Generalized teleparallel theory
Eur. Phys. J. C
Generalized teleparallel theory
Ednaldo L. B. Junior 1 2
Manuel E. Rodrigues 0 2
0 Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Campus Universitário de Abaetetuba , Abaetetuba, Pará CEP: 68440-000 , Brazil
1 Faculdade de Engenharia da Computação, Universidade Federal do Pará, Campus Universitrio de Tucuruí , Tucuruí, Pará CEP: 68464-000 , Brazil
2 Faculdade de Física, PPGF, Universidade Federal do Pará , Belém, Pará 66075-110 , Brazil
We construct a theory in which the gravitational interaction is described only by torsion, but that generalizes the teleparallel theory still keeping the invariance of local Lorentz transformations in one particular case. We show that our theory falls, in a certain limit of a real parameter, under f (R¯ ) gravity or, in another limit of the same real parameter, under modified f (T ) gravity; on interpolating between these two theories it still can fall under several other theories. We explicitly show the equivalence with f (R¯ ) gravity for the cases of a Friedmann-Lemaître-Robertson-Walker flat metric for diagonal tetrads, and a metric with spherical symmetry for diagonal and non-diagonal tetrads. We study four applications, one in the reconstruction of the de Sitter universe cosmological model, for obtaining a static spherically symmetric solution of de Sitter type for a perfect fluid, for evolution of the state parameter ωDE, and for the thermodynamics of the apparent horizon.
1 Introduction
One of the most important findings in modern physics is that
our universe has accelerated expansion [1–3]. However, a
plausible common explanation for this is using the model
of a very exotic fluid called dark energy, which has
negative pressure. Another well-known possibility is to modify
Einstein’s general relativity (GR) [4], making the action of
the theory depend on a function of the curvature scalar R,
but at a certain limit of parameters the theory falls under
GR. This way to explain the accelerated expansion of our
universe is known as modified or generalized gravity.
Considering that the gravitational interaction is described only by
the curvature of space-time, we can generalize the Einstein–
Hilbert action through an analytic function of scalars of the
theory, as for example f (R¯ ) gravities [5–9], with R¯ being
the Ricci scalar or curvature scalar, f (R¯ , ) [10–13], with
being the trace of energy-momentum tensor, or yet f (G)
[14–18], f (R¯ , G) [19–24] and f (R¯ , , R¯μν μν ) [25], with
μν being the energy-momentum tensor.
An alternative to consistently describe the gravitational
interaction is one which only considers the torsion of
spacetime, thus canceling out any effect of the curvature. This
approach is known as teleparallel theory (TT) [26–29], which
is demonstrably equivalent to GR. In order to describe not
only the gravitational interaction, but also the accelerated
expansion of our universe, Ferraro and Fiorini [30] proposed
a possible generalization of the TT, which became known as
f (T ) gravity [31–62], which up to now has provided good
results in both cosmology and local phenomena of
gravitation. A key problem in f (T ) gravity is that it breaks the
invariance under local Lorentz transformations
complicating the interpretation of the relationship between all inertial
frames of the tangent space to the differentiable manifold
(space-time) [63,64]. This problem may lead to the
emergence of spurious new degrees of freedom, which are
responsible for the breakdown of the local Lorentz symmetry [65].
A consequence of the formulation of the theory using a scalar
which is not invariant under local Lorentz transformations,
the torsion scalar T in this case, is that instead of the theory
presenting differential equations of motion of fourth order,
as in the case of f (R¯ ) gravity, it has second-order differential
equations. That seems like a benefit but it is a consequence of
the fact of the local Lorentz symmetry. This generalization
of the TT still is not equivalent to a generalization f (R¯ ) for
RG.
This is the main reason why we will address the
construction of a theory that generalizes the TT, but which still keeps
the local Lorentz symmetry in a particular case. Therefore,
it is clear that we must build the function of the action with
dependence on a scalar that at some limit is invariant under
local Lorentz transformations. This will be shown soon.
The paper is organized as follows. In Sect. 2 we present a
review of f (T ) gravity, introducing the functional variation
method used in this work, obtaining the equations of motion
of this theory, noting a poorly treated point at the limit to GR.
In Sect. 3 we propose the action of generalized teleparallel
theory, we obtain the equations of motion through functional
variation of the same and compare with f (T ) gravity. We
show the equivalence of our theory with f (R¯ ) gravity, in the
case of cosmology for the line element of flat FLRW metric
in Sect (...truncated)