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How does time dilation become noticeable at speeds near the speed of light?

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Brian Greene sits in a living room, holding a pen, with handwritten equations m = E/c^2 and E = mc^2 on a whiteboard beside him, highlighting c^2 as the speed of light. — relativity Brian Greene kicking off his *Your Daily Equation* video series with Albert Einstein's famous equation E = mc².

In the theory of special relativity, time dilation refers to the phenomenon where a clock in motion is observed to run slower than a clock at rest relative to the observer. This effect is governed by the factor $\sqrt{1 - v^{2} / c^{2}}$ , where $v$ is the relative velocity and $c$ is the speed of light.12

As an object's speed approaches the speed of light, the value of $v^{2} / c^{2}$ increases, causing the time dilation factor to become significantly smaller, which results in a more pronounced "slowing down" of time for the moving object as measured by a stationary observer. While these effects are negligible at everyday speeds, they become substantial at velocities near $c$ .231

This has been confirmed through observations of elementary particles like muons, which travel at nearly the speed of light and exhibit extended lifetimes compared to those at rest. It is important to note that an observer never experiences these changes within their own reference frame; the effects are only apparent when comparing measurements between different frames of motion.243