2.5   Time Dilation

• Time dilation is a key consequence of special relativity:  a clock moves more slowly for a moving observer than for a static one.

We have our starting point, then:  the speed of light is the same, for any two observers, regardless of the relative motion of those observers. This is often stated formally as “the light’s speed is the same in any inertial frame of reference". A frame of reference is a coordinate system with the observer at its origin. So if the two observers are moving relative to each other, so are their two frames of reference. So long as there are no external forces acting on the observers, the two reference frames will have a constant relative velocity. We can then call them inertial reference frames. We’re going to restrict ourselves to this situation.

 OK, so now we need to round up Nefertiti and Cormorant again, because we’re going to set up another scenario, which will let us explore the consequences of this idea:

 First, you and Nefertiti are going to play another sardine-throwing game. You can see the setup in the animation below. You’re on the beach, and Cormorant is lurking overhead. You draw two perpendicular lines in the sand and Nefertiti stands at the point where they intersect. You stand at the other end of one line, poised to receive incoming fish. Now Cormorant is going to fly overhead, following the perpendicular line. At the instant he is level with Nefertiti she will let fly with the sardine, hurling it directly at you. Some time later, it will hit you, square between the eyes. By this time, Cormorant – flying with constant velocity - will have progressed further along his perpendicular path. The second part of the diagram shows how it looks to Cormorant. Remember that in his frame of reference, he is static at the centre of everything while you and Nefertiti are moving. Now, we are going to think about the velocity of the Sardine:

In Nefertiti’s frame of reference: 

…….. and in Cormorant’s frame of reference:

So there we have it: if you worked out Question 2-6, you have the confirmation that the sardine is flying faster in Cormorant’s frame of reference than it is in Nefertiti’s and that is exactly what we expect, since the sardine has two components to its velocity in Cormorant’s frame of reference: one arising from the movement of you and Nefertiti, relative to him, and one arising from Nefertiti’s throw. This is Galileo’s version of relativity in action.

Now for the second part of the game.  Exact same set up, except this time at the instant Cormorant draws level, Nefertiti - having exhausted her sardine supply - is going to fire a pulse of light at you instead, using the new pulsed laser she has just won in a poker game. So you go through the same analysis to figure out the velocity of the light in Nefertiti’s frame of reference and in Cormorant’s and, by direct analogy with the sardine, you figure that the light moves faster in Cormorant’s frame of reference because it travels further in the same time. Which is all fine, except that it isn’t:  there are Uncle Albert at one shoulder, and Willem de Sitter at the other, each gently wagging a finger and reminding you that it doesn’t work like that:  the speed of the light must be the same in both frames of reference, regardless of the relative movement of those frames. So how can we reconcile these things? 

Have a look at figure II-vi, below, which summarises the situation when the light pulse has just hit you:

In the right hand picture, the light has travelled a distance y in Nefertiti's reference frame but a distance r in Cormorant's frame. Always on the lookout for a right-angled triangle, Pythagoras is once again ready to lend a hand in analysing this situation. Let him lead you through the next question.

If we are to keep Uncle Albert and the Universe happy, and maintain a constant light speed, you can see that – since speed = distance/time - either the distance travelled by the light must have been shorter or the time taken for its journey must have been longer in Cormorant’s frame of reference than in Nefertiti’s. In general, either - or both - of these possibilities need to be considered but this experiment has been cunningly designed: we’ll establish later in this chapter that distances in a direction perpendicular to the relative movement of the two observers are the same in both reference frames. So that means it’s the time that’s the issue in this case.

This is now seriously weird. We are saying that the same event – movement of the light from Nefertiti to you – took a different amount of Cormorant’s time than it did of your own time. Can this be true? Nefertiti suggests we repeat the experiment but this time make some timings. She and Cormorant both use their stopwatches to time how long the light pulse takes to move from her laser to you.  These experimental timings will confirm the result: more time passes during the flight of the light when Cormorant measures it than when Nefertiti does . To him, it looks like Nefertiti’s stopwatch is running slowly. This effect is called time dilation. We can express it more generally, in seabird-free terms, as:

               A clock runs slower for a moving observer than for a static one.

It’s really important to grasp that this is not just some trick of perception. Time does not just “appear” to advance more slowly if the observer is moving, it really does.  All kinds of ramifications, paradoxes and general weirdness surround time dilation. Not least, there’s the tricky question of who is the one moving. From your point of view it’s Cormorant who’s moving, so he sees your clock as running slow. From his point of view, though, he is static at the origin of his frame of reference and it is you who is moving. So from his perspective it is your clock that’s going slow. These things can’t both be true. Except that they are. The universe does not work like your head thinks it does.

You could easily spend weeks sorting all of this out – or longer, as measured by someone watching you from a moving train – and it is something you should consider doing one day - but for now let’s keep our eyes on the prize and remember that time dilation is just a waystation on our route to other things. So we'll press on. Nefertiti is eager to express what we’ve discovered in mathematical form.

Focussing on the same experimental setup, let’s see if you can derive an expression for tC (the time elapsed for Cormorant, relative to tN (the time elapsed for you and Nefertiti), while the light is making its way from Nefertiti to you.  We will do this by constructing expressions for the three distances x, y and r.

The answer you should have arrived at is:

This just needs a bit of generalising, and then we'll have the key equation of special relativity. In this Gedankenexperiment, Nefertiti's and Cormorant's frames of reference were coincident at time 0, when the light was released but then moved apart, in the x-direction. The event whose time is being measured is the arrival of the light at you. Now the key thing is that, as Nefertiti sees it, the path of the light involves no movement in the x-direction: it is stationary in her frame of reference. We therefore call this the rest frame of the thing we're measuring (the light's time of flight along this path). Henceforth and forever, we will denote the rest frame as frame 0 and hence the flight time as Nefertiti measured it as t0. 

Cormorant, on the other hand is a moving observer of the event. This means, as we have seen, that he will experience more time while the same event unfolds, than Nefertiti did in the rest frame.  This will be true for any moving frame of reference, not just Cormorant's one.  In our generalisation we'll denote any such moving frame as frame 1 and the time measured there t1. 

We can now rewrite the equation we derived for Nefertiti and Cormorant in general terms: