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Time Dilation

Time Dilation Derivation:

We have already seen that there is a problem with simultaneous events if the finite speed of light is taken into account. We will see below that time itself appears differently, to relatively moving observers.

Consider two observers Bonnie and Clyde. Clyde is in one rocket ship and he observes Bonnie on another rocket ship moving past his rocket ship with speed v (v must be near c for the effect to be important in practice).

Bonnie has a clock on the rocket ship with her and a device consisting of two mirrors. A pulse of light (photon) leaves mirror #1 and is reflected by mirror #2 back toward mirror #1.

Bonnie can use the clock to measure the time between the photon leaving mirror #1 and returning back to mirror #1. This time interval represents a "click" of Bonnie’s clock. The time interval between two "clicks" of Bonnie’s clock is given by

(1)

where we used the 2nd postulate that says the speed of light c is the same in all inertial reference frames. is a proper time interval. A proper time interval requires only only one clock to measure it. (This may seem obvious but below, we will see a case where it is not true.) The mirrors and Bonnie’s clock are at rest with respect to Bonnie’s rocketship. The device of two mirrors and pulse of light is used as a clock as it is simple to visualize what is going on when time is measured. Note is the time between two events: event #1 is the flash of light leaving mirror #1 and event #2 is the flash of light returning to mirror #1. How do the same two events appear to Clyde who is on the other rocketship? While Bonnie needs only one clock (or timer) to determine the time between the two events, Clyde on the other hand needs two clocks as Clyde sees Bonnie’s two mirrors move between the two events.

Observation: The time interval Clyde observes between event #1 and event #2 is

(2)

where again we used the 2nd postulate that the light speed in the same in all inertial frames of reference. Both of Clyde’s clocks are synchronized according to Clyde and Clyde must bring the two clocks together for comparison of the times on the clocks to determine . Mirror #1 moves a horizontal distance between event #1 and event #2. It is immediately clear that the time interval observed by Clyde is longer than the time interval Bonnie observes because

Exactly how much larger can easily be calculated using the Pythagorean theorem.

(3)

Using eqt. (1) and (2) in eqt. (3) we have

(4)

Canceling the 2 in each term yields

(5)

and rearranging slightly

(6) .

Solving for Clyde’s time interval

(7)

so at last

(8) .

This is the connection between Clyde’s measurement of the time between the two events and Bonnie’s measurement. This effect is called time dilation or stretching as . There is nothing wrong with either Bonnie’s clock or Clyde’s clocks, they get different times because the method of measurement is different. One is measuring the time interval with a clock at rest (Bonnie), the other (Clyde) is measuring the time of a moving clock.

Time Dilation for Particles:

Particle processes have an intrinsic clock that determines the half-life of a decay process. However, the rate at which the clock ticks in a moving frame, as observed by a static observer, is slower than the rate of a static clock. Therefore, the half-life of a moving particles appears, to the static observer, to be increased by the factor gamma.

For example, let's look at a particle sometimes created at SLAC known as a tau. In the  frame of reference where the tau particle is at rest, its lifetime is known to be approximately 3.05 x 10^-13 s. To calculate how far it travels before decaying, we could try to use the familiar equation distance equals speed times time. It travels so close to the speed of light that we can use c = 3x10⁸ m/sec for the speed of the particle. (As we will see below, the speed of light in a vacuum is the highest speed attainable.) If you do the calculation you find the distance traveled should be 9.15 x 10^-5 meters.

d = v t

d = (3 x 10⁸ m/sec)( 3.05 x 10^-13 s) = 9.15 x 10^-5 m

Here comes the weird part - we measure the tau particle to travel further than this!

Pause to think about that for a moment. This result is totally contradictory to everyday experience. If you are not puzzled by it, either you already know all about relativity or you have not been reading carefully.

What is the resolution of this apparent paradox? The answer lies in time dilation. In our laboratory, the tau particle is moving. The decay time of the tau can be seen as a moving clock. According to relativity, moving clocks tick more slowly than static clocks.

We use this fact to multiply the time of travel in the taus moving frame by gamma, this gives the time that we will measure. Then this time times c, the approximate speed of the tau,  will give us the distance we expect  a high energy  tau to travel.

What is gamma in this case? It depends on the tau's energy. A typical SLAC tau particle has a gamma = 20. Therefore, we detect the tau to decay in an average distance of 20 x (9.15 x 10^-5 m) = 1.8 x 10^-3 m or approximately 1.8 millimeters. This is 20 times further than we expect it to go if we use classical rather than relativistic physics. (Of course, we actually observe a spread of decay times according to the exponential decay law and a corresponding spread of distances. In fact, we use the measured distribution of distances to find the tau half-life.)

Observations particles with a variety of velocities have shown that time dilation is a real effect. In fact the only reason cosmic ray muons ever reach the surface of the earth before decaying is the time dilation effect.