To derive time dilation and length contraction using Lorentz transformations, we can start with the Lorentz transformation equations that relate space and time coordinates between two inertial reference frames in special relativity. Let's consider two frames of reference: the "primed" frame (moving with velocity v relative to the "unprimed" frame).
The Lorentz transformation equations for coordinates in the x-direction are:
x' = γ(x - vt) x = γ(x' + vt')
where γ (gamma) is the Lorentz factor:
γ = 1/√(1 - v^2/c^2)
Here, c is the speed of light, x' and x are the coordinates in the primed and unprimed frames respectively, and t and t' are the corresponding time coordinates.
Now, let's derive time dilation and length contraction using these equations.
- Time Dilation: Consider an event occurring at a certain point in spacetime in the primed frame. We want to find the time interval between the event as measured in the unprimed frame.
Start with the Lorentz transformation for time:
t' = γ(t - vx/c^2)
Rearrange the equation to solve for t:
t = γ(t' + vx'/c^2)
Comparing this equation to the original time transformation equation, we see that the time interval t is greater than t' (t > t') for any non-zero velocity v. This indicates time dilation, meaning that an observer in the unprimed frame will measure a longer time interval compared to an observer in the primed frame.
- Length Contraction: Consider a rod at rest in the primed frame, with a length L'. We want to find the length of the rod as measured in the unprimed frame.
Start with the Lorentz transformation for coordinates:
x = γ(x' + vt')
The length of the rod in the unprimed frame is given by the difference between its coordinates:
L = x2 - x1 = γ(x2' + vt') - γ(x1' + vt')
Simplifying the equation, we get:
L = γ(L' + vt' - vt')
L = γL'
Comparing this equation to the original length transformation equation, we see that the length L is shorter than L' (L < L') for any non-zero velocity v. This indicates length contraction, meaning that an observer in the unprimed frame will measure a shorter length for the rod compared to an observer in the primed frame.
These derivations demonstrate the effects of time dilation and length contraction predicted by special relativity using Lorentz transformations. These effects arise due to the fundamental postulate of the constancy of the speed of light in all inertial frames of reference.