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According to special relativity, a properly functioning clock moving relative to you will tick slower than your clock, assuming that measurements are made in inertial reference frames. The moving clock will show a smaller number of seconds have passed if it is used to measure the duration of the same event that your clock is used to measure. We sometimes speak of time dilation by saying time itself is 'slower' or dilated, but time isn't going slower in any absolute sense, only relative to some other frame of reference. Time doesn't actually have a rate.
Time dilation is not an illusion of perception; and it's not a matter of the second having different definitions in different reference frames. Also, it's not a Doppler effect. For example, if a flashing green light is accelerating rapidly away from you, then each pulse of light is redder and more delayed than the one sent out before it. So, the time between pulses appears to be longer than it properly is. If the flashing light were, instead, moving toward you, the effect would be reversed. The light would be bluer and the pulses closer together. However, the red-shifts and blue shifts due to the Doppler effect are not examples of time dilation. Time dilation isn't affected by the direction of motion, only by speed.
Time dilation due to difference in constant speeds is described by Einstein's special theory of relativity. The general theory of relativity describes a second kind of time dilation, one due to different accelerations and different gravitational influences.
Newton's physics describes duration as an absolute property, implying it is not relative to the reference frame. However, he describes the speed of light as being relative to the frame. Einstein's special theory of relativity reverses both of these aspects of time. For inertial frames, it implies the speed of light is not relative to the frame, but duration is relative to the frame. In general relativity, however, the speed of light can vary within one reference frame if matter and energy are present.
To quantitatively illustrate time dilation due to motion, consider a properly functioning clock moving with a constant velocity v in an inertial frame. The time which elapses between two ticks of its second hand is not really the one second it has when it's at rest in the frame, but is the longer time of 1/square root(1-vē/cē) seconds. The moving clock takes longer to tick. Its second lasts longer, and so we observers at rest in the frame judge the clock's ticking to be 'dilated' or spread out and thus slowed down relative to our clock. The moving clock is still accurate, though. Time really is going slower in moving inertial frames than in stationary ones.
Time dilation due to motion is relative in the sense that if your spaceship moves past mine so fast that I measure your clock to be running at half speed, then you will measure my clock to be running at half speed also, provided both of us are in inertial frames. If one of us is affected by a gravitational field or undergoes acceleration, then that person isn't in an inertial frame and the results are different.
Both types of time dilation by a significant role in time-sensitive satellite navigation systems such as the Global Positionining System. The atomic clocks on the satellites must be programmed to compensate for the relativistic dilation effects of both gravity and motion.
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