Gödel-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.

21 pages