A recent study by Posselt et al. (2017) reported the deepest X-ray investigation of the Geminga pulsar wind nebula (PWN) by using emph{Chandra X-ray Observatory}. In comparison with previous studies of this system, a number of new findings have been reported and we found these suggest the possible variabilities in various components of this PWN. This motivates us to carry out a dedicated search for the morphological and spectral variations of this complex nebula. We have discovered variabilities on timescales from a few days to a few months from different components of the nebula. The fastest change occurred in the circumstellar environment at a rate of 80 per cent of the speed of light. One of the most spectacular results is the wiggling of a half light-year long tail as an extension of the jet, which is significantly bent by the ram pressure. The jet wiggling occurred at a rate of about 20 per cent of the speed of light. This twisted structure can possibly be a result of a propagating torsional Alf`{v}en wave. We have also found evidence of spectral hardening along this tail for a period of about nine months.
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