Violation of causality in $f(T)$ gravity

In the standard formulation, the f(T) field equations are not invariant under local Lorentz transformations, and thus the theory does not inherit the causal structure of special relativity. Actually, even locally violation of causality can occur in this formulation of f(T) gravity. A locally Lorentz covariant f(T) gravity theory has been devised recently, and this local causality problem seems to have been overcome. The non-locality question, however, is left open. If gravitation is to be described by this covariant f(T) gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its field equations allow homogeneous Godel-type solutions, which necessarily leads to violation of causality on non-local scale. Here, to look into the potentialities and difficulties of the covariant f(T) theories, we examine whether they admit Godel-type solutions. We take a combination of a perfect fluid with electromagnetic plus a scalar field as source, and determine a general Godel-type solution, which contains special solutions in which the essential parameter of Godel-type geometries, \(m^2\), defines any class of homogeneous Godel-type geometries. We show that solutions of the trigonometric and linear classes (\(m^2 < 0\) and \(m=0\)) are permitted only for the combined matter sources with an electromagnetic field matter component. We extended to the context of covariant f(T) gravity a theorem which ensures that any perfect-fluid homogeneous Godel-type solution defines the same set of Godel tetrads \(h_A^{~\mu }\) up to a Lorentz transformation. We also showed that the single massless scalar field generates Godel-type solution with no closed time-like curves. Even though the covariant f(T) gravity restores Lorentz covariance of the field equations and the local validity of the causality principle, the bare existence of the Godel-type solutions makes apparent that the covariant formulation of f(T) gravity does not preclude non-local violation of causality in the form of closed time-like curves.