Time dilation is a phenomenon in special relativity where time appears to run differently for objects or observers in relative motion. It states that the passage of time is perceived to be slower for objects moving at high speeds relative to a stationary observer.
To understand how time dilation is derived using Lorentz transformations, let's consider two inertial reference frames: the stationary frame S and the moving frame S', which is moving at a constant velocity v relative to S.
In the stationary frame S, there is a clock at rest, and in the moving frame S', there is a clock moving at velocity v. We want to compare the time measured by the stationary clock with the time measured by the moving clock.
Using the Lorentz transformations, we can relate the coordinates and time measurements between the two frames. For simplicity, let's consider motion along the x-axis.
The Lorentz transformations for time and position are as follows:
t' = γ(t - (v/c^2)x) x' = γ(x - vt)
Where t' and x' are the coordinates measured in the moving frame S', t and x are the coordinates measured in the stationary frame S, γ is the Lorentz factor, v is the relative velocity between the frames, and c is the speed of light in a vacuum.
Now, let's consider the time difference between two events as measured by the stationary observer. Δt is the time interval measured by the stationary clock at position x1 and x2 in frame S.
Δt = t2 - t1
In the moving frame S', the time difference between the same events is given by:
Δt' = t'2 - t'1
Substituting the Lorentz transformations for time, we have:
Δt' = γ(t2 - (v/c^2)x2) - γ(t1 - (v/c^2)x1) = γ[(t2 - t1) - (v/c^2)(x2 - x1)] = γΔt - (v/c^2)(x2 - x1)
Now, let's consider the case where Δx = x2 - x1 is the distance traveled by the moving clock between the two events. In the stationary frame S, this distance is given by:
Δx = x2 - x1
In the moving frame S', the distance is given by:
Δx' = x'2 - x'1
Substituting the Lorentz transformations for position, we have:
Δx' = γ(x2 - vt2) - γ(x1 - vt1) = γ[(x2 - x1) - v(t2 - t1)] = γΔx - vΔt
Now, combining the expressions for Δt' and Δx', we have:
Δt' = γΔt - (v/c^2)Δx Δx' = γΔx - vΔt
From these equations, we can see that Δt' is larger than Δt due to the presence of the γ factor. This means that the time interval measured by the moving clock is dilated or stretched compared to the time interval measured by the stationary clock. This is the phenomenon of time dilation.
Experimental evidence, such as measurements involving high-speed particles, muons, and atomic clocks on fast-moving satellites, has confirmed the predictions of time dilation in special relativity. These experiments have demonstrated that the time measured by clocks in motion relative to a stationary observer runs slower, consistent with the Lorentz transformations and the concept of time dilation.