To derive time dilation and length contraction using Lorentz transformations, we start with the fundamental postulates of special relativity:
- The laws of physics are the same in all inertial reference frames.
- The speed of light in a vacuum is the same in all inertial reference frames.
Let's consider two inertial reference frames, denoted as the "primed" frame (moving frame) and the "unprimed" frame (rest frame). The Lorentz transformations describe the relationships between the coordinates and times measured in these frames.
Time Dilation:
Let's suppose there are two events, Event 1 and Event 2, that occur at the same location in the unprimed frame (x' = 0). In the primed frame, these events are separated by a time interval Δt'. We want to find the corresponding time interval Δt measured in the unprimed frame.
The Lorentz transformation for time is given by:
Δt = γ(Δt' - (v/c^2)Δx')
Where: Δt' is the time interval measured in the primed frame. Δx' is the spatial separation of the two events in the primed frame. v is the relative velocity between the two frames. c is the speed of light. γ is the Lorentz factor, defined as γ = 1/√(1 - v^2/c^2).
If Δx' = 0 (events occur at the same location in the primed frame), then the equation simplifies to:
Δt = γΔt'
This equation shows that the time interval Δt measured in the unprimed frame is dilated compared to the time interval Δt' measured in the primed frame. This is known as time dilation.
Length Contraction:
Now let's consider an object that is at rest in the primed frame, and its length is measured in the unprimed frame. Let the object have a proper length L' in the primed frame. We want to find the corresponding length L measured in the unprimed frame.
The Lorentz transformation for length is given by:
L = L' / γ
This equation shows that the length L measured in the unprimed frame is contracted compared to the proper length L' measured in the primed frame. This is known as length contraction.
In summary, the Lorentz transformations provide the mathematical framework for understanding the effects of time dilation and length contraction in special relativity. Time dilation refers to the stretching of time intervals in a moving frame relative to a stationary frame, while length contraction refers to the shortening of lengths measured in a moving frame relative to a stationary frame.