Exact solutions in modified massive gravity and off-diagonal wormhole deformations
Sergiu I. Vacaru
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1
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Theory Division
, CERN,
1211 Geneva 23, Switzerland
1
Rector's Office,
Alexandru Ioan Cuza University
, Alexandru Lapusneanu street, nr. 14, UAIC, Corpus R, office 323, Iasi 700057,
Romania
We explore 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, or may not, possess 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 wormhole-like objects matching an external black ellipsoidde 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.
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The bulk of physically important exact solutions in gravity
theories (for instance, defining black holes and wormholes)
are described by metrics with two Killing symmetries, see for
summaries of the results the monographs [1, 2]. For such
solutions, there are certain canonical frames of reference, when
the coefficients of fundamental geometric/physical objects
depend generically on one or two (from maximum four, in
four dimensional, 4-d, theories) spacetime coordinates. This
class of metrics can be diagonalized by coordinate
transformations or contain off-diagonal terms generated by rotations.
To construct generic off-diagonal solutions parameterized by
metrics with six independent coefficients depending
generically on three and/or, in general, on all spacetime
coordinates is a very difficult technical and geometric task and the
a e-mail: ;
b Associated visiting researcher at CERN
physical meaning of such generalized/modified, or Einstein,
spacetimes is less clear.
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. For deformations on a small
parameter, the new classes of target solutions may preserve certain
important physical properties of a prime metric (for instance,
of a black hole/ring one, or for a wormhole) but may also
possess new characteristics related to anisotropic polarizations
of constants, nonlinear off-diagonal interactions with new
symmetries etc.
Wormhole configurations with spacetime handles
(shortcuts), non-trivial topology and exotic matter [5] have attracted
attention for theoretical probes of foundations of gravity
theories and as possible objects of nature (for reviews, see [68]
and references therein). Such solutions are determined in
reverse direction when some tunneling metrics of prescribed
(for instance, spherical and/or conformal) symmetry are
considered and then one could try to find some corresponding
exotic matter sources. A number of interesting and/or
peculiar solutions were found when time-like curves and
respective causality violations are allowed, for stressenergy
tensors with possible violation of the null energy conditions.
The wormhole subjects were revived some times in
connection to black hole solutions, coupling with gauge interactions,
singularities, generalized/modified gravity theories etc.
We studied locally anisotropic wormhole and/or flux tubes
in five dimensional (5-d) gravity [911]. Such objects can be
determined by extra dimensional or warped/ trapped
configurations and/or possible ellipsoidal, toroidal, bipolar,
solitonic etc. gravitational polarizations of vacuum and/or
gravitational constants. The corresponding spacetime geometries
are described by generic off-diagonal metrics1 with
coeffi
1 These cannot be diagonalized by coordinate transformations.
cients depending on three or four coordinates and various
types of (pseudo) Riemannian or non-Riemannian
connections.
In the present paper, 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. We shall work with two equivalent
connections (the LeviCivita and an auxiliary one) defined
by the same metric structure and apply and extend the
anholonomic frame deformation method (AFDM, see details in
[3, 4], and references therein) of constructing exact solutions
in gravity theories.
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 space (...truncated)