G\"odel-type universes in Lorentz-violating Kalb-Ramond gravity with Ricci- and Riemann-tensor nonminimal couplings

Abstract

This paper studies homogeneous \Godel-type geometries in a gravitational model where local Lorentz symmetry is spontaneously broken by the vacuum expectation value of an antisymmetric Kalb-Ramond two-form. We consider the Einstein-Hilbert action supplemented by two independent nonminimal curvature couplings: a Ricci-tensor coupling of the form $B_{a}{}^{c}B_{bc}R^{ab}$ and a Riemann-tensor coupling of the form $B^{ab}B^{cd}R_{abcd}$. In the homogeneous vacuum sector, where the Kalb-Ramond field strength and the first derivatives of the symmetry-breaking potential vanish, the field equations reduce to a closed algebraic system in the tetrad frame of the \Godel-type metric. We derive the curvature tensors, the reduced metric equations, and the Kalb-Ramond consistency equations for several inequivalent orientations of the vacuum two-form. Particular attention is given to the pseudo-magnetic background $B_{12}=b$, because it preserves the symmetry of the rotation plane and yields a transparent deformation of the usual bumblebee result. For this sector the Kalb-Ramond equation fixes $\frac{m^2}{\omega^2}=\frac{2(\xi_1+3\xi_2)}{\xi_1+2\xi_2}$, where $\xi_1$ and $\xi_2$ are the Ricci and Riemann couplings. The Ricci coupling alone reproduces the noncausal \Godel value $m^2=2\omega^2$, while the Riemann coupling shifts the chronology bound and allows the critical causal value $m^2=4\omega^2$ for $\xi_2=-\xi_1$. We also show that the Riemann coupling can move the critical radius for closed timelike curves, but ordinary positive-energy matter severely restricts completely causal solutions. Finally, we discuss perfect fluid, scalar field, and electromagnetic sources.