To demonstrate the time dilation effect for a clock moving at a relativistic velocity v relative to an observer on the ground, we can use the time dilation formula derived from special relativity. The formula relates the proper time (Δt_0) experienced by the moving clock to the time (Δt) measured by the observer on the ground.
The time dilation formula is given by:
Δt = γ * Δt_0
where γ (gamma) is the Lorentz factor defined as:
γ = 1 / √(1 - v²/c²)
In this case, we assume the observer on the ground measures the time interval Δt for the moving clock, while the moving clock itself experiences the proper time interval Δt_0.
Let's consider an example where the clock moves at a significant fraction of the speed of light, v. We'll assume v is given in terms of the speed of light, c. We'll also assume that the observer on the ground measures a time interval Δt, and we want to find the proper time interval Δt_0 experienced by the moving clock.
Using the time dilation formula, we have:
Δt = γ * Δt_0
Rearranging the equation to solve for Δt_0, we get:
Δt_0 = Δt / γ
Substituting the Lorentz factor γ, we have:
Δt_0 = Δt / (1 / √(1 - v²/c²))
Simplifying further, we can rewrite it as:
Δt_0 = Δt * √(1 - v²/c²)
This equation shows that the proper time interval Δt_0 experienced by the moving clock is related to the time interval Δt measured by the observer on the ground through the factor √(1 - v²/c²).
From this equation, we can see that as the velocity v approaches the speed of light c, the term inside the square root (√(1 - v²/c²)) approaches zero. This means that the proper time Δt_0 experienced by the moving clock becomes significantly smaller than the time Δt measured by the observer on the ground. This effect is what we refer to as time dilation, where time appears to slow down for the moving clock relative to the observer on the ground.
Experimental evidence, such as measurements involving atomic clocks on high-speed vehicles or satellites, confirms the predictions of time dilation in special relativity. These experiments have observed that clocks moving at relativistic velocities indeed experience time at a different rate compared to stationary clocks.