To derive time dilation and length contraction using Lorentz transformations, we can start with the basic postulates of special relativity and the Lorentz transformation equations. The postulates are:
- The laws of physics are the same in all inertial reference frames.
- The speed of light in a vacuum is constant for all observers, regardless of the motion of the source or the observer.
Now, let's consider two reference frames, denoted as the "primed" frame (moving with velocity v relative to the "unprimed" frame). We will derive the time dilation and length contraction formulas using these transformations.
- Time Dilation:
We'll consider a clock in the primed frame, which is moving relative to the unprimed frame. Let t' be the time measured by the clock in the primed frame and t be the time measured in the unprimed frame.
The Lorentz transformation for time is given by:
t' = γ(t - (v/c^2)x),
where γ is the Lorentz factor:
γ = 1 / sqrt(1 - (v^2 / c^2)).
For the clock at rest in the unprimed frame (x = 0), we have:
t = γt'.
Dividing the above equations, we get:
t/t' = γ,
which shows that the time measured in the unprimed frame (t) is dilated compared to the time measured in the primed frame (t'). This is known as time dilation.
- Length Contraction:
Let's consider a rod at rest in the primed frame, with a length L'. The length of the rod measured in the unprimed frame is L.
The Lorentz transformation for length is given by:
L = L' / γ.
This equation shows that the length of an object measured in the unprimed frame (L) is contracted compared to the length measured in the primed frame (L'). This is known as length contraction.
So, using the Lorentz transformations, we can derive the formulas for time dilation (t/t' = γ) and length contraction (L = L' / γ). These formulas accurately describe the effects of special relativity, where time appears to slow down and lengths appear to contract for objects moving relative to an observer.