Exact solutions in modified massive gravity and off-diagonal wormhole deformations

Sergiu I. Vacaru

doi:10.1140/epjc/s10052-014-2781-y

Abstract

There are explored off-diagonal deformations of "prime" metrics in Einstein gravity (for instance, for wormhole configurations) into "target" exact solutions in f(R,T)-modified and massive/ bi-metric gravity theories. The new classes of solutions may posses, or not, Killing symmetries and can be characterized by effective induced masses, anisotropic polarized interactions and cosmological constants. For nonholonomic deformations with (conformal) ellipsoid/ toroid and/or solitonic symmetries and, in particular, for small eccentricity rotoid configurations, we can generate wormholes like objects matching external black ellipsoid - de Sitter geometries. We conclude that there are nonholonomic transforms and/or non-trivial limits to exact solutions in general relativity when modified/ massive gravity effects are modeled by off-diagonal and/or nonholonomic parametric interactions.

Highlights

In our work, see [3,4] and references therein, we elaborated a geometric method which allows us to deform nonholonomically any ‘prime’ diagonal metric into various classes of ‘target’ off-diagonal solutions with one Killing and/or nonKilling symmetries
We address the problem of constructing deformations of prime wormhole metrics in general relativity, GR, resulting in generic off-diagonal solutions in modified gravity, MG, and theories with nonholonomically induced torsion, effective masses and bi-metric and bi-connection structures
Modified gravity theories with functional dependence on curvature and other traces of energy–momentum tensors for matter fields, torsion sources etc. and/or with contributions by massive/bi-metric and generalized connection terms for Lagrangians belong to the most active research area oriented to a solution of important problems in modern cosmology and particle physics

Summary

Introduction

See [3,4] and references therein, we elaborated a geometric method which allows us to deform nonholonomically any ‘prime’ diagonal metric into various classes of ‘target’ off-diagonal solutions with one Killing and/or nonKilling symmetries. The idea of the AFDM is to find certain classes of nonholonomic (equivalently, anholonomic/ non-integrable) frames with conventional 2 + 2 + · · · , or 3 + 2 + · · · , splitting of dimensions on (pseudo) Riemannian spacetime when the (in general, modified) Einstein equations decouple for a correspondingly defined ‘auxiliary’ connection This results in systems of nonlinear partial differential equations (PDE) which can be integrated in very general forms. It is possible to elaborate realistic physical models with nonholonomically constrained nonlinear off-diagonal gravitational and matter field interactions if we consider deformations on a small parameter (for instance, small eccentricities for ellipsoid/rotoid configurations) This allows us to construct new classes of off-diagonal solutions determining parametric deformations of wormhole and black hole physical objects resulting in new observable physical effects and more complex spacetime configurations.

Geometric preliminaries

The coefficients α βγ

Decoupling and integrability of MG field equations

Decoupling with respect to N -adapted frames

Generating off-diagonal solutions

Formal integration via polarization functions

Levi–Civita deformations in massive gravity

Induced torsion in massive gravity

Small f -modifications and massive gravity

Off-diagonal deformations of wormhole metrics

Prime metrics for 4-d wormholes

Off-diagonal de Sitter deformations in massive gravity