No, according to the special theory of relativity, clocks do not run at the same rate regardless of whether they are moving or not. One of the key principles of the special theory of relativity is that the passage of time can be affected by relative motion between observers.
According to this theory, if two observers are in relative motion to each other, their measurements of time can differ. Specifically, an observer in motion relative to another observer will measure the other observer's clock to be running slower than their own. This effect is known as time dilation.
Time dilation occurs because the speed of light in a vacuum is constant for all observers, regardless of their relative motion. As a result, when one observer is in motion relative to another, their measurement of time becomes dilated or stretched compared to the stationary observer.
However, it's important to note that time dilation effects become noticeable only when objects are moving at speeds comparable to the speed of light. For everyday speeds we experience on Earth, the differences in time measurements due to time dilation are incredibly small and negligible.
In summary, according to the special theory of relativity, clocks do not run at the same rate for observers in relative motion. Time dilation causes the measurements of time to differ between observers moving at different velocities.