Observing time dilation directly with just your body and a watch would be challenging because the effects of time dilation at everyday velocities and gravitational fields are extremely small and difficult to measure without precise equipment. However, I can describe a thought experiment that illustrates the concept:
Imagine you have a twin sibling, and one of you stays on Earth while the other embarks on a high-speed space journey, traveling near the speed of light. After a significant duration of time, the traveling twin returns to Earth.
Upon reuniting, you would notice a discrepancy in the ages of both twins. The twin who traveled at high speed would have aged less compared to the twin who stayed on Earth. This is due to the time dilation effect caused by the relative velocity between the two twins. The twin in motion experienced time passing more slowly relative to the stationary twin on Earth.
In this thought experiment, the time dilation effect is observable when the traveling twin returns and the two compare their ages. However, it's important to note that this scenario requires extremely high speeds, close to the speed of light, which are currently beyond our technological capabilities for human space travel.
To measure time dilation with precision, advanced instruments and techniques are used, such as atomic clocks and particle accelerators. These experiments involve comparing the behavior of precise timekeeping devices in different reference frames or gravitational fields.
In summary, while you cannot directly perceive time dilation in everyday life without specialized equipment, experiments and observations conducted by scientists have provided evidence for its existence.