Before you read this, I suggest you read posts 16.12 and 26.7.
I introduced the concept of time in post 16.12. However, the special theory of relativity leads to an unexpected property of time – it depends on the state of motion of an observer; time does not pass equally in different inertial frames of reference. The more quickly you move, the slower time passes. We don’t normally notice this effect because it only becomes appreciable at speeds that are comparable to the speed of light in a vacuum, c. This effect is called time dilation.
Is there an experimental test of time dilation? Yes. In 1940, the American physicists Bruno B Rossi (1905-1993) and David B Hall (I can’t find anything more about him) measured the rate of decay of a subatomic particle called the muon. They found that muons decayed more slowly when travelling at a speed close to c.
What is the explanation of time dilation?
The picture above shows a section through two parallel planes that appear as lines containing the points S and O. I will define an orthogonal Cartesian coordinate system, with its origin at S, whose x-axis lies along the lower line. O is vertically above S, so that SO defines the y-axis. The length of SO is L. The two planes do not move with respect to each other. If a burst of light takes time, t, to travel from S to O then, from the definition of speed
We will now suppose that O is the position of an observer.
Now let’s suppose that the upper plane moves at a speed v, relative to S, in the x-axis direction, as shown in the picture above. When the burst of light leaves S, O is at O’, a distance –vt’ from O, where t’ is the time the burst of light takes to reach O. Why have I now defined this time to be t’ and not t? Because things are in motion and I can’t be sure of the effects of this motion.
Now let’s think of the motion relative to the observer. The observer is now considered to be stationary (see post 16.12) so that S is considered to be moving in the minus x-axis direction. When the light reaches the observer, its source appears to be S’, a distance –vt’ from S in the x-axis direction. In the frame of reference of the observer, the burst of light has travelled from S’ to O. Remember that, since the observer is now considered to be at rest, O and O’ are now the same point. If L’ is the distance from S’ to O equation 1 becomes
remembering that, according to Einstein’s special theory of relativity, c is constant in all inertial frames of reference.
Applying Pythagoras’ theorem to the triangle OSS’ gives the result that
Substituting the expressions for L and L’, from equations 1 and 2, into equation 3 and dividing by c2 gives
so that
It follows that the relationship between t’ and t is given by
Equation 4 is sometimes written in the form
where the Lorentz factor, γ, is given by
When v is much less than c, which is usually the case, v2/c2 is so small that it is negligible and we don’t notice time dilation. But when v approaches c, t’ is less than t so that time passes more slowly.
Going back to our second picture, when v = c, the value of t’ is infinity so that the observer never sees the burst of light. This is what we would expect because, if S’ appears to be moving away from O’ with the speed of light, so a burst of light can never reach it.
Related posts
26.7 Einstein’s special theory of relativity
16.12 Measuring movement
16.9 Does the sun move around the earth?
16.4 Movement
