bcrowell said:The Lorentz transformation of time has the form [itex]t'=(\ldots)t+(\ldots)x[/itex]. To me, "time dilation" has the connotation of referring to the fact that the coefficient of the first term on the right doesn't have its Galilean value of 1, while "relativity of simultaneity" focuses more on the fact that the second coefficient doesn't have its Galilean value of 0. I haven't heard "relativity of time" as much, so I wouldn't hazard a guess as to its connotations; it might be a convenient blanket term for all such relativistic effects.
Time dilation or time relativity is a phenomenon in which time appears to pass at different rates for objects in different frames of reference, based on their relative velocities and gravitational fields.
Time dilation is caused by a difference in velocity or gravitational potential between two objects. According to Einstein's theory of relativity, time slows down as an object's velocity approaches the speed of light and as it gets closer to a massive object.
In our everyday life, the effects of time dilation are very small and are only noticeable at extremely high velocities or in strong gravitational fields. For example, the time dilation experienced by astronauts in orbit around Earth is only about 0.007 seconds per day.
Yes, time dilation has been observed and measured in numerous experiments and observations, such as the famous Hafele-Keating experiment and the measurements of atomic clocks on GPS satellites. The effects may be small, but they have been proven to exist.
No, time dilation is not the same as time travel. Time dilation only affects the passage of time for an object relative to another, while time travel would involve traveling through time and experiencing different points in time. Time dilation does not allow for traveling back in time, as time only appears to slow down for the object itself, not for external observers.