Instability of tachyon supertube in type IIA Gödel spacetime

We study the tachyon supertube probes in a type IIA supergravity background which is a stringy-like Gödel spacetime and contains closed timelike curve. In the case of small value of f, which is a parameter of the background, we use the Minahan-Zwiebach tachyon action to obtain a single regular tube solution and argue that the tube is a BPS D2-brane. However, we find that the fluctuation around the tube configuration has a negative-energy mode. This means that the tachyon supertube, despite being a BPS configuration, develops an instability in the pathological spacetime with closed timelike curve which violates the causality.