Yes, men working deep in mines do experience a slight time dilation effect due to their proximity to the Earth's center of gravity. However, the magnitude of this time dilation is extremely small and negligible in most practical situations.
According to the theory of general relativity, gravitational time dilation occurs when an object is in a gravitational field or experiences a gravitational potential difference. Time runs slightly slower in regions with a stronger gravitational field compared to regions with a weaker gravitational field.
In the case of men working deep in mines, the gravitational field they experience is indeed slightly stronger compared to someone on the Earth's surface. However, the effect is so minuscule that it is typically not noticeable or measurable in everyday circumstances.
To put it into perspective, for a person working at a depth of 1 kilometer (about 0.62 miles) below the Earth's surface, the time dilation effect would be on the order of a few nanoseconds (billionths of a second) per year. This is an incredibly tiny difference that would not be perceivable in any practical sense.
In contrast, time dilation effects experienced by astronauts in orbit around the Earth are relatively more significant due to their high speeds and the weaker gravitational field they encounter in space. Astronauts in orbit experience a time dilation effect that is measurable and has been confirmed through precise experiments.
Therefore, while men working deep in mines do experience a slight gravitational time dilation effect, its practical implications are negligible and would not lead to noticeable differences in their subjective experience of time.