In the context of special relativity, which deals with the effects of motion at high speeds, time dilation is caused by relative motion between observers. More specifically, time dilation occurs when an observer is moving at a significant fraction of the speed of light relative to another observer.
The motion that leads to time dilation is not restricted to translational motion alone. It can include any type of motion, as long as it involves a change in velocity relative to another observer. This includes rotational motion or motion in curved paths.
If you were rotating around a fixed point, your motion would still cause time dilation relative to a stationary observer. This effect is known as "rotational time dilation" or "gravitational time dilation," depending on the context. When you rotate, your velocity is constantly changing direction, which creates acceleration. According to general relativity, acceleration is equivalent to the effects of gravity, and gravity is known to cause time dilation. Therefore, even in the case of rotational motion, there would be a small time dilation effect compared to a stationary observer.
However, it's worth noting that the time dilation resulting from rotational motion alone would generally be much smaller compared to the time dilation caused by high-speed translational motion. The effects of rotational time dilation would typically require very high rotational speeds or extremely precise measurements to be observable.