We present a class of curved-spacetime vacuum solutions which develope closed timelike curves at some particular moment. We then use these vacuum solutions to construct a time-machine model. The causality violation occurs inside an empty torus, which constitutes the time-machine core. The matter field surrounding this empty torus satisfies the weak, dominant, and strong energy conditions. The model is regular, asymptotically-flat, and topologicallytrivial. Stability remains the main open question.The problem of time-machine formation is one of the outstanding open questions in spacetime physics. Time machines are spacetime configurations including closed timelike curves (CTCs), allowing physical observes to return to their own past. In the presence of a time machine, our usual notion of causality does not hold. The main question is: Do the laws of nature allow, in principle, the creation of a time machine from "normal" initial conditions? (By "normal" I mean, in particular, initial state with no CTCs.)Several types of time-machine models were explored so far. The early proposals include Godel's rotating-dust cosmological model The above models, however, all suffer from one or more severe problems. In some of them [3] [6] the weak energy condition (WEC) [7] is violated, indicating unrealistic matter-energy content. The WEC states that for any physical (timelike) observer the energy density is non-negative, which is the case for all known types of (classical) matter fields. In other models the CTCs are either pre-existing [1] [2] [4] and/or "come from infinity" [4] (see [8] [9]), and/or there is a curvature singularity [2].The only of the above models which does not violate the WEC, and in which CTCs evolve from normal initial data (and within a compact region of space) is that of Ref. [5]. But this model, too, is not satisfactory, for the following reason: The energy-momentum source occupying the time-machine core, though consistent with the WEC (and also with the dominant energy condition [10]), does not fit any known type of matter field. Therefore this configuration (spacetime plus matter) is not a solution of any prescribed set of field equations. Even if we assumed there exist physical matter fields that yield the desired 1