When Newton developed his laws of motion, he assumed that the passage of time was independent of the reference frame. In other words, he claimed that all the mechanical laws that govern an object remain the same in any frame moving at constant velocity.
Albert Einstein showed that this theory was not applicable for objects moving at high velocities. His theory of special relativity explains the origins of time dilation. If you imagine two clocks, one stationary and one moving, the moving clock will appear to be slower compared to the stationary clock. Furthermore, the faster the clock is moving, the more slowly it will pass time.
To summarize, when a net force is applied to an object, and the object begins to accelerate, it will experience time dilation as a result of its increased velocity. It is the velocity, not the acceleration, which is responsible for time dilation.
Here is an example which illustrates the effect of time dilation. Think about a car moving at 50 miles per hour and let's compare the time dilation experienced by a clock inside the car compared to one not moving. At this velocity, the moving clock would feel time go slower by approximately one part in 1015. This means that after 1015 seconds (or 30 million years), the clock inside the car would only be one second slower than the one outside!! So while time dilation is a real phenomenon and has been experimentally confirmed, it really only matters when the velocities involved are a significant fraction of the speed of light.
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