing ground for the IRPG for a simple case, which will be extended to the full theory in future work.
We take a closer and new look at the effects of tidal forces on the free fall of a quantum particle inside a spherically symmetric gravitational field. We derive the corresponding Schrodinger equation for the particle by starting from the fully relativistic Klein-Gordon equation in order (i) to briefly discuss the issue of the equivalence principle and (ii) to be able to compare the relativistic terms in the equation to the tidal-force terms. To the second order of the nonrelativistic approximation, the resulting Schrodinger equation is that of a simple harmonic oscillator in the horizontal direction and that of an inverted harmonic oscillator in the vertical direction. Two methods are used for solving the equation in the vertical direction. The first method is based on a fixed boundary condition, and yields a discrete-energy spectrum with a wavefunction that is asymptotic to that of a particle in a linear gravitational field. The second method is based on time-varying boundary conditions and yields a quantized-energy spectrum that is decaying in time. Moving on to a freely-falling reference frame, we derive the corresponding time-dependent energy spectrum. The effects of tidal forces yield an expectation value for the Hamiltonian and a relative change in time of a wavepackets width that are mass-independent. The equivalence principle, which we understand here as the empirical equivalence between gravitation and inertia, is discussed based on these various results. For completeness, we briefly discuss the consequences expected to be obtained for a Bose-Einstein condensate or a superfluid in free fall using the nonlinear Gross-Pitaevskii equation.
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