36.6 Time Dilation

The principle of relativity has driven us to the logical conclusion that time is not the same for two reference frames moving relative to each other. Our analysis thus far has been mostly qualitative. It’s time to start developing some quantitative tools that will allow us to compare measurements in one reference frame to measurements in another reference frame.

FIGURE 36.18a shows a special clock called a light clock. The light clock is a box with a light source at the bottom and a mirror at the top, separated by distance h. The light source emits a very short pulse of light that travels to the mirror and reflects back to a light detector beside the source. The clock advances one “tick” each time the detector receives a light pulse, and it immediately, with no delay, causes the light source to emit the next light pulse.

The figure labeled ("a") illustrates a light clock.

The figure labeled (b) illustrates a light clock in two different frames of reference.

Full Alternative Text

Our goal is to compare two measurements of the interval between two ticks of the clock: one taken by an experimenter standing next to the clock and the other by an experimenter moving with respect to the clock. To be specific, FIGURE 36.18b shows the clock at rest in reference frame $S'$ . We call this the rest frame of the clock. Reference frame $S'$ , with the clock, moves to the right with velocity v relative to reference frame S.

Relativity requires us to measure events, so let’s define event 1 to be the emission of a light pulse and event 2 to be the detection of that light pulse. Experimenters in both reference frames are able to measure where and when these events occur in their frame. In frame S, the time interval $Δ t = t_{2} - t_{1}$ is one tick of the clock. Similarly, one tick in frame $S'$ is $Δ t' = t_{2}^{'} - t_{1}^{'}$ .

To be sure we have a clear understanding of the relativity result, let’s first do a classical analysis. In frame $S'$ , the clock’s rest frame, the light travels straight up and down, a total distance 2h, at speed c. The time interval is $Δ t' = 2 h / c$ .

FIGURE 36.19a shows the operation of the light clock as seen in frame S. The clock is moving to the right at speed v in S, thus the mirror moves distance $\frac{1}{2} v (Δ t)$ during the time $\frac{1}{2} (Δ t)$ in which the light pulse moves from the source to the mirror. The distance traveled by the light during this interval is $\frac{1}{2} u_{light} (Δ t)$ , where $u_{light}$ is the speed of light in frame S. You can see from the vector addition in FIGURE 36.19b that the speed of light in frame $S$ is $u_{light} = {(c^{2} + v^{2})}^{1 / 2}$ . (Remember, this is a classical analysis in which the speed of light does depend on the motion of the reference frame relative to the light source.)

The figure labeled ("a") illustrates the operation of the light clock as seen in frame S.

The figure labeled (b) illustrates the vector addition of clock and light velocities in frame S.

Full Alternative Text

The Pythagorean theorem applied to the right triangle in Figure 36.19a is

\begin{array}{l} h^{2} + {(\frac{1}{2} ν Δ t)}^{2} & = & {(\frac{1}{2} u_{light} Δ t)}^{2} & = & {(\frac{1}{2} \sqrt{c^{2} + v^{2}} Δ t)}^{2} \\ = & {(\frac{1}{2} c Δ t)}^{2} + {(\frac{1}{2} v Δ t)}^{2} \end{array}

(36.4)

The term ${(\frac{1}{2} v Δ t)}^{2}$ is common to both sides and cancels. Solving for $Δ t$ gives $Δ t = 2 h / c$ , identical to $Δ t'$ . In other words, a classical analysis finds that the clock ticks at exactly the same rate in both frame S and frame $S'$ . This shouldn’t be surprising. There’s only one kind of time in classical physics, measured the same by all experimenters independent of their motion.

The principle of relativity changes only one thing, but that change has profound consequences. According to the principle of relativity, light travels at the same speed in all inertial reference frames. In frame $S'$ , the rest frame of the clock, the light simply goes straight up and back. The time of one tick,

Δ t' = \frac{2 h}{c}

(36.5)

is unchanged from the classical analysis.

FIGURE 36.20 shows the light clock as seen in frame S. The difference from Figure 36.19a is that the light now travels along the hypotenuse at speed c. We can again use the Pythagorean theorem to write

h^{2} + {(\frac{1}{2} v Δ t)}^{2} = {(\frac{1}{2} c Δ t)}^{2}

(36.6)

Solving for $Δ t$ gives

Δ t = \frac{2 h / c}{\sqrt{1 - v^{2} / c^{2}}} = \frac{Δ t'}{\sqrt{1 - v^{2} / c^{2}}}

(36.7)

The time interval between two ticks in frame S is not the same as in frame $S'$ .

The figure illustrates a light clock analysis in which the speed of light is the same in all reference frames.

FIGURE 36.20 Full Alternative Text

It’s useful to define $β = v / c$ , the velocity as a fraction of the speed of light. For example, a reference frame moving with $v = 2.4 \times 10^{8} m / s$ has $β = 0.80$ . In terms of $β$ , Equation 36.7 is

Δ t = \frac{Δ t'}{\sqrt{1 - β^{2}}}

(36.8)

NOTE The expression ${(1 - v^{2} / c^{2})}^{1 / 2} = {(1 - β^{2})}^{1 / 2}$ occurs frequently in relativity. The value of the expression is 1 when $v = 0$ , and it steadily decreases to 0 as $v \to c$ (or $β \to 1$ ). The square root is an imaginary number if $v > c$ , which would make $Δ t$ imaginary in Equation 36.8. Time intervals certainly have to be real numbers, suggesting that $v > c$ is not physically possible. One of the predictions of the theory of relativity, as you’ve undoubtedly heard, is that nothing can travel faster than the speed of light. Now you can begin to see why. We’ll examine this topic more closely in Section 36.9. In the meantime, we’ll require v to be less than c.

36.6 Time Dilation

FIGURE 36.18 The ticking of a light clock can be measured by experimenters in two different reference frames.

FIGURE 36.19 A classical analysis of the light clock.

FIGURE 36.20 A light clock analysis in which the speed of light is the same in all reference frames.