Time Dilation

Consider a clock that is synchronized by bouncing a light ray, that propagates through a vacuum, off a reflector that is located a distance $l_0$ from the clock. Thus, the time taken for the light ray to travel from the clock to the reflector, and back again,

$$t_0= \frac{2\,l_0}{c},$$ (3.66)

corresponds to one tick of the clock.

Figure 3.6: Time dilation.

Suppose that we observe the aforementioned clock in a frame of reference that moves with velocity ${\bf v}$ with respect to the clock's rest frame, where the direction of ${\bf v}$ is perpendicular to the path of the light ray in the rest frame. See Figure 3.6. Let $t_1$ be the time required for a light ray to travel from the clock to the reflector, and back again, in the moving frame. In the moving frame, the clock moves a parallel (to $-{\bf v}$) distance $v_1\,t$ in this time interval. Note that the transverse distance, $l_0$, of the reflector from the clock is the same in both reference frames. (See Section 3.2.2.) It is clear, by symmetry, that in traveling from the clock to the reflector, the light ray in the moving frame has moved a transverse distance $l_0$ and a parallel (to $-{\bf v}$) distance $v\,t_1/2$. Moreover, the ray travels the same transverse and parallel distances in traveling from the reflector back to the clock. Hence, the net path-length of the light ray is

$$L= 2\left(\frac{v^2\,t_1^{\,2}}{4}+l_0^{\,2}\right)^{1/2}.$$ (3.67)

Now, because the light ray travels at the speed $c$ in the moving frame, according to Einstein's second postulate, we have

$$t_1 =\frac{L}{c},$$ (3.68)

which implies that

$$t_1 = \frac{2}{c}\left(\frac{v^2\,t_1^{\,2}}{4}+l_0^{\,2}\right)^{1/2},$$ (3.69)

or

$$t_1^{\,2} = \frac{v^2\,t_1^{\,2}}{c^2} + \frac{4\,l_0^{\,2}}{c^2}= \frac{v^2\,t_1^{\,2}}{c^2} + t_0^{\,2},$$ (3.70)

where use has been made of Equation (3.66). The previous expression can be rearranged to give

$$t_1 = \frac{t_0}{\sqrt{1-v^2/c^2}}.$$ (3.71)

Thus, we conclude that $t_1>t_0$. Given that a tick of our clock corresponds to the time required for a light ray to travel from the clock to the reflector, and back again, we conclude that the clock ticks more slowly in the moving reference frame than it does in its rest frame. This effect is known as time dilation.

We can also conclude that any type of clock, not just a light-clock, will tick more slowly in a moving reference frame than in its rest frame, by the same factor as our light clock, otherwise the same experiment (i.e., measuring the time it takes a light ray to travel a distance $2\,l_0$ in vacuum using the former type of clock) would produced different results in different inertial frames, which is forbidden by Einstein's first postulate.

Let us define the so-called Lorentz factor,

$$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}.$$ (3.72)

Note that $\gamma\geq 1$. The time dilation law, (3.71), can be written

$$t_1 = \gamma\,t_0.$$ (3.73)

In other words, time is dilated by the Lorentz factor in a moving frame of reference.

Muons are unstable particles that have measured lifetimes of $2.917\,\mu{\rm s}$ in their rest frame. However, when the Earth's atmosphere is struck by cosmic-ray particles, very energetic muons that move at 98% of the speed of light are produced. The measured lifetimes of these cosmic-ray muons is indeed about five times longer than the rest-frame lifetime of a muon, in accordance with the previous two equations.