No, time dilation does not imply that time can run backwards in certain parts of the universe. Time dilation, as described by special relativity, refers to the relative slowing down or speeding up of time between different observers in different frames of reference. However, it does not allow for the reversal of the fundamental arrow of time.
The concept of the arrow of time is related to the asymmetry of physical processes and the observed irreversibility of certain phenomena, such as the increase of entropy in closed systems. The arrow of time points in the direction of increasing entropy or disorder.
Time dilation, on the other hand, does not alter the fundamental arrow of time. It may cause time to appear to run slower or faster for different observers depending on their relative motion, but it does not reverse the overall direction of time or allow for time to flow backward.
The idea of time reversal or time running backward is not supported by our current understanding of physics. While some physical processes may appear to be reversible on small scales, at a macroscopic level, the arrow of time remains consistent, and the fundamental laws of physics are time-symmetric.
It's important to note that time dilation is a well-established phenomenon that has been experimentally observed and verified. It is a consequence of the geometry of spacetime in special relativity. However, it does not imply the possibility of time running backwards or the reversal of the arrow of time.