Proper time is a concept in special relativity that represents the time experienced by an observer who is moving along with a clock or an object. It is also referred to as the "invariant" or "rest" time because it is independent of the motion of other observers.
The Lorentz factor, denoted by γ (gamma), is a fundamental quantity in special relativity that appears in the equations describing the relationship between space and time for objects moving at relativistic speeds. It is given by the equation:
γ = 1 / √(1 - v²/c²)
where v is the velocity of an object relative to an observer, and c is the speed of light in a vacuum. The Lorentz factor describes how time, length, and mass change with relative motion.
When an object is moving at speeds comparable to the speed of light, classical Newtonian physics no longer accurately describes its behavior. Instead, Einstein's theory of special relativity provides a more accurate description. According to special relativity, the passage of time is not absolute, but rather depends on the relative motion between observers.
The proper time, denoted by Δt, experienced by an object or a clock moving at relativistic speeds can be calculated by integrating the infinitesimal time intervals along its path. This integration involves multiplying the time intervals experienced by the moving object with the Lorentz factor. By incorporating the Lorentz factor, the proper time takes into account the effects of time dilation, which means that time appears to run slower for objects in motion relative to a stationary observer.
Utilizing the Lorentz factor in the calculation of proper time allows us to account for the observed phenomena of time dilation and ensure consistency with the principles of special relativity. It provides a framework to reconcile the different perceptions of time between observers in relative motion and helps establish a consistent understanding of the nature of time in a relativistic context.