Yes, astronauts traveling at high speeds relative to people on Earth do experience time differently due to the phenomenon of time dilation, which is described by the Lorentz factor in the theory of special relativity.
The Lorentz factor, denoted by γ (gamma), is a mathematical factor that accounts for the effects of time dilation, length contraction, and relativistic mass increase as an object approaches the speed of light. The Lorentz factor is given by the equation:
γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the object and c is the speed of light.
As an object's velocity approaches the speed of light (c), the Lorentz factor increases. This means that time dilation becomes more pronounced at higher velocities.
For astronauts traveling at high speeds in space, such as in orbit around the Earth, their relative velocity with respect to observers on Earth is significant. Therefore, the Lorentz factor comes into play, and astronauts experience time dilation.
From the perspective of an astronaut in a high-speed spacecraft, time appears to pass more slowly compared to an observer on Earth. Conversely, from the viewpoint of an observer on Earth, time for the astronaut seems to be dilated, meaning it passes more slowly.
However, it's important to note that the time dilation effect experienced by astronauts in space is relatively small and becomes more significant as velocities approach the speed of light. For spacecraft in Earth's orbit, the time dilation effect is measurable but negligible in day-to-day activities. It would require highly precise measurements and comparisons between clocks in space and on Earth to detect these small differences.
In summary, due to their high speeds relative to observers on Earth, astronauts in space do experience time differently. The Lorentz factor, which accounts for time dilation in special relativity, plays a role in this difference.