To derive time dilation and length contraction using Lorentz transformations, we start with the fundamental equations of special relativity. The Lorentz transformations describe how measurements of time and space intervals between two inertial reference frames (observers moving at constant velocity relative to each other) are related.
Let's consider two inertial reference frames: Frame S (stationary observer) and Frame S' (moving observer with a relative velocity 'v' along the x-axis with respect to Frame S). We will assume that the frames coincide at time t = t' = 0.
Time Dilation: Time dilation is the phenomenon where time appears to pass differently for observers moving at different velocities. It is given by the Lorentz transformation for time. The time 't' measured by the stationary observer (Frame S) and the time 't'' measured by the moving observer (Frame S') are related as follows:
t' = γ(t - (v/c^2)x)
where: γ = 1/√(1 - v^2/c^2) is the Lorentz factor, 'c' is the speed of light in a vacuum, 'x' is the position of the event in the x-axis in Frame S.
Length Contraction: Length contraction is the phenomenon where an object's length appears shorter along the direction of motion when measured by a moving observer compared to its length at rest. It is given by the Lorentz transformation for length. The length 'L' measured in Frame S' (moving observer) and the length 'L0' measured in Frame S (stationary observer) are related as follows:
L = L0 / γ
where: γ = 1/√(1 - v^2/c^2) is the Lorentz factor, 'L0' is the proper length (length measured in Frame S when the object is at rest).
Deriving these formulas involves solving the Lorentz transformation equations, which are beyond the scope of a simple explanation. The Lorentz transformations are a consequence of the postulates of special relativity, which assume the constancy of the speed of light for all observers and the equivalence of inertial frames of reference.
It's important to note that these phenomena have been experimentally verified and are a fundamental aspect of the theory of special relativity. They have significant implications for understanding the behavior of time, space, and motion at relativistic speeds.