The connection between Lorentz transformations and time dilation is a fundamental aspect of special relativity, which is a theory that describes the behavior of objects moving at high speeds, close to the speed of light.
Lorentz transformations are mathematical equations that relate the space and time coordinates of events as observed in different inertial reference frames. In special relativity, the laws of physics are required to be invariant (i.e., the same) in all inertial reference frames. Lorentz transformations allow us to calculate how measurements of space and time in one reference frame translate into another frame moving at a constant velocity relative to the first.
Time dilation, on the other hand, is a consequence of the relativistic relationship between time and space. According to special relativity, time is not absolute but depends on the relative motion between observers. When an object is moving relative to an observer, time appears to pass more slowly for the moving object as compared to a stationary observer.
The Lorentz transformations incorporate this phenomenon of time dilation. They describe how the time coordinate of an event changes when observed from different reference frames moving at relative velocities. The Lorentz factor, which is a term that appears in the Lorentz transformations, accounts for time dilation.
Mathematically, time dilation is expressed as t' = γt, where t' is the time interval as measured in the moving frame, t is the time interval as measured in the stationary frame, and γ (gamma) is the Lorentz factor. The Lorentz factor depends on the relative velocity between the frames and is given by γ = 1/√(1 - v²/c²), where v is the relative velocity and c is the speed of light.
In summary, Lorentz transformations provide the mathematical framework to relate space and time coordinates between different inertial reference frames. Time dilation arises from these transformations, showing that time intervals measured in moving frames appear dilated or stretched relative to time intervals measured in stationary frames.