The aether-modified gravity and the Godel metric

We formulate the nonminimal aether-modified gravity whose action represents itself as a sum of the usual Einstein-Hilbert action and the CPT-even Lorentz-breaking aether-like gravity term proposed by Carroll. For this theory, we show that the G\"{o}del metric solves the modified Einstein equations under a corresponding modification of the matter, while, for the small Lorentz-breaking corrections, the matter turns out to be usual. On the other side, if the matter is suggested to be the same as in the usual general relativity, the G\"{o}del metric is modified by additive aether terms.