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In the previous section, we discussed the postulates of special relativity. In this section, we will look at one of the consequences that arise from these postulates.
Time dilation can be summarised fairly succinctly. Consider an observer who is stationary in a particular frame of reference, which we will call frame A. This could be as simple as someone standing still on a flat road (in this case, the road is the frame of reference). Suppose that this person has a clock. Now consider a second observer in a frame of reference (frame B) that is moving with respect to our original observer (e.g. someone in a car moving at constant speed). The second observer also has a clock.
The time dilation effect states that the observer in frame A sees the clock in frame B as running more slowly than his own clock. Further, and this is where things start to get interesting, the effect is reciprocal - because, to the observer in B, frame A is moving relative to frame B, the observer in B observes that the clock in A appears to run more slowly than the clock in frame B.
The time dilation effect, from the perspective of the observer in A, is illustrated in the following diagram. The reciprocity of the effect leads to some interesting paradoxical situations, one of which (the twin paradox) we will discuss in a later article.
Throughout these series of articles, it is not my intention to give detailed derivations of the phenomena and effects that I am describing. However, I think it is of interest to show (or at least try to show) how the time dilation effect comes about, and to look at some of the consequences, and indeed why it is not an important effect in our everyday lives.
Derivation of Time Dilation Formula
To see how the time dilation effect arises, we will consider how a simple clock functions in stationary and moving frames of reference. For this purpose, we will consider a clock that is assumed to consist of a source of light that fires a light pulse up to a mirror, which then reflects the light back down to the source, where it can be detected. Each "tick" of this clock consists of the up and down motion of the light pulse.
At this point, you might argue that such a clock as this bears no resemblance to the clocks that we are used to, with hands and numbers on a dial. You will have to trust me that it can be shown that the arguments we are going to discuss here apply to any type of clock. The type of clock we have chosen here is just particularly convenient for developing the time dilation arguments.
(The argument to prove this is based on a contradiction - it can be shown that if two different types of clock led to different behaviour then it would violate the first postulate of relativity).
Let us assume that two such identical "light pulse" clocks are present in frames A and B. Let us first of all consider our observer in frame B, and how that observer sees the frame B clock functioning. This is quite simple: the observer sees the light pulse travelling vertically upwards and then back down again after being reflected. The following diagram illustrates this.
The crux of the time dilation effect is that the observer in frame A does NOT see the clock functioning in this way (remember, frame B is moving with respect to frame A). Instead, the observer in A would see the light following a diagonal path as frame B moves during a "tick". It is easier to visualise this if we first think about what we would see if the observer in B were to throw a ball vertically upwards, with respect to frame B. The following diagram shows why the observer in frame A would see the ball follow a diagonal path.
The same argument applies to the light pulse on our simple clock, and each "tick" of the clock in frame B appears, to an observer in frame A, as shown below.
Having determined that the path taken by the light pulse is different for the observers in frames A and B, we are in a position to see how the time dilation effect comes about. The second postulate of relativity states that the speed of light is the same for all observers. However, to the observer in frame A, the path taken by the light pulse during each "tick" of the clock is longer. Therefore, to the observer in frame A, the clock appears to tick more slowly than it does to the observer in frame B, because the light pulse has a longer path to follow.
Through this line of argument, we have arrived at the conclusion that a clock in a moving frame of reference (frame B) appears to tick more slowly than a clock in a stationary frame of reference (frame A). This is the time dilation effect that we have been seeking.
It is very easy to show (using Pythagoras' Theorem on the diagrams above) that the time required for a "tick" to the observer in frame A is related to the time required for a "tick" in frame B through the equation
where v is the velocity with which frame B is moving, and C is the velocity of light.
Application of the Formula
This expression is very easy to evaluate. Let us consider a couple of examples to illustrate this effect. Let us assume that to the observer in frame B, a "tick" lasts for one second in that frame (remember, the arguments apply to any type of clock). This means that TB is equal to one second in the expression above. We want to evaluate the length of a tick of the clock in frame B, as seen from an observer in frame A. That is, we want to evaluate TA. The velocity of light C is equal to 186,000 miles per second. Let us pick some values for the velocity of frame B, and see how long the tick is to a stationary observer:
We can see from these numbers that at "normal" speeds, the time dilation effect is an extremely small effect, and very difficult to detect. It has been done, however - experiments have been undertaken in which extremely accurate atomic clocks have been placed on jumbo jets that have been flown around the world, and it has been found that the clocks placed on the jets did indeed run more slowly than those that remained at rest on the ground.
However, as speeds start to approach that of the speed of light, the effect becomes more noticeable, and indeed we will discuss a situation where this is very important in the next section.
A moving clock runs more slowly than one that is stationary with respect to the person observing the clocks. At normal speeds, the effect is very small, but as speeds approach those of the speed of light, the effect becomes more pronounced.