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Why do moving clocks appear to run slower in special relativity?

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In special relativity, the phenomenon where moving clocks appear to run slower is known as time dilation. This effect occurs because simultaneity—the determination of which events occur at the same time—is an observer-dependent notion.

When two observers are in relative inertial motion, they disagree on which events are simultaneous. Because a consistent definition of simultaneity is required to compare the rates of clocks, each observer finds that the other's clock runs slower than their own. Specifically, if an observer measures a clock moving at a relative velocity $v$ , they will observe that clock running slower by a factor of $\sqrt{1 - v^{2} / c^{2}}$ , where $c$ is the speed of light.1

This is not a mechanical error but a fundamental property of spacetime. It has been confirmed through experiments involving atomic clocks and the observation of unstable elementary particles that exhibit increased lifetimes when traveling at high speeds.1