Classical and quantum Bianchi type III vacuum Hořava–Lifshitz cosmology

A diagonal Bianchi type III spacetime is treated, both at the classical and quantum level, in the context of Hořava–Lifshitz   gravity. The system of the classical equations of motion is reduced to one independent Abel’s equation of the first kind. Closed form solutions are presented for various values of the coupling constants appearing in the action. Due to the method used, solutions of Euclidean, Lorentzian and neutral signatures are attained. The main general solution for λ=1λ=1 is seen to contain a curvature singularity at t=0t=0, while in its Einsteinian limit it reproduces the known diagonal solution (with and/or without cosmological constant). The rest of the solutions correspond to somewhat isolated cases, as they emerge from either setting the coefficient of the lapse to zero or taking λ=1/3λ=1/3. They are singularity free, but they all have a singular limit as the other constants approach their Einsteinian values. At the quantum level, the resulting Wheeler–DeWitt equation is explicitly solved for λ=1,σ=0 and λ=13. The ensuing wave functions diverge in the Einsteinian limit.